CS计算机代考程序代写 python Bayesian finance algorithm Financial Econometrics - Slides-01: RETURN PROPERTIES Part I

Financial Econometrics – Slides-01: RETURN PROPERTIES Part I

Introduction Asset Return

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Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

Financial Econometrics
Slides-01: RETURN PROPERTIES Part I

Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

Introduction

Financial time series (FTS) analysis is concerned with theory and practice of
asset valuation over time.
Comparison with other Time Series analysis: similarity and difference? Highly
related, but with some added uncertainty, because FTS must deal with the
ever-changing business & economic environment and the fact that volatility is
not directly observed. Objective of the course

• to learn ways to get financial information from web directly and to process
the information.

• to provide some basic knowledge of financial time series data such as
skewness, heavy tails, and measure of dependence between asset returns

• to introduce some statistical tools & econometric models useful for
analyzing these series.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

Examples of FTS

• to analyze high-dimensional asset returns, including co-movement

Examples of financial time series

1. Daily log returns of Apple stock: 2004 to 2013 (10 years)

2. The VIX index

3. CDS spreads: Daily 3-year CDS spreads of JP Morgan from July

20, 2004 to September 19, 2014.

4. Quarterly earnings of Coca-Cola Company: 1983-2009

Seasonal time series useful in

• earning forecasts
• pricing weather related derivatives (e.g. energy)
• modeling intraday behavior of asset returns

5. US monthly interest rates (3m & 6m Treasury bills)

Relations between the two series? Term structure of interest

rates

6. Exchange rate between US Dollar vs Euro

Fixed income, hedging, carry trade

7. Size of insurance claims

Values of fire insurance claims (×1000 Krone) that exceeded 500
from 1972 to 1992.

8. High-frequency financial data:

Tick-by-tick data of Caterpillars stock: January 04, 2010.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

2004 2006 2008 2010 2012 2014

−0
.2

0
−0

.1
0

0.
00

0.
05

0.
10

year

lo
g−

rtn

Daily log returns of Apple stock

Figure 1: Daily log returns of Apple stock from 2004 to 2013

2006 2008 2010 2012 2014
0

.0
0

0
0

.0
0

5
0

.0
1

0
0

.0
1

5
0

.0
2

0
year

sp
re

a
d

3
y

CDS of JPM: 3−yr spread

Figure 3: Time plot of daily 3-year CDS spreads of JPM: from July 20, 2004 to September
19, 2014.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

VIXCLS [2004−01−02/2014−03−07]

Last 14.11

Jan 02 2004 Jan 03 2007 Jan 04 2010 Jan 02 2013

Figure 4: CBOE Vix index: January 2, 2004 to March 7, 2014.

Time

1985 1990 1995 2000 2005 2010
0

.0
0

.2
0

.4
0

.6
0

.8
1

EPS of Coca Cola: 1983−2009

Figure 5: Quarterly earnings per share of Coca-Cola Company

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return

year

e
u

2000 2002 2004 2006 2008 2010

0
.8

1
.0

1
.2

1
.4

1
.6

Dollars per Euro

Figure 6: Daily Exchange Rate: Dollars per Euro

year

rt
n

2000 2002 2004 2006 2008 2010
−

0
.0

2
0
.0

0
.0

2
0
.0

ln−rtn: US−EU

Figure 7: Daily log returns of FX (Dollar vs Euro)

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Asset Returns: Definition

Let Pt be the price of an asset at time t, and assume no dividend.

• One-period simple return:
Gross return

1 + Rt =
Pt

Pt−1

Pt = Pt−1(1 + Rt)

Simple return:

Rt =
Pt − Pt−1

Pt−1
=

Pt
Pt−1

− 1

• Multiperiod simple return

1 + Rt(k) =
Pt

Pt−k

= (1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1)
= Π

k−1
j=0 (1 + Rt−j)

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Asset Returns: Example

Table below gives five daily closing prices of Apple stock in December 2011.
12/8 to 12/9 1 + Rt = 393.62/390.66 ≈ 1.0076 so that the daily
simple return is 0.76%, which is (393.62− 390.66)/390.66.

Date 12/02 12/05 12/06 12/07 12/08 12/09

Price($) 389.70 393.01 390.95 389.09 390.66 393.62

Time interval is important! Default is one year.

Annualized (average) return:

Annualized[Rt(k)] =



k−1∏

j=0
(1 + Rt−j)




1/k

− 1.

An approximation:

Annualized[Rt(k)] ≈
1

k−1∑

j=0
Rt−j.

Continuously compounding: Illustration of the power of compound-

ing (int. rate 10% per annum)

Type #(payment) Int. Net

Annual 1 0.1 $1.10000

Semi-Annual 2 0.05 $1.10250

Quarterly 4 0.025 $1.10381

Monthly 12 0.0083 $1.10471

Weekly 52 0.1
52

$1.10506

Daily 365 0.1
365

$1.10516

Continuously ∞ $1.10517

A = C exp[r × n]
where r is the interest rate per annum, C is the initial capital, n is

the number of years, and exp is the exponential function.

• The 1-day simple return of holding the stock from 12/8 to 12/9:
0.76%

• The 3-day simple return for holding the stock from 12/02 to 12/07:
−0.15%
• The 5-day simple return for holding the stock from 12/02 to 12/09:

Answer?

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Annulalized Asset Returns

12/8 to 12/9 1 + Rt = 393.62/390.66 ≈ 1.0076 so that the daily
simple return is 0.76%, which is (393.62− 390.66)/390.66.

Date 12/02 12/05 12/06 12/07 12/08 12/09

Price($) 389.70 393.01 390.95 389.09 390.66 393.62

Time interval is important! Default is one year.

Annualized (average) return:

Annualized[Rt(k)] =



k−1∏

j=0
(1 + Rt−j)




1/k

− 1.

An approximation:

Annualized[Rt(k)] ≈
1

k−1∑

j=0
Rt−j.

Continuously compounding: Illustration of the power of compound-

ing (int. rate 10% per annum)

Type #(payment) Int. Net

Annual 1 0.1 $1.10000

Semi-Annual 2 0.05 $1.10250

Quarterly 4 0.025 $1.10381

Monthly 12 0.0083 $1.10471

Weekly 52 0.1
52

$1.10506

Daily 365 0.1
365

$1.10516

Continuously ∞ $1.10517

A = C exp[r × n]
where r is the interest rate per annum, C is the initial capital, n is

the number of years, and exp is the exponential function.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Annulalized Asset Returns

Present value:

C = A exp[−r × n]
Continuously compounded (or log) return

rt = ln(1 + Rt) = ln
Pt
Pt−1

= pt − pt−1,

where pt = ln(Pt).

Multiperiod log return:

rt(k) = ln[1 + Rt(k)]

= ln[(1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1)]
= ln(1 + Rt) + ln(1 + Rt−1) + · · · + ln(1 + Rt−k+1)
= rt + rt−1 + · · · + rt−k+1.

Example Consider again the Apple stock price.

1. What is the log return from 12/8 to 12/9:

A: rt = ln(393.62)− ln(390.66) = 7.5%.
2. What is the log return from day 12/2 to 12/9?

A: rt(4) = ln(393.62)− ln(389.7) = 1%.

Portfolio return: N assets

Rp,t =
N∑

i=1
wiRit

Example: An investor holds stocks of IBM, Microsoft and Citi-

Group. Assume that her capital allocation is 30%, 30% and 40%.

Use the monthly simple returns in Table 1.2 of the text. What is the

mean simple return of her stock portfolio?

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Annulalized Asset Returns

Example Consider again the Apple stock price.

• What is the log return from 12/8 to 12/9?
A:

7.5%

• What is the log return from day 12/2 to 12/9?
A:

• What is the log return from day 12/6 to 12/8?
A:

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Market Index and Return

• Market index: Pm,t =
∑N

i=1
witPi,t, t = 1, 2, · · ·

weight wit depends on outstanding shares of stock i, etc

• Log return:

rm,t = 100%× ln
(

Pm,t
Pm,t−1

)

Topic 1. Features of Some Financial Time Series

• Financial time series

– Market index and return

• Market index: � ,� = ∑ “#,��#,�
$
#%� , � = 1,2…

weight “#,� depends on outstanding shares of stock &, etc

• Log return: � ,� = 100% × ln
�’,�
�’,��

eg. S&P/ASX200 Index and Return

-12

-8

-4

2000

3000

4000

5000

6000

7000

02 03 04 05 06 07 08 09 10 11

RETURN PRICE

UNSW Business School,

Economics
Slides-01, Financial Econometrics 10

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Portfolio Return

An investor holds stocks of IBM, Microsoft and Citi- Group. Assume that her
capital allocation is 30%, 30% and 40%. What is the mean simple return of
her stock portfolio?
Assume monthly simple returns for IBM, microsoft and Citi-Group, 1.35%,
2.62% and 1.17% respectively.
Answer: 1.66%

• Portfolio Return: Rp,t =
∑N

i=1
witRi,t, t = 1, 2, · · · , where N is the

number of assets held by investor and wit is wealth allocation.

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Adjusted Returns

1 Adjusting for dividends (Total Returns)

rt = ln (1 + Rt) = ln

(
Pt + Dt
Pt−1

)
= ln(Pt + Dt)− ln(Pt−1)

2 Adjusting for inflation (Real Returns)

r
Real
t = ln

(
1 + R

Real
t

)
= ln

(
Pt

Pt−1

CPIt−1
CPIt

)
3 Adjusting for Risk (Excess Returns)

Zt = Rt −Rft
zt = ln(Zt) = ln(Rt −Rft) 6= rt − rft

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Dividends, Excess returns

Answer: E(Rt) = 0.3× 1.35 + 0.3× 2.62 + 0.4× 1.17 = 1.66.
Dividend payment:

Rt =
Pt + Dt
Pt−1

− 1, rt = ln(Pt + Dt)− ln(Pt−1).

Excess return: (adjusting for risk)

Zt = Rt −R0t, zt = rt − r0t
where r0t denotes the log return of a reference asset (e.g. risk-free

interest rate).

Relationship:

rt = ln(1 + Rt), Rt = e
rt − 1.

If the returns are in percentage, then

rt = 100× ln(1 +
Rt
100

), Rt = [exp(rt/100)− 1]× 100.

Temporal aggregation of the returns produces

1 + Rt(k) = (1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1),
rt(k) = rt + rt−1 + · · · + rt−k+1.

These two relations are important in practice, e.g. obtain annual

returns from monthly returns.

Example: If the monthly log returns of an asset are 4.46%, −7.34%
and 10.77%, then what is the corresponding quarterly log return?

Answer: 4.46− 7.34 + 10.77 = 7.89%.
Example: If the monthly simple returns of an asset are 4.46%,

−7.34% and 10.77%, then what is the corresponding quarterly simple
return?

Answer: R = (1+0.0446)(1−0.0734)(1+0.1077)−1 = 1.0721−1
= 0.0721 = 7.21%

Rachida OuysseSchool of Economics Financial Econometrics

Introduction Asset Return Simple/Gross Return Annulaized/Compounded Retun Portfolio Return Adjusted return

Answer: E(Rt) = 0.3× 1.35 + 0.3× 2.62 + 0.4× 1.17 = 1.66.
Dividend payment:

Rt =
Pt + Dt
Pt−1

− 1, rt = ln(Pt + Dt)− ln(Pt−1).

Excess return: (adjusting for risk)

Zt = Rt −R0t, zt = rt − r0t
where r0t denotes the log return of a reference asset (e.g. risk-free

interest rate).

Relationship:

rt = ln(1 + Rt), Rt = e
rt − 1.

If the returns are in percentage, then

rt = 100× ln(1 +
Rt
100

), Rt = [exp(rt/100)− 1]× 100.

Temporal aggregation of the returns produces

1 + Rt(k) = (1 + Rt)(1 + Rt−1) · · · (1 + Rt−k+1),
rt(k) = rt + rt−1 + · · · + rt−k+1.

These two relations are important in practice, e.g. obtain annual

returns from monthly returns.

Example: If the monthly log returns of an asset are 4.46%, −7.34%
and 10.77%, then what is the corresponding quarterly log return?

Answer: 4.46− 7.34 + 10.77 = 7.89%.
Example: If the monthly simple returns of an asset are 4.46%,

−7.34% and 10.77%, then what is the corresponding quarterly simple
return?

Answer: R = (1+0.0446)(1−0.0734)(1+0.1077)−1 = 1.0721−1
= 0.0721 = 7.21%

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns

Financial Econometrics
Slides-01: RETURN PROPERTIES Part II

Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characteristics: Population

Let Xt be a random variable with pdf f(x)

µ = E[Xt] : center

σ
2

= var(Xt) = E[(Xt − µ)2] : spread

skewness(Xt) = S(X) = E

[
(Xt − µ)3

σ3

]
: symmetry

kustosis(Xt) = K(X) = E

[
(Xt − µ)4

σ4

]
: tail thickness

K(X)− 3 : Excess kurtosis

Note: The kth moment and central moment of Xt are:

m
′
k = E[X

k
t ]

mk = E[(Xt − µ)k]

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characteristics of Random Variable

• Why are the mean and variance of returns important?
They are concerned with long-term return and risk, respectively.

• Why is return symmetry of interest in financial study?
Symmetry has important implications in holding short or long financial
positions and in risk management.

• Why is kurtosis important?
Related to volatility forecasting, efficiency in estimation and tests, etc.

High kurtosis implies heavy (or long) tails in distribution.

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Examle: Normal Random Variable

Normal Distribution

 ∼ ( 2)

() =
1

√
22

exp

Ã
−
(− )2
22

!
 −∞ ≤  ≤ ∞

[] = 

var() = 2

skew() = 0

kurt() = 3

 = 0 for  odd

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characteristics: Sample moments

Sample moments

Let {     } denote a random sample of size  where  is a realization
of the random variable ̃

̂ =
1



X
=1

 ̂
2 =

 − 1

X
=1

( − ̂)2 = ̂2

dskew = ̂3
̂3

 dkurt = ̂4
̂3

̂ =
1

 − 1

X
=1

( − ̂)

Note: we divide by  − 1 to get unbiased estimates. Check software to see
how moments are computed.

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characteristics: Visually

Skewness

3
1

( )1 µ

σ=

−
= ∑

T
t

x
S

Topic 1. Features of Some Financial Time Series

UNSW Business School,

Economics
Slides-01, Financial Econometrics 20

Kurtosis

• Often is reported as a deviation from Normal K=3:

4
1

( )1 µ

σ=

−
= ∑

T
t

x
K

σ=

−
= −∑

4
1

( )1
3

T
dev t

x
K

Topic 1. Features of Some Financial Time Series

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Economics
Slides-01, Financial Econometrics 21

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Testing for normality

• QQ-plot: plot standardized empirical quantiles vs. theoretical quantiles
from specified distribution. Note: Shapiro-Wilks (SW) test for normality:
correlation coefficient between values used in QQ-plot

• Jarque-Bera (JB) test for normality

JB =
T

(
ˆskew

2
+

( ˆkurt− 3)2

)
∼A χ2(2)

Note: if rt is N(µ, σ
2) then:

√
T ˆskew ∼ N(0, 6), and

√
T ( ˆkurt− 3) ∼ N(0, 24)

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characterirtics: Normality test

The null hypothesis:
H0 : Data (the return) Xt are Normally distributed.

1 Skewness test: Zsk =
ˆskew√
6/T
∼ N(0, 1)

Reject H0 if |zsk| is too large (> 1.96, at 5%).

2 Kurtosis test: Zkt =
ˆkurt−3√
24/T

∼ N(0, 1)

Reject H0 if |zkt| is too large (> 1.96, at 5%).
3 Jaque-Bera test: JB = Z2ks + Z

2
kt ∼ χ

2
2

Reject JB is too large (> 5.99 at 5%)

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Example: Descriptive Statistics

Topic 1. Features of Some Financial Time Series

• Descriptive statistics

eg. NYSE index prices: (19950103-20020830)

Composite, Industrial,

Trans, Utility, Finance.

Descriptive statistics of log returns.

Correlations of log returns

100

200

300

400

500

600

700

800

900

250 500 750 1000 1250 1500 1750

COMP
INDU
TRAN

UTIL
FINA

Composite Industrial Trans Utility Finance

Mean 0.035 0.034 0.031 0.007 0.052

Std. Dev. 1.006 1.009 1.320 1.087 1.310

Skewness -0.316 -0.386 -1.044 -0.275 -0.042

Kurtosis 7.224 7.755 18.103 5.637 5.772

Composite Industrial Trans Utility Finance

Composite 1

Industrial 0.983 1

Trans 0.731 0.708 1

Utility 0.769 0.711 0.505 1

Finance 0.885 0.800 0.668 0.623 1

Portfolio variance and

diversification:

a =
�

5
(@ + *),

Var a =
1

4
[Var *

+ Var @
+ 2Cov(*, @)]

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Economics
Slides-01, Financial Econometrics 23

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Example: Descriptive Statistics

Topic 1. Features of Some Financial Time Series

• Descriptive statistics

– Normality test
eg. Comp. index log return

time series plot

histogram

100

200

300

400

500

-6 -4 -2 0 2 4

Series: RC

Sample 1 1931

Observations 1930

Mean 0.035300

Median 0.052285

Maximum 5.178704

Minimum -6.791142

Std. Dev. 1.006207

Skewness -0.315728

Kurtosis 7.224376

Jarque-Bera 1467.129

Probability 0.000000

-8

-6

-4

-2

250 500 750 1000 1250 1500 1750

p-value =

�(| 5
5

> z{)

UNSW Business School,

Economics
Slides-01, Financial Econometrics 25

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Stylized Fact: Large kurtosis

Topic 1. Features of Some Financial Time Series

• Descriptive statistics

– Some stylised facts about index return series

• concentration around zero with a few large “outliers”

• large standard deviations (volatile)

• negative skewness (longer tail at the negative side)

• large kurtosis (tail probabilities larger than normal)

• large variation followed by large ones (clustering)

100

200

300

400

500

-6 -4 -2 0 2 4

Series: RC

Sample 1 1931

Observations 1930

Mean 0.035300

Median 0.052285

Maximum 5.178704

Minimum -6.791142

Std. Dev. 1.006207

Skewness -0.315728

Kurtosis 7.224376

Jarque-Bera 1467.129

Probability 0.000000
-8

-6

-4

-2

250 500 750 1000 1250 1500 1750

leptokurtic

Histogram of RC

D
e
n
s
it
y

-6 -4 -2 0 2 4

0
.0

0
.1

0
.2

0
.3

0
.4

0
.5

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Economics
Slides-01, Financial Econometrics 27

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Descriptive statistics: Autocorrelation

• Predictability
• We say Xt+1 is predictable if information at t , eg. {Xt, Xt−1. · · · , }, helps

to improve our prediction of Xt+1.
• In particular, Xt+1 is predictable if Xt+1 is correlated with Xt−j for some
j > 0 (ie. Cov(Xt+1, Xt−j) 6= 0).

• Autocorrelation Function (ACF)
• Autocovariance: γj = Cov(Xt, Xt−j) = Cov(Xt, Xt+j)

Sample autocovariance: γ̂j =
1
T

∑T
t=j+1(Xt −X)(Xt−j −X)

• Autocorrelation: ρj =
γj
γ0

Sample Autocorrelation: ρ̂j =
γ̂j
γ̂0

• Partial autocorrelation (PAC)
• PAC pj is a measure of the direct relation between Xt and Xt−j for
j = 1, 2, · · ·

• pj is the correlation between Xt and Xt−j after controlling for the effects
of Xt and Xt−1 · · ·Xt−j+1

• p̂1 = φ̂11 in Xt = φ10 + φ11Xt−1 + e1t
• p̂2 = φ̂11 in Xt = φ20 + φ21Xt−1 + φ22Xt−2 + e2t, · · ·

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Test for autocorrelation

The null hypothesis: H0: There is no autocorrelation (White noise process)

1 Autocorrelation test:
√
T ρ̂j ∼ N(0, 1) under the null hypothesis

Reject if |ρ̂j | is too large (> 1.96/
√
T , at 5% significance level)

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Joint Hypothesis Tests

• We can also test the joint hypothesis that all m of the ρk correlation
coefficients are simultaneously equal to zero using the Q-statistic
developed by Box and Pierce:

Q = T

m∑
k=1

ρ̂
2
k

where T=sample size, m=maximum lag length

• The Q-statistic is asymptotically distributed as a χ2m.

• However, the Box Pierce test has poor small sample properties, so a
variant has been developed, called the Ljung-Box statistic:

Q
∗
= T (T + 2)

m∑
k=1

ρ̂2k
T − k

∼ χ2m

• This statistic is very useful as a portmanteau (general) test of linear
dependence in time series.

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

An ACF Example

• Question:
Suppose that a researcher had estimated the first 5 autocorrelation
coefficients using a series of length 100 observations, and found them to
be (from 1 to 5): 0.207, -0.013, 0.086, 0.005, -0.022.

Test each of the individual coefficient for significance, and use both the
Box-Pierce and Ljung-Box tests to establish whether they are jointly
significant.

Solution

A coefficient would be significant if it lies outside (−0.196,+0.196) at the
5% level, so only the first autocorrelation coefficient is significant.

Q = 5.09 and Q∗ = 5.26
Compared with a tabulated χ2(5)=11.1 at the 5% level, so the 5
coefficients are jointly insignificant.

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Example: ACF/PACF

Topic 1. Features of Some Financial Time Series

• Descriptive statistics
eg. NYSE composite return

AC test at 5% level:

1.96/ d = 0.04462,

rs is rejected at

� = 1,2,5,12

LB test at 5% level:

rs is rejected for

all �, as all p-values

are less than 0.05.

UNSW Business School,

Economics
Slides-01, Financial Econometrics 32

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Example: ACF/PACF of squared Returns

Topic 1. Features of Some Financial Time Series

• Descriptive statistics

– What about squared returns?

Usually strongly correlated.

– Why squared returns?

7 ��
5 ≈ �G�(��)

eg. NYSE Composite

return squared

UNSW Business School,

Economics
Slides-01, Financial Econometrics 33

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Summary of stylized Facts

KEY stylised facts about financial return series

1 the returns have small, often non-significant autocorrelations (no linear
return predictability)

2 the squared returns have strong positive autocorrelations (predictability in
volatility, volatility clustering)

3 large kurtosis (heavy tails, tail probabilities larger than normal)

Rachida OuysseSchool of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Summary

• Characterizing Financial time series:
• asset price and returns
• stylised facts about index return series

• Normality tests Zks, Zkt, JB
• Predictability in returns

• Autocovariance and autocorrelation
• Tests for autocorrelation: AC test and Qm

• Next week: Application of linear regression in Finance (asset pricing)

Rachida OuysseSchool of Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression model

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R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression model

Financial Econometrics

Slides-02

Linear Regression

Review and Applications in Finance

R. Ouysse
Economics1

1 Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression model

Linear Regression

• A model where one variable Yt is linearly explained by a group of variables
(X1t, · · · , Xkt), t = 1, · · · , T

• Easy to Implement
• Versatile for financial data analysis
• Foundation for more advanced models

• General formulation
• Yt = �1 + �2Xt1 + �3Xt2 + · · ·+ �KXtK + µt, t = 1 · · · , T
• Yt: dependent variable
• Xt1, · · ·XtK : explanatory variables, regressors
• �1,�2, · · · ,�K : parameters (to be estimated)
• µt: error term
• T : number of observations

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model

Application 1: Capital Asset Pricing Model aka CAPM

One of the most important problems of modern financial economics is the
quantification of the tradeo↵ between risk and expected return. Common
sensesuggests risky investments (stock market) will generally yield higher
returns than investments free of risk!

• Markowitz (1995) casts the investor’s portfolio selection problem in terms
of expected return and variance of the return.

! Investors optimally hold a mean-variance e�cient portfolio: a portfolio
with the highest expected return for a given level of variance.

=) The E�cient Frontier & Capital Market Line
• Capital Asset Pricing Model is concerned with the pricing of assets in

equilibrium. In equilibrium, all assets must be held by someone.

�! How investors determine the expected returns—and thereby asset
prices—as a function of risk.

=) The Security Market Line

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model

CAPM

• Given that: some risk can be diversified, diversification is easy and
costless, and rational investors diversify

• There should be no premium associated with diversifiable risk.
• The question becomes: What is the equilibrium relation between

systematic risk and expected return in the capital markets?

• The CAPM is the best-known and most-widely used equilibrium model of
the risk/return (systematic risk/return) relation.

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model

CAPM Intuition

What would be a ”fair” expected return on any stock?

• E(Rit) = Rft (risk free)+ Risk Premium
• Risk free assets earn the risk-free rate (think of this as a rental rate on

capital). The risk free compensate for time.

• If the asset is risky, we need to add a risk premium.
• The size of the risk premium depends on the amount of systematic risk for

the asset (stock, bond, or investment project) and the price per unit risk.

• Rit �Rft: Excess return

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model

CAPM Intuition Formalized

E [Rit] = Rft +
Cov (Rit, Rmt)
V ar (Rmt)

[E [Rmt]�Rft]

E [Rit] = Rft + �i [E [Rmt]�Rft]

The expression above is referred to as the ”Security Market Line”.

• E [Rmt]�Rft Market Risk premiun (compensation for risk) or the price
per unit of risk

• �i: number of units of systematic risk
• �i > 1 (or < 1): the asset is more (less) risky than the market portfolio • �i < 0 : the asset is a hedge against the market portfolio • �i how sensitive the asset to market movement R. Ouysse Economics Financial Econometrics Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model CAPM Formalized Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. Three inputs are required: (i) An estimate of the risk free interest rate. The current yield on short term treasury bills is one proxy. Practitioners tend to favor the current yield on longer-term treasury bonds but this may be a fix for a problem we don’t fully understand. (ii) An estimate of the market risk premium, E [Rmt]�Rft. Expectations are not observable. Generally use a historically estimated value. The market is defined as a portfolio of all wealth including real estate, human capital, etc. In practice, a broad based stock index, such as the S&P 500 or the portfolio of all NYSE stocks, is generally used. (iii) An estimate of beta. R. Ouysse Economics Financial Econometrics Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model CAPM: Econometric model Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. Let Xmt = Rmt �Rft and Xit = Rit �Rft and consider the econometric model: Xit = ↵i + �iXmt + µit • The CAPM can be examined by testing H0 : ↵i = 0 • If ↵i > 0, asset i beats the market by earning more than �iE [Xmt]
• This has been used to test the performance of mutual funds (application

in the Brooks textbook)

R. Ouysse Economics Financial Econometrics

Linear Regression Applications In Finance Review of Linear Regression modelCapital Asset Pricing Model The term structure of interest rates Present Value model

CAPM: Application

What detremines the expected return of an asset?
Example: Mobil (a US petroleum firm), 1978:01-1987:12 with T = 120.

Topic 2. Linear Regression & Applications in Finance

School’of’Economics,’UNSW’ Slides<02,'Financial'Econometrics' 7' -.3 -.2 -.1 .0 .1 .2 .3 .4 78 79 80 81 82 83 84 85 86 87 MARKET RISKFREE MOBIL -.2 -.1 .0 .1 .2 .3 .4 -.3 -.2 -.1 .0 .1 .2 market m ob il Scatter Plot Dependent'Variable:'E_MOBIL ' ' Method:'Least'Squares ' ' Sample:'1978M01'1987M12 ' ' Included'observaOons:'120 ' ' Variable Coefficient Std.'Error t ✏

⌘
! 0 as T goes to 1.

Intuitively �̂ gets closer and closer to � as T ! 1.
(does not imply unbiaseness, may sill be that E(�̂) 6= �)

3 Asymptotically normal (Why?): �̂ = � + (X 0X)�1X 0µ

(or Exat normality if µ are normally distributed)

�̂ ⇠ N
�
�,�

2
(X

0
X)

�1�

4 E�cient among linear estimators:

OLS has smallest variance among linear estimators

R. Ouysse Economics Financial Econometrics

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Financial Econometrics
Slides-03: Linear Regression with Time Series

Diagnostics Tests, Robust Inference& Model Stability

Dr. Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Testing the CAPM: Mobil Exxon

The CAPM implies that the market rewards investors for the market risk

E(Ri)−Rf = βi [E(Rm −Rf )]
where Ri is the return on an asset i, and Rm is the return on the market index.

• To estimate the CAPM: Run an OLS regression of excess returns on asset
i, Xi,t, on the market excess return Xm,t

Xi,t = αi + βiXm,t + µt

• If the CAPM holds, the null hypothesis H0 : αi = 0 Ha : αi 6= 0
(two-tailed test)

Linear Regression Applications In Finance Review of Linear Regression model

Capital Asset Procing Model

CAPM: Application

What detremines the expected return of an asset?
Example: Mobil (a US petroleum firm), 1978:01-1987:12 with T = 120.

Topic 2. Linear Regression & Applications in Finance

School’of’Economics,’UNSW’ Slides<02,'Financial'Econometrics' 7' -.3 -.2 -.1 .0 .1 .2 .3 .4 78 79 80 81 82 83 84 85 86 87 MARKET RISKFREE MOBIL -.2 -.1 .0 .1 .2 .3 .4 -.3 -.2 -.1 .0 .1 .2 market mo bil Scatter Plot Dependent'Variable:'E_MOBIL ' ' Method:'Least'Squares ' ' Sample:'1978M01'1987M12 ' ' Included'observaOons:'120 ' ' Variable Coefficient Std.'Error t F̂

)
= 0.0046 =⇒decision: Reject the null. PythonCode

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Arbitrage Pricing Theory (APT)

What determines the expected return of an asset?

Excess returns: Xi,t = Ri,t −Rf,t and Xm,t = Rm,t −Rf,t
1 CAPM:

E(Xi,t) = αi + βiE(Xm,t)

RPi = αi + βiRPm

where RPi: risk premium for asset i, RPm: market risk premium

2 APT (Arbitrage Pricing Theory)

E(Xi,t) = RPi = αi + βiRPm + βotherRPotherfactors

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Arbitrage Pricing Theory (APT)

What determines the expected return of an asset?

• APT (Arbitrage Pricing Theory): if there are r risk factors priced in
the fiunancial market, then:

RPi = αi + βiRPm + βi,1RP1 + · · ·βi,rRPr

• RPj is the risk premium for exposure to factor j risk;, j = 1, · · · , r.
• βi,j is the sensitivity of the asset to factor j; it also measures asset i’s

exposure to the factor risk j

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

So what are the other risk factors in the APT?

A well established APT Model in the finance literature is the Fama&French
three factor model: FamaFrench

RPi = αi + βi.mRPm + βi.sRPs + βi.hRPh + βi.uRPu (1)

• RPm is the market risk premium
• RPs is the size factor risk premium (small market capitalisation)
• RPh is the value factor risk premium (high book-to-market stocks)
• RPu is the momentum risk factor premium (prior gains)
• βi.m, βi.s, βi.h and βi.u are the betas for the market risk, size factor, value

factor and momentum respectively

Dr. Rachida OuysseSchool of Economics Slides-03

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Example 1: Expected Return

Based on the following data and a risk-free rate of return of 2%, compute
expected return under APT model.

Beta of each factor Factor Risk Premium
βi.m 1.2 RPm 5.1
βi.s 0.8 RPs 0.5
βi.h 0.2 RPh 0.95
βi.u -0.1 RPu 2.5

Solution:

E(Ri) = Rf + αi + βi.mRPm + βi.sRPs + βi.hRPh + βi.uRPu

= 2%+ 1.2∗5.1% + 0.8∗0.5% + (0.2)∗0.95% + (−0.1)∗2.5%
E(Ri) = 8.46%

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Question 1: NIKE

Given the risk-free rate of return of 1.0%, average return of Nike (i.e: S&P 500
company with small market cap) of 15.88% p.a and the data provided in table
below:
Q1(a) Compute the expected return of the NIKE under APT model;
Q1(b) Determine the alpha return of the NIKE;
Q1(c) Construct a portfolio comprising S&P500 index fund (market portfolio),
Wilshire 5000 index fund, Russell 1000 value index fund and US T-Bills to
replicate the expected return of Nike.

Table 1: Factor beta, returns and risk premium
Factor Beta R Risk Premium
RPm 0.7877 14.5% 13.49%
RPs 0.6701 14.65% 0.15%
RPv -0.0288 10.38% -4.12%

(i) RPm is the market risk premium, i.e: excess of S&P500 return over the risk-free rate of return;
(ii) RPs is the size factor risk premium , i.e: excess of Wilshire 5000 index returns over the S&P500 returns (iii) RPv is the value
factor risk premium , i.e: excess of Russell 1000 index returns over the S&P500 returns.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Solution to Question 1

E(RNKE) = Rf + βNKE,m(Rm −Rf ) + βNKE,s(Rs −Rm) + βNKE,v(Rv −Rm)

(i) E(RNKE) =
1.0% + (0.7877)(13.49%) + (0.6701)(0.15%) + (−0.0288)(−4.12%) = 11.85%

(ii) αNKE = Actual return – Expected return = 15.88%− 11.85% = 4.03%

(iii) Replicating portfolios’ weights:

E(Ri) = Rf + βi,mE
(
Rm −Rf

)
+ βi,sE (Rs −Rm) + βi,vE (Rv −Rm)

= Rf (1− βi,m) + (βi,m − βi,s − βi,v)E(Rm) + βi,sE(Rs) + βi,vE(Rv)
= wi,RfRf + wi,mE(Rm) + wi,sE(Rs) + wi,vE(Rv)

wi,Rf = 1− βm = 1− 0.7877 = 0.2123
wi,m = βi,m − βi,s − βi,v = 0.7877− (0.6701)− (−0.0288) = 0.1464
ws = βi,s = 0.6701
wv = βi,v = −0.0288

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Solution to Question 1 continued

Replicating portfolio: (a) long position :21.23% in US T-Bills,
(b) long position: 14.64% in market portfolio (or S&P500 market index fund),
(c) long position: 67.01% in Wilshire 5000 index fund, and
(d) short 2.88% in Russell 1000 value index fund.

Computing the expected return of replicating portfolio:
Rf = 1.0% WRf = 0.2123
E(Rm) = 14.5% Wm = 0.1464
E(Rs) = 14.65% Ws = 0.6701
E(Rv) = 10.38% Wv = −0.0288
E(RAAPL)=
0.2123∗1.0% + 0.1464∗14.5% + 0.6701∗14.65%− 0.0288∗10.38%
= 11.85%

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Estimating & Testing the APT: Exxon Example

. What determines the expected return of Exxon Mobil?

The APT extends the CAPM to allow for additional risk factors Xu,t(eg.
unexpected macro events , unexpected changes in firm profits, etc)

Xu,t = (INF,OIL) H0 : γINF = γOIL = 0,

Do we reject?

Topic 2. Linear Regression & Applications in Finance

• Applications in finance

– Arbitrage pricing theory (APT)
• What determines the expected return of an asset?

– Excess returns: L.,$ = &.,$ − &M,$ and LN,$ = &N,$ − &M,$
CAPM: L.,$ = ; + ,LN,$ + P.,$

– APT extends CAPM: L.,$ = ; + ,LN,$ + QLR,$ + P.,$ ,
to include further risk factors LR,$ (eg. unexpected macro
events, unexpected changes in firm profits, etc).

eg. Mobil,

Xu,t = INF, OIL,
01: QVWX = QYVZ = 0 ,
Do we reject?

School of Economics, UNSW Slides-02, Financial Econometrics 5

Variable Coefficient Std. Error t-Statistic Prob.

C 0.004 0.006 0.721 0.472

E_MKT 0.713 0.086 8.271 0.000

INF 0.440 0.641 0.687 0.494

OIL 0.341 0.637 0.536 0.593

Test Statistic Value df Probability

F-statistic 0.6965 (2, 116) 0.5004

1 PythonCodeAPT

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Test for Autocorrelation

. H0 : No autocorrelation in the error term µt

1 Durbin-Watson test (DW):

Reject H0 if DW is too different from 2.

2 LM test for autocorrelation (Breush-Godfrey):
• Run OLS on the original regression

Yt = β0 + β1X1t + · · ·+ βKXKt + µt (2)

and save residuals et
• Run OLS on the auxiliary regression

et = γ0 + γ1X1t + · · ·+ γKXKt (3)
+δ1et−1 + · · ·+ δqet−q + errort, (4)

and save R−squared R2a;
• Reject H0 if (T − q)R2a > χ2q−critical value.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Test for Heteroskedasticity

. H0 : Homoskedasticity of the error term µt
1 LM test (White)

• Suppose the original regression has only two regressors.
• Run OLS on the original regression

Yt = β0 + β1X1t + β2X2t + µt (5)

and save residuals et
• Run OLS on the auxiliary regression

e2t = γ0 + γ1X1t + γ2X2t (6)

+δ1X
2
1t + δ2X

2
2t + δ3X1tX2t + errort, (7)

and save R−squared R2a;
• Reject H0 if TR2a > χ2m−critical value, where m is the number of

regressors in the auxiliary regression, here (m = 5)

Notice the problem of m increasing with K (m = 2K +
K(K−1)

2
)

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Test for Heteroskedasticity

2 Alternative method for White LM test
• Run OLS on the original regression

Yt = β0 + β1X1t + · · ·+ βKXKt + µt (8)

and save residuals et, and predicted values Ŷt
• Run OLS on the auxiliary regression

e2t = γ0 + γ1Ŷt + γ2Ŷ
2
t + errort, (9)

and save R−squared R2a;
• Reject H0 if TR2a > χ22−critical value.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Example:Mobil

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Diagnostic statistics
eg. CAPM: Mobil

– No evidence for AC in the error term (large p-value/Do not Reject).

– Strong evidence for heteroskedasticity (small p-value/Reject).

– Strong evidence for non-normality (small p-value/Reject).

School of Economics, UNSW Slides-02, Financial Econometrics 8

Dependent Variable: E_MOBIL

Method: Least Squares

Sample: 1978M01 1987M12

Included observations: 120

Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005881 0.721087 0.4723

E_MARKET 0.714695 0.085615 8.347761 0.0000

R-squared 0.371287 Mean dependent var 0.009353

Adjusted R-squared 0.365959 S.D. dependent var 0.080468

S.E. of regression 0.064074 Akaike info criterion -2.641019

Sum squared resid 0.484452 Schwarz criterion -2.594561

Log likelihood 160.4612 F-statistic 69.68511

Durbin-Watson stat 2.087124 Prob(F-statistic) 0.000000
0

-0.1 -0.0 0.1 0.2 0.3

Series: Residuals
Sample 1978M01 1987M12
Observations 120

Mean 1.27e-18
Median 0.000819
Maximum 0.278652
Minimum -0.145562
Std. Dev. 0.063805
Skewness 0.788429
Kurtosis 5.152737

Jarque-Bera 35.60378
Probability 0.000000

White Heteroskedasticity Test:

F-statistic 3.587532 Probability 0.030751

Obs*R2 6.933821 Probability 0.031213

Breusch-Godfrey Serial Correlation LM Test:

F-statistic 0.229380 Probability 0.795386

Obs*R2 0.472709 Probability 0.789501

• No evidence for AC in the error term (large p-value).
• Strong evidence for heteroskedasticity (small p-value).
• Strong evidence for non-normality (small p-value).

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Robust Standard Errors

• The key assumption is E(µt|Xt) = 0
(which my be weakened to Cov(Xt, µt) = 0).

Can we test for this ’key assumption’? How would the test look like?

• Even when there is heteroskedasticity or autocorrelation in µt, the OLS
estimators are still consistent. However, the standard errors of the
estimators are incorrect and MUST be corrected.
• In practice, we should always use robust standard errors that correct the

effect of heteroskedasticity and/or autocorrelation:
1 White standard errors (correct heteroskedasticity)
2 Newey-West (HAC) standard errors (correct jointly heteroskedasticity and

autocorrelation.)

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Example:Mobil

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Robust standard errors
eg. Mobil

School of Economics, UNSW Slides-02, Financial Econometrics 10

OLS s.e.
Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005881 0.721087 0.4723

E_MKT 0.714695 0.085615 8.347761 0.0000

White s.e.
Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005620 0.754602 0.4520

E_MKT 0.714695 0.086243 8.287035 0.0000

Newey-West s.e.
Variable Coefficient Std. Error t-Statistic Prob.

C 0.004241 0.005130 0.826596 0.4101

E_MKT 0.714695 0.090799 7.871135 0.0000

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability SummaryRobust Standard Errors Miscellaneous issues

Miscellaneous issues

• Dynamics:the lags of Yt may be included in the RHS of the regression
eg. Mobil” Xi,t = α+ βXm,t + γXi,t−1 + µi,t
• Dummy variable

• Stock market event:

Yt = β0 + β1Xt + β2DtXt + µt, (10)

Dt = 0 pre crisis and Dt = 1 post crisis.
The effect of Xt on Yt is β1 before the crisis but becomes (β1 + β2) after
the crisis.

• Day-of-the-week effects: Frit =
{

0, t is not on a Friday
1, t is on a Friday

}

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Model Stability

.
Model stability: Does Its structure changes over time?

1 Recursive parameter estimates

Monitor changes in parameter estimates over time.
• Start from an initial sample of size τ , estimate the model, get β̂(τ),
• add one observation to the sample, estimate the model, get the β̂(τ + 1),
• Continue recursively until last estimate with full sample β̂(T )

– eg. Mobil: Stability of the CAPM Model Xi,t = α+ βXm,t + µi,t

Recursive estimates of the market beta β:

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Model stability: Its structure changes over time?
• Recursive parameter estimates

Monitor changes in parameter estimates over time.

!”,… , !o , !”,… , !op” , …, !”,… , !’
()(q), ()(q + 1), …, ()(/)

eg. Mobil:

L.,2 = ; + (LN,2 + P.,2
Recursive estimates of (

School of Economics, UNSW Slides-02, Financial Econometrics 12

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

79 80 81 82 83 84 85 86 87

Recursive C(2) Estimates ± 2 S.E.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Model Stability

2 Recursive residuals
• Estimate recursively the model parameters:
β̂(τ), β̂(τ + 1), · · · , β̂(T ),

• estimate recursive residuals: eτ+1|τ = Yτ+1 −Xτ+1β̂(τ)
eτ+1|τ , eτ+2|τ+1, · · · , eT |T−1
If the model is correct (stable),

Wτ+1|τ =
eτ+1|τ

se(eτ+1|τ )
∼ N(0, 1) (11)

Mobil Recursive Residuals (CAPM)

Topic 2. Linear Regression & Applications in Finance

• Linear regression

– Model stability: Its structure changes over time?

• Recursive residuals: [op$|o = ‘op$ − *op$*+ q

‘$, … , ‘o , ‘$,… , ‘op$ , …, ‘$,… , ‘/”$
[op$|o, [op1|op$, …, [/|/”$

If the model is correct,

sop$|o =
tuvw|u

xy(tuvw|u)
∼ {(0,1).

eg. Mobil:

Recursive residuals

School of Economics, UNSW Slides-02, Financial Econometrics 13

-.2

-.1

78 79 80 81 82 83 84 85 86 87

Recursive Residuals ± 2 S.E.

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Model Stability

3 CUSUM Test (cumulative sum of standardised recursive residuals)

CUSUMt =

t∑
τ=K+1

Wτ+1|τ , (12)

t = K + 1,K + 2, · · · , T − 1

Reject Stability if it goes outside the 95% bands

eg. Mobil CUSUM test:

Topic 2. Linear Regression & Applications in Finance

• Linear regression
– Model stability: Its structure changes over time?

• CUSUM test (cumulative sum of standardised recursive residuals)

CUSUM% = ∑ sop+|o
%
o-#p+ , 5 = 0 + 1,0 + 2,… , 6 − 1 .

Reject “stability” if it goes outside the 95% bands.

eg. Mobil:

CUSUM test

• Eviews
View/Stability Tests/Recursive Estimates

after a linear regression is estimated.

• Stata: See Moodle (CUSUM6)

School of Economics, UNSW Slides-02, Financial Econometrics 14

-40

-30

-20

-10

78 79 80 81 82 83 84 85 86 87

CUSUM 5% Significance

Dr. Rachida OuysseSchool of Economics Slides-03

Testing the CAPM Application: Arbitrage Pricing Theory Model Test for autocorrelation Test for Heteroskedasticity Model Stability Summary

Summary

1 Linear regression
• What are the basic assumptions about linear regression
• What are OLS estimators and their properties
• What are the diagnostic statistics we have covered
• Why we should use robust standard errors
• What are recursive estimates of β
• What is the CUSUM test

2 Applications in finance
• CAPM is about the relationship of · · ·
• APT is an extension of · · ·
• These can be evaluated with a · · · model.

Dr. Rachida OuysseSchool of Economics Slides-03

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Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 1/ 23

Financial Econometrics
Slides-04: ARMA models

Genera; Linear Process and characterization

Dr. Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not
be removed from this material.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 2/ 23

Plan.

Time Series Models (Mainly Theoretical Aspects)

• View time series as stochastic processes
• Notions of stationarity (Covariance Stationary)
• Models for stationary time series

• General linear process (GLP): useful representation especially for computing
expectations…

• Characteristics of models
• Patterns in the AC and PAC of a model: White noise

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 3/ 23

Motivation

� Describe empirically relevant patterns in the data ??
� Obtain the distribution of future values, conditional on the past, in order to

forecast the future values and evaluate the likelihood of certain events ??

� Provide insight in possible sources of non-stationarity

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 4/ 23

Characteristics of a Time Series

Univariate Time Series Analysis: ARIMA models

Introduction

Characteristics of a Time Series

A time series y1, . . . , yT is a sequence of values a specific variable
y has taken on at equal distances (e.g. daily, quarterly, yearly, …)
over some period of time.

These observations will be considered as being generated by some
stochastic Data Generating Process (DGP)

I A time series y1, . . . , yT is generated by a stochastic process
yt , for t = 1, . . . , T .

I A time series y1, . . . , yT is a collection of realizations of a
random variable yt ordered in time.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 5/ 23

Univariate Time Series Models

Univariate Time Series Analysis: ARIMA models

Introduction

Univariate Time Series Models

A time series model tries to describe the stochastic process yt by
a relatively simple model. Univariate time series models are a
class of models where one attempts to model and predict
(economic) variables using only information contained in their own
past values and possibly current and past values of an error term.

These models are (mainly) a-theoretical:

I not based upon any underlying theoretical model

I attempt to capture empirically relevant patterns in the data

, Structural models
I generally based upon any underlying theoretical model

I attempt to model a variable from the current and/or past
values of other explanatory variables (suggested by theory)

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 6/ 23

Defining stationarity and non-stationarity

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Stationary versus Non-stationary Stochastic Processes

Defining stationarity and non-stationarity

A series yt is strictly stationary if the distribution of its values is
not a↵ected by an arbitrary shift along the time axis:

f (yt) = f (yt+k) 8k (1)

! The entire distribution of yt is not a↵ected by an arbitrary shift
along the time axis. See for example Figure 1.

A series yt is covariance or weakly stationary if it satisfies:

I E (yt) = µ < 1 I Var (yt) = E (yt � µ)2 = �2 < 1 I Cov (yt , yt�k) = E (yt � µ) (yt�k � µ) = �k 8k ! The first and the second moment of the distribution of yt are finite and not a↵ected by an arbitrary shift along the time axis. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 7/ 23 Defining stationarity and non-stationarity ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Stationary versus Non-stationary Stochastic Processes I After being hit by a shock, a stationary series tends to return to its mean (called mean reversion) and fluctuations around this mean (measured by the variance) will have a broadly constant amplitude. I If a time series is not stationary in the sense defined above, it is called non-stationary, i.e. non-stationary series will have a time-varying mean and/or a time-varying variance and/or time-varying covariances. I Non-stationarity can have di↵erent sources: linear trend, structural break, unit root, ... Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 8/ 23 Defining stationarity and non-stationarity ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Stationary versus Non-stationary Stochastic Processes Figure 5 : A non-stationary process (structural break) Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 9/ 23 Stationary Time Series ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � If the dependence structure is stable (stationary), it can be learned from historical data. � Strict Stationarity • A time series is strictly stationary (SS) if its joint distribution at any set of points in time is invariant to any time shift. eg. dist(yt1 , yt2) = dist(yt1+s, yt2+s) � Covariance Stationarity • A time series is covariance stationary (CS) if its mean, variance and autocovariance are all independent of the time index t, and its variance is finite. E(yt) = µ, V ar(yt) = γ0 <∞, Cov(yt, yt−j) = γj for all j Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 10 / 23 Autocorrelation and Partial Autocorrelation Function ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autocorrelation and Partial Autocorrelation Function Autocorrelation and Partial Autocorrelation Function Assuming covariance stationarity, particular useful tools when building ARMA models are the so-called Autocorrelation and Partial Autocorrelation Function. In general, the joint distribution of all values of yt is characterised by the so-called autocovariances, i.e. the covariances between yt and all of its lags yt�k . The sample autocovariances �k can be obtained as �k = cov(yt , yt�k), k = 1, 2, . . . (2) = 1 T � k TX t=k+1 (yt � y) (yt�k � y), (3) where y = T�1 PT t=1 yt is the sample mean. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 11 / 23 Autocorrelation and Partial Autocorrelation Function ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autocorrelation and Partial Autocorrelation Function As the autocovariances are not independent of the units in which the variables are measured, it is common to standardize by defining autocorrelations ⇢k as ⇢k = cov (yt , yt�k) var (yt) = �k �0 . (4) Note that ⇢0 = 1 and �1  ⇢k  1. The autocorrelations ⇢k considered as a function of k are referred to as the autocorrelation function (ACF) or correlogram of the series yt . The ACF provides useful information on the properties of the DGP of a series as it describes the dependencies among observations. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 12 / 23 Autocorrelation Function ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autocorrelation and Partial Autocorrelation Function If the data are generated from a stationary process, it can be shown that under the null hypothesis: H0 : ⇢k = 0 8k > 0

the sample autocorrelation coe�cients are asymptotically
normally distributed with mean zero and variance 1

T
.

Therefore, in finite sample it holds:

⇢k ⇠ N
�
0, T�1

�

The individual significance of an autocorrelation coe�cient can
be tested by constructing the 95% confidence interval:

h
�1.96/

p
T ; 1.96/

p
T
i

! see dashed lines in Figures 6 and 7.
Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 13 / 23

Autocorrelation Function

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autocorrelation and Partial Autocorrelation Function

Looking at a large number of autocorrelations, we will see that
some exceed two standard deviations as a result of pure chance
even though the true values in the DGP are zero (Type I error).

The joint significance of a group of m autocorrelation coe�cients
can be tested by the so-called Box-Pierce Q-statistic:

Q = T
Xm

k=1
⇢2k (5)

If the data are generated from a stationary process, Q is
asymptotically �2 distributed with m degrees of freedom.

Superior small sample performance is obtained by modifying the q
statistic (reported in EViews output):

Q⇤ = T (T + 2)
Xm

k=1
⇢2k/(T � k) (6)

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 14 / 23

Partial Autocorrelation Function

� An alternative piece of information is provided by the so-called partial
autocorrelation function (PACF). The partial autocorrelation pj s the
correlation between yt and yt?k conditional on yt?1, · · · , yt?k+1. It measures
the dependency between yt and yt?k keeping constant in-between values.

� The sample partial autocorrelations can be calculated from OLS regressions:
• p̂1 = φ̂11 in yt = φ10 + φ11yt−1 + e1t
• p̂2 = φ̂22 in yt = φ20 + φ21yt−1 + φ22yt−2 + e2t
• p̂3 = φ̂33 in yt = φ30 + φ31yt−1 + φ32yt−2 + phi33yt−2 + e3t
· · ·

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 15 / 23

Defining a White Noise Process

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

White Noise Process

Defining a White Noise Process

A series yt is called a white noise process if its DGP has a
constant mean, a constant variance and is serially uncorrelated.
Formally:

E (yt) = E (yt�1) = … = µ

Var (yt) = Var (yt�1) = … = �
2

Cov (yt , yt�k) = Cov (yt�j , yt�j�k) =
⇢

�2 if k = 0
0 otherwise

yt is a zero-mean white noise process if µ = 0.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 16 / 23

Defining a White Noise Process

� A time series �t is a white noise if its is covariance stationary with zero mean
and no autocorrelation.
• By definition:

E(�t) = 0, V ar(�t) = σ
2
, Cov(�j , �t−j) = 0, for allj 6= 0.

• A white noise is denoted as: yt ∼WN(0, σ2)
A white noise is not necessarily i.i.d (independent and identically distributed)

• An i.i.d white noise is denoted as: i.i.d WN(0, σ2)
• , White noises are building blocks of time series models.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 17 / 23

Test whether a time series is white noise

• Key feature of a WN(0, σ2) is H0: no autocorrelation
• The sampling distribution of the ACF and PACF for a WN is approximately
N(0, 1/T )

B Reject H0
if either ACF or PAC is outside the ±1.96/

√
T bands; or

the Ljung-Box Q-stats have small p-values.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 18 / 23

Test whether a time series is white noise

eg. NYSE Composite return squared r2t .

Topic 3. Time Series Models

• White noise process
– Test whether a time series is white noise

• Key feature of a WN is 𝐻𝐻0: no autocorrelation
• The sampling distribution of AC and PAC for a WN is

approximately N(0,1/T).
• Reject 𝐻𝐻0
if either AC or PAC is outside the ±1.96/ 𝑇𝑇 bands; or
if the Ljung-Box Q-stats (Slides-01) have small p-values.

eg. NYSE Composite return squared

School of Economics, UNSW Slides-04, Financial Econometrics 8

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 19 / 23

General linear Process

� Why general linear process?
B Wold decomposition. Any covariance stationary process can be expressed as

a general linear process.

yt = µ+

∞∑
i=0

bi�t−i, �t ∼WN(0, σ2)

• Because bi → 0 as i→∞,it is possible to use finite parameters to characterise
CS time series. This leads to practical (parsimonious) models (ARMA).

� Will mainly consider the cases with iid WN in this topic, for which
”conditional”=”unconditional”
• E(�t|�t−j) = E(�t) (�t is not predictable)
• V ar(�t|�t−j) = V ar(�t) for all j = 1, 2, , 3, · · ·

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 20 / 23

General linear Process

� Conditional Expectations
• The general linear process with i.i.d WN :

yt = µ+

∞∑
i=0

bi�t−i, �t ∼WN(0, σ2)

• Let Ωt be the information set based on

{yt, yt−1, · · · , �t, �t− 2, · · ·}

� Conditional mean and variance of yt+h for h = 1, 2, · · · :
• E(yt+h|Ωt) = µ+

∑∞
i=h

bi�t+h−i,
• V ar(yt+h|Ωt) = σ2

∑h−1
i=0

b2i
• What happens when h→∞?
B Limited memory: info at t is not relevant to remote future.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 21 / 23

General linear Process: Conditional Expectations

Topic 3. Time Series Models

• General linear process
– Conditional expectations

� 𝑦𝑦𝑡𝑡+ℎ = 𝜇𝜇 + 𝑏𝑏0𝜀𝜀𝑡𝑡+ℎ + ⋯+ 𝑏𝑏ℎ−1𝜀𝜀𝑡𝑡+1
not in Ω𝑡𝑡

+ 𝑏𝑏ℎ𝜀𝜀𝑡𝑡 + 𝑏𝑏ℎ+1𝜀𝜀𝑡𝑡−1 + ⋯
in Ω𝑡𝑡

� 𝐸𝐸 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 = 𝜇𝜇 + ∑ 𝑏𝑏𝑖𝑖𝜀𝜀𝑡𝑡+ℎ−𝑖𝑖
∞
𝑖𝑖=ℎ ,

� Var 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 = 𝜎𝜎2 ∑ 𝑏𝑏𝑖𝑖
2ℎ−1

𝑖𝑖=0 .

eg. When ℎ = 2,

� 𝑦𝑦𝑡𝑡+2 = 𝜇𝜇 + 𝑏𝑏0𝜀𝜀𝑡𝑡+2 + 𝑏𝑏1𝜀𝜀𝑡𝑡+1
not in Ω𝑡𝑡

+ 𝑏𝑏2𝜀𝜀𝑡𝑡 + 𝑏𝑏3𝜀𝜀𝑡𝑡−1 + ⋯
in Ω𝑡𝑡

� 𝐸𝐸 𝑦𝑦𝑡𝑡+2 Ω𝑡𝑡 = 𝜇𝜇 + 𝑏𝑏2𝜀𝜀𝑡𝑡 + 𝑏𝑏3𝜀𝜀𝑡𝑡−1 + ⋯ ,
� Var 𝑦𝑦𝑡𝑡+2 Ω𝑡𝑡 = 𝜎𝜎2(𝑏𝑏0

2 + 𝑏𝑏1
2) .

• Conditional variance is smaller than unconditional.

Variance being constant, not ideal to capture the
“clustering” in return series. Need ARCH-type models.

School of Economics, UNSW Slides-04, Financial Econometrics 12

-8

-6

-4

-2

250 500 750 1000 1250 1500 1750

𝜀𝜀𝑡𝑡+1 = 1-step forecast error

Conditional variance is smaller than unconditional variance. Variance being
constant, not ideal to capture the clustering in return series. Need ARCH-type
model.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 22 / 23

General linear Process: Forecast Based on Ωt

Topic 3. Time Series Models

• General linear process (GLP)
– Forecast based on Ω𝑡𝑡

• Use the information set Ω𝑡𝑡 to forecast 𝑦𝑦𝑡𝑡+ℎ for ℎ ≥ 1.
Let 𝑓𝑓𝑡𝑡+ℎ|𝑡𝑡 be the forecast based on Ω𝑡𝑡.

• Choose 𝑓𝑓𝑡𝑡+ℎ|𝑡𝑡 to minimise the MSFE

MSFE = 𝐸𝐸[ 𝑦𝑦𝑡𝑡+ℎ − 𝑓𝑓𝑡𝑡+ℎ|𝑡𝑡
2

|Ω𝑡𝑡].

• The optimal point forecast is
𝑓𝑓𝑡𝑡+ℎ|𝑡𝑡

∗ = 𝐸𝐸 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 .

• If 𝜇𝜇, 𝑏𝑏𝑖𝑖, 𝜎𝜎2 are known, the 2-se interval forecast is
𝐸𝐸 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 ± 2 Var 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 or

(𝜇𝜇 + ∑ 𝑏𝑏𝑖𝑖𝜀𝜀𝑡𝑡+ℎ−𝑖𝑖
∞
𝑖𝑖=ℎ ) ± 2 𝜎𝜎

2 ∑ 𝑏𝑏𝑖𝑖
2ℎ−1

𝑖𝑖=0
1/2

School of Economics, UNSW Slides-04, Financial Econometrics 13

Forecast error:
𝑒𝑒𝑡𝑡+ℎ|𝑡𝑡 = 𝑦𝑦𝑡𝑡+ℎ − 𝑓𝑓𝑡𝑡+ℎ|𝑡𝑡

∗ ,
Var 𝑒𝑒𝑡𝑡+ℎ|𝑡𝑡 Ω𝑡𝑡 = Var 𝑦𝑦𝑡𝑡+ℎ Ω𝑡𝑡 .

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 23 / 23

Summary: What to take from this lecture?

1 White noise is the building block of time series models

2 In order to model the dynamics of a time series, use the white noise process
to piece together the dynamics: GLP

3 GLP useful representation: compute expectations, variance and ACF

4 Special models: AR, MA

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-04 ©UNSW 24 / 23

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 1 / 27

Financial Econometrics
Slides-05: Time Series Analysis using ARMA models

Part 2

Dr. Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not
be removed from this material.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 2 / 27

Plan.

Time Series Models (Mainly Theoretical Aspects)

• MA process
• AR process

• Wold Decomposition
• AF and PACF patterns
• Impulse response function

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 3 / 27

Defining Moving Average Process MA(q)

� Moving average models
• In Wold decomposition, bi → 0 as i→∞. A simple approximation to the GLP

is to restricting
bi = 0 for all i > q.

• The result is MA(q) model:

yt = µ+ �t + θ1�t−1 + · · ·+ θq�t−q, �t ∼ i.i.d WN(0, σ2),

where yt is the ”average”of the current shock and its q recent lags. The shock
�t and its lags are unobservable.

• Use lag operator L: Lzt = zt−1 to write MA(q):

yt = µ+ Θ(L)�t,

where
Θ(L) = 1 + θ1L+ θ2L

2
+ · · ·+ θqLq.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 4 / 27

MA(1) model

� MA(1) model
• MA(1) model (as a data generating process)

yt = µ+ �t + θ1�t−1, �t ∼ i.i.d WN(0, σ2),

• MA(1):
yt = µ+ Θ(L)�t,

where Θ(L) = 1 + θ1L.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 5 / 27

MA(1) model: Unconditional moments

� MA(1) model: Characteristics
• It is always stationary.
E(yt) = µ, V ar(yt) = (1 + θ

2
1)σ

γj = Cov(yt, yt−j) =

{
θ1σ

2, j = 1
0, j > 1

}

ρj =
γj
γ0

{
θ1/(1 + θ

2), j = 1
0, j > 1

}
(AC cutoff at j = 1).

• If the estimated ρ̂j has a cutoff at j = 1, the time series may be fitted in an
MA(1) model.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 6 / 27

MA(1) model: Conditional moments

� MA(1) model: Conditional moments
• Conditional on Ωt = {�t, �t−1, · · · ; yt, yt−1, · · · }

E (yt+h|Ωt) =
{
µ+ θ1�t, h = 1
µ, h > 1

}
,

V ar (yt+h|Ωt) =
{
σ2, h = 1
(1 + θ21)σ

2, h > 1

}
.

• Conditional variance ≤unconditional variance (why?)

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 7 / 27

MA(1) model: Dynamic Behavior

� MA(1) model: Impulse response function
• the effect on yt+h of a one-std-deviation increase in ]�t:

σ
δyt+h
δ�t

{
σθ1, h = 1
0, h > 1

}
.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 8 / 27

MA(1) model: Invertibility

� MA(1) model: Invertibility
• Can we back out a unique θ1 from:

ρj =
γj
γ0

{
θ1/(1 + θ

2
1), j = 1

0, j > 1

}
.

Can we get to know {�t, �t−1, · · · } based on {yt, yt−1, · · · }?
• Yes if MA is invertible

– The MA(q) process yt = µ+ Θ(L)�t is invertible if the roots of Θ(z) = 0 are
all oytside the unit circle.

– For MA(1), the root of 1 + θ1z = 0 is z = −1/θ1. Hence, MA(1) is invertible
when | − 1/θ1| > 1 or |θ1| < 1. - Invertible in the sense that Θ(L)−1 exists properly. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 9 / 27 MA(1) model: Invertibility ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � MA(1) model: Invertible - When MA is invertible, the shock may be recovered from the observable: �t = Θ(L) −1(yt − µ). For MA(1), when invertible, Θ(L) −1 = (1 + θ1L) −1 = 1 + (−θ1)L+ (−θ1)2L2 + · · · , (1) �t = yt − µ+ ∞∑ i=1 (−θ1)i(yt−i − µ) (2) = yt + ∞∑ i=1 (−θ1)iyt−i − µ/(1 + θ1). (3) Hint. Use expansion: 1/(1− x) = 1 + x+ x2 + · · · - Parameters can be estimated by minimizing ∑T t=1 �2t - The alternative expression: yt = µ/(1 + θ1)− ∑∞ i=1 (−θ1)iyt−i + �t indicates that the PAC function of invertible MA(1) has no cutoffs and decays exponentially. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 10 / 27 MA(1) model: Example ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material MA(1): simulated and fitted Topic 3. Time Series Models • MA models – MA(1) model eg. time series plots of simulated MA(1) 𝜌𝜌1 = 𝜃𝜃1/(1 + 𝜃𝜃1 2) eg. NYSE comp return School of Economics, UNSW Slides-04, Financial Econometrics 19 -5 -4 -3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 MA(1): theta = -0.9 -4 -3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 MA(1): theta = 0, White Noise -4 -3 -2 -1 0 1 2 3 25 50 75 100 125 150 175 200 MA(1): theta = 0.9 Variable Coefficient Std. Error t-Statistic Prob. C 0.035311 0.02457 1.43729 0.1508 MA(1) 0.075177 0.02271 3.31031 0.0009 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 11 / 27 MA(q) model: Dynamic Behaviour ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material �Dynamic Behaviour of a Moving Average Process MA(q) An MA process is simply a linear combination of white noise error terms ?. These error terms can be seen as impulses or innovations or shocks while the MA model describes the dynamic impact of these shocks on the series yt. The impulse response function, i.e. the dynamic impact of an impulse �t on yt, yt+1, · · · is given by δyt/δ�t = 1 δyt/δ�t = θ1 · · · δyt+q/δ�t = θq δyt+q+k/δ�t = 0, for k > 0

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 12 / 27

MA(q) model: Properties

�General Properties of a Moving Average Process MA(q)
I E(yt) = µ
I γ0 = (1 + θ21 + θ22 + · · ·+ θ2q)σ2
I The ACF:

γk = (θk + θk−1θ1 + θk+2θ2 + · · ·+ θqθq−k)σ2, fpr k = 1, · · · , q.
γk = 0, for k > q.

I The PACF? pk 6= 0 ∀k dies out slowly
�Stationarity conditions for an MA process:
I γ0 is finite
I γk is finite

=⇒ a finite order MA process will always be stationary.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 13 / 27

MA(q) Conclusions

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Moving Average Process

Conclusions

I As the ACF cuts o↵ after q lags, the order of an MA process
can be determined from an inspection of the sample ACF.

I It can be shown (see below) that the PACF dies out slowly.

I A finite order MA process is stationary by construction, as
it is a weighted sum of a fixed number of white noise
processes, i.e. the mean, variance and autocovariances don’t
depend on time!

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 14 / 27

Autoregressive Process: Definition

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Defining an Autoregressive Process

Let “t be a white noise process. Then:

yt = ↵0 + ↵1yt�1 + ↵2yt�2 + … + ↵pyt�p + “t (12)

= ↵0 +
Xp

i=1
↵iyt�i + “t

is an autoregressive process of order p, denoted AR(p).
! yt depends on its own lagged values and on the current value of
a white noise disturbance term “t .

The model can conveniently be rewritten in so-called lag operator
notation as

yt = ↵0 +
Xp

i=1
↵iL

iyt + “t with L
iyt = yt�i

↵ (L) yt = ↵0 + “t (13)

where ↵ (L) = 1 � ↵1L � ↵2L2 � … � ↵pLp is a lag polynomial of
order p

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 15 / 27

AR Process: Impulse response function

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Dynamic Behaviour of an AR(1) Process

In an AR process, the value for yt is simply a linear combination of
past values plus a white noise error term “t . Again, these error
terms can be seen as impulses or innovations or shocks while the
AR model describes the dynamic impact of these shocks on the
series yt .

In order to trace out the dynamic impact of an impulse “t on
yt , yt+1, . . ., it is very convenient to first ‘solve’ the AR model in
terms of the ” sequence. For notational convenience, first consider
an AR(1) process

yt = ↵0 + ↵1yt�1 + “t

where “t is a white noise process.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 16 / 27

AR Process: Impulse response function

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

The easiest way to express yt as a function of the ” sequence is by
backward substitution. This implies substituting

yt�1 = ↵0 + ↵1yt�2 + “t�1

in the equation for yt to obtain

yt = ↵0 + ↵1 (↵0 + ↵1yt�2 + “t�1) + “t

= (1 + ↵1)↵0 + ↵
2
1yt�2 + ↵1″t�1 + “t

Next substitute

yt�2 = ↵0 + ↵1yt�3 + “t�2

in the equation for yt to obtain

yt =
�
1 + ↵1 + ↵

2
1

�
↵0 + ↵

3
1yt�3 + ↵

2
1″t�2 + ↵1″t�1 + “t

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 17 / 27

AR Process: Impulse response function

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

After repeating this t � 1 times, we obtain

yt =
�
1 + ↵1 + . . . + ↵

t�1
1

�
↵0 + ↵

t
1y0 + ↵

t�1
1 “1 + . . . + ↵1″t�1 + “t

= ↵0
Xt�1

i=0
↵i1 + ↵

t
1y0 +

Xt�1
i=0

↵i1″t�i (14)

where y0 is the initial condition or the value for y in period 0.

The impulse response function can now easily be obtained

dyt/d”t = ↵
0
1 = 1

dyt+1/d”t = ↵1

dyt+2/d”t = ↵
2
1

dyt+3/d”t = ↵
3
1

…

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 18 / 27

AR Process: Convergence

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Note that whether an AR(1) series is mean-reverting after being
hit by a shock depends on the particular value for ↵1. Two cases
can be distinguished:

I The convergence case |↵1| < 1 A shock a↵ects all future observations but with a decreasing e↵ect, i.e. the AR(1) process is mean-reverting. I The non-convergence case |↵1| � 1 A shock a↵ects all future observations but with an equal impact (↵1 = 1) or with an increasing impact (↵1 > 1), i.e.
the AR(1) series is not mean-reverting.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 19 / 27

Properties of AR(1) Process: Unconditional mean

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Properties of an AR(1) Process

Let t ! 1 in eq. (14):

yt = ↵0
X1

i=0
↵i1 + ↵

1
1 y0 +

X1
i=0

↵i1″t�i (15)

I The expected value of yt is given by

E (yt) = E
⇣�

1 + ↵1 + ↵
2
1 + …

�
↵0 + ↵

1
1 y0 +

X1
i=0

↵i1″t�i
⌘

= E
��

1 + ↵1 + ↵
2
1 + …

�
↵0 + ↵

1
1 y0

�

! if |↵1| < 1 : E (yt) converges to ↵0 (1 � ↵1) ! if |↵1| � 1 : E (yt) is time-dependent Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 20 / 27 Properties of AR(1) Process: Unconditional Variance ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process I The variance of yt is given by V (yt) = E (yt � E (yt))2 = E ⇣X1 i=0 ↵i1"t�i ⌘2 = E � "2t + ↵ 2 1" 2 t�1 + ↵ 4 1" 2 t�2 + . . . + cross-products � = E � "2t � + ↵21E � "2t�1 � + ↵41E � "2t�2 � + . . . = � 1 + ↵21 + ↵ 4 1 + . . . � �2 ! if |↵1| < 1 : V (yt) converges to �2� 1 � ↵21 � ! if |↵1| � 1 : V (yt) is time-dependent Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 21 / 27 Properties of AR(1) Process: ACF ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process I The autocovariances �k are given by �1 = cov (yt , yt�1) = E ((yt � E (yt)) (yt�1 � E (yt�1))) = E �� "t + ↵1"t�1 + ↵ 2 1"t�2 + . . . � � "t�1 + ↵1"t�2 + ↵ 2 1"t�3 + . . . �� = E � ↵1" 2 t�1 + ↵ 3 1" 2 t�2 + ↵ 5 1" 2 t�3 + . . . + cross-products � = ↵1E � "2t�1 � + ↵31E � "2t�2 � + ↵51E � "2t�3 � + . . . = � ↵1 + ↵ 3 1 + ↵ 5 1 + . . . � �2 = ↵1 � 1 + ↵21 + ↵ 4 1 + . . . � �2 ! if |↵1| < 1 : �1 converges to ↵1 �2� 1 � ↵21 � ! if |↵1| � 1 : �1 is time-dependent Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 22 / 27 Properties of AR(1) Process: ACF ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process �2 = cov (yt , yt�2) = E ((yt � E (yt)) (yt�2 � E (yt�2))) = E �� "t + ↵1"t�1 + ↵ 2 1"t�2 + . . . � � "t�2 + ↵1"t�3 + ↵ 2 1"t�4 + . . . �� = E � ↵21" 2 t�2 + ↵ 4 1" 2 t�3 + ↵ 6 1" 2 t�4 + . . . + cross-products � = ↵21E � "2t�2 � + ↵41E � "2t�3 � + ↵61E � "2t�4 � + . . . = � ↵21 + ↵ 4 1 + ↵ 6 1 + . . . � �2 = ↵21 � 1 + ↵21 + ↵ 4 1 + . . . � �2 ! if |↵1| < 1 : �2 converges to ↵21 �2� 1 � ↵21 � ! if |↵1| � 1 : �2 is time-dependent Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 23 / 27 Properties of AR(1) Process: ACF ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process �k = cov (yt , yt�k) = E ((yt � E (yt)) (yt�k � E (yt�k))) ! if |↵1| < 1 : �k converges to ↵k1 �2� 1 � ↵21 � ! if |↵1| � 1 : �k is time-dependent I The ACF (for stationary series!) is given by ⇢1 = �1 /�0 = ↵1 ⇢2 = �2 /�0 = ↵ 2 1 ... ⇢k = �k /�0 = ↵ k 1 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 24 / 27 AR Process: Stationary Conditions ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process Stationarity conditions for an AR(1) process I ↵11 = 0 I � 1 + ↵1 + ↵ 2 1 + ... � is finite I � 1 + ↵21 + ↵ 4 1 + . . . � is finite I ↵1 � 1 + ↵21 + ↵ 4 1 + . . . � is finite I . . . ! an AR(1) process is stationary is |↵1| < 1. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 25 / 27 AR Process: Conclusions ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process Conclusions: I The PACF cuts o↵ after 1 lag. I The ACF is infinite in extent (but dies out for covariance stationary processes). I The properties of an AR(1) process crucially depend on the value for ↵1 I If |↵1| < 1 the AR(1) process can be written as a stable infinite MA process (the so-called MA representation): yt = ↵0 1 � ↵1 + X1 i=0 ↵i1"t�i . In this case the series is stationary as it has finite constant mean, variance and autocovariances. I If |↵1| � 1 no stable MA representation exists. In this case the series is non-stationary as the mean, variance and autocovariances are time-varying. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 26 / 27 AR(1) Example: Simulated and Fitted ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 3. Time Series Models • AR models – AR(1) model eg. time series plots of simulated AR(1) 𝜌𝜌𝑗𝑗 = 𝜙𝜙1 𝑗𝑗 eg. NYSE comp return: ‘c’ below is in fact 𝜇𝜇 = 𝑐𝑐/(1 − 𝜙𝜙1) School of Economics, UNSW Slides-04, Financial Econometrics 25 -4 -3 -2 -1 0 1 2 3 4 25 50 75 100 125 150 175 200 AR(1): phi = 0, White Noise -3 -2 -1 0 1 2 3 25 50 75 100 125 150 175 200 AR(1): phi = 0.5 -6 -4 -2 0 2 4 6 25 50 75 100 125 150 175 200 AR(1): phi = 0.9 -10 -5 0 5 10 25 50 75 100 125 150 175 200 AR(1): phi = 1 Variable Coefficient Std. Error t-Statistic Prob. C 0.035159 0.024547 1.43235 0.1522 AR(1) 0.068401 0.022727 3.00976 0.0026 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-05 ©UNSW 27 / 27 Copyright©Copyright University of New South Wales 2020. All rights reserved. Course materials subject to Copyright UNSW Sydney owns copyright in these materials (unless stated otherwise). The material is subject to copyright under Australian law and overseas under international treaties. The materials are provided for use by enrolled UNSW students. The materials, or any part, may not be copied, shared or distributed, in print or digitally, outside the course without permission. Students may only copy a reasonable portion of the material for personal research or study or for criticism or review. Under no circumstances may these materials be copied or reproduced for sale or commercial purposes without prior written permission of UNSW Sydney. Statement on class recording To ensure the free and open discussion of ideas, students may not record, by any means, classroom lectures, discussion and/or activities without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the student?s own private use. WARNING: Your failure to comply with these conditions may lead to disciplinary action, and may give rise to a civil action or a criminal offence under the law. THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 1 / 33 Financial Econometrics Slides-06: Generalizing to ARMA and Forecasting Dr. Rachida Ouysse School of Economics1 1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 2 / 33 Plan. ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • General AR(p) • Wold Decomposition • AF and PACF patterns • Impulse response function • Yule-Walker equations • AR & MA mix- ARMA models • AF and PACF patterns • Impulse response function • Estimation of ARMA • Forecasting in ARMA Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 3 / 33 Stationarity of AR(2) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process The conditions for stationarity/invertibility of an AR(1) process can be extended to higher order AR processes. I First consider an AR(2) process � 1 � ↵1L � ↵2L2 � yt = ↵ (L) yt = ↵0 + "t . In general, the polynomial ↵ (L) can be rewritten as � 1 � ↵1L � ↵2L2 � = (1 � �1L) (1 � �2L) . where �1 and �2 can be solved from �1 + �2 = ↵1 and ��1�2 = ↵2 The conditions for invertibility of the second order polynomial are just the conditions that both the first order polynomials (1 � �1L) and (1 � �2L) are invertible, i.e. |�1| < 1 and |�2| < 1. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 4 / 33 Stationarity of AR(2) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process A more common way of presenting these conditions is in terms of the so-called characteristic equation � 1 � ↵1z � ↵2z2 � = 0, or (1 � �1z) (1 � �2z) = 0. This equation has two solutions, denoted z1 and z2 z1, z2 = ↵1 ± q ↵21 + 4↵2 �2↵2 , referred to as the characteristic roots of the ↵ (L) polynomial. The requirement |�i | < 1 corresponds to |zi | > 1. If any solution
satisfies |zi |  1, the polynomial ↵ (L) is non-invertible. A solution
that equals unity is referred to as a unit root.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 5 / 33

General Conditions for Stationarity for an AR(p)

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

Calculating the roots of a higher order AR process is
computationally not a trivial job. In most circumstances there is
little need to directly calculate the characteristic roots, though, as
there are some useful simple rules for checking
stationarity/invertibility of higher order processes

I Necessary condition:
Pp

i=1 ↵i < 1 I Su�cient condition: Pp i=1 |↵i | < 1 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 6 / 33 Useful representations ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process As, under appropriate conditions, an AR(p) process has an MA(1) representation and an MA(q) has an AR(1) representation, there is no fundamental di↵erence between AR and MA models. I The MA representation is convenient to derive the properties (mean, variance, ...) of a series I The AR representation is convenient for making predictions conditional upon the past When estimating time series models (cf. below), the choice is simply a matter of parsimony. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 7 / 33 What is the AR process is stationary? ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process For a stationary AR(p) process, it is more convenient to derive the properties from imposing that the mean, variance and autocovariances do not depend on time. For computational convenience consider an AR(2) process. I The unconditional mean of yt can be solved from E (yt) = ↵0 + ↵1E (yt�1) + ↵2E (yt�2) which, assuming that E (yt) does not depend on time allows us to write E (yt) = ↵0 /(1 � ↵1 � ↵2) I The variance of yt can be solved by defining xt = yt � E (yt) which yields xt = ↵1xt�1 + ↵2xt�2 + "t (18) Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 8 / 33 What is the AR process is stationary? ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process The variance of yt can be obtained by multiplying both sides by xt and taking expectations V (yt) = �0 = E (↵1xtxt�1 + ↵2xtxt�2 + xt"t) = ↵1�1 + ↵2�2 + E (xt"t) = ↵1�1 + ↵2�2 + � 2 (19) where E (xt"t) = � 2 is obtained from multiplying both sides of (18) by "t and taking expectations. Multiplying both sides by xt�1 and xt�2 and taking expectations we obtain �1 = ↵1�0 + ↵2�1 (20) �2 = ↵1�1 + ↵2�0 (21) These equations can be solved for �0 to obtain �0 = (1 � ↵2) (1 + ↵2) (1 � ↵1 � ↵2) (1 + ↵1 � ↵2) �2 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 9 / 33 What is the AR process is stationary? ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models Autoregressive Process I The autocorrelation coe�cients ⇢1 and ⇢2 can be obtained by dividing (20) and (21) by �0 ⇢1 = ↵1 + ↵2⇢1 ⇢2 = ↵1⇢1 + ↵2 and solving to obtain ⇢1 = ↵1 /(1 � ↵2) ⇢2 = ↵ 2 1 /(1 � ↵2) + ↵2 It is easily verified that the higher-order autocorrelation coe�cients are given by ⇢k = ↵1⇢k�1 + ↵2⇢k�2 Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 10 / 33 Yule Walker Equations ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � The beauty of the yule Walker Equations! xt = α1xt−1 + · · ·+ αpxt−1 + �t xtxt−1 = α1xt−1xt−1 + · · ·+ αpxt−pxt−1 + �txt−1 E(xtxt−1) = α1E(xt−1xt−1) + · · ·+ αpE(xt−pxt−1) + E(�txt−1) γ1 = α1γ0 + α2γ1 + · · ·+ αpγp−1 · · · xtxt−j = α1xt−1xt−j + · · ·+ αpxt−pxt−j + �txt−j E(xtxt−j) = α1E(xt−1xt−j) + · · ·+ αpE(xt−pxt−j) + E(�txt−j) γ|j| = α1γ|j−1| + α2γ|j−2| + · · ·+ αpγ|p−j|, · · · γ = Γα Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 11 / 33 Defining an ARMA Process ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Defining an ARMA Process Let "t be a white noise process. Then: ↵ (L) yt = ↵0 + � (L) "t (22) with ↵ (L) an AR polynomial of order p and � (L) an MA polynomial of order q, is an autoregressive moving average process with orders p and q, denoted ARMA(p, q). ! yt depends on its own lagged values and on current and past values of a white noise disturbance term "t . Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 12 / 33 Dynamic Behaviour ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Dynamic Behaviour and Impulse Response Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Dynamic Behaviour of an ARMA(p, q) Process If the AR polynomial ↵ (L) is invertible, the ARMA(p, q) process can be written as a stable MA(1) process of the form yt = ↵ (L) �1 ↵0 + ↵ (L) �1 � (L) "t = ↵00 + ✓ (L) "t where ↵00 = ↵0 � 1 � Pp i=1 ↵i and ✓ (L) = ↵ (L) �1 � (L) = 1+ P1 i=1 ✓iL i , with ✓i = undetermined coe�cients. Even if the AR polynomial is non-invertible, we can still solve for the " sequence but this solution will not be a stable MA process, i.e. yt = f (t) + ✓ (L) "t where f (t) indicates that the mean is a function of time. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 13 / 33 Dynamic Behaviour ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Dynamic Behaviour and Impulse Response Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process The impulse response function can be obtained from the MA representation. Note that as a finite order MA process is stationary by construction, an ARMA process is stationary if the AR component is stationary (i.e. if the AR polynomial is invertible). I In the stationary case the impact of shocks gradually dies out (i.e. P1 i=1 ✓i is finite) I In the non-stationary case the impact of a shock never vanishes (i.e. P1 i=1 ✓i is infinite) Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 14 / 33 General Properties of an ARMA(p,q) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material � Unconditional Moments of an ARMA(p, q) Univariate Time Series Analysis: ARIMA models Building ARIMA models An ARMA Process Properties of an ARMA(p, q) Process If the AR polynomial ↵ (L) is non-invertible the mean, variance and covariances are time-varying. If the AR polynomial ↵ (L) is invertible, the AR process can be rewritten as the stable infinite MA process. The properties of a stationary AR process can easily be derived from this MA representation. I Unconditional mean: E (yt) = ↵0 �� 1 � Pp i=1 ↵i � I Unconditional variance: V (yt) = � 2 P1 i=0 ✓ 2 i I Covariances: �k = (✓k + ✓1✓k+1 + ✓2✓k+2 + . . .)� 2 As an ARMA(p, q) process includes both an AR and an MA component, both the ACF and the PACF do not cut o↵ at some point. As such, it is di�cult to determine the order of an ARMA model from the ACF and PACF. Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 15 / 33 Maximum Likelihood Estimation: Intuitive Illustration ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Maximum Likelihood Estimation Binary dependent variable model The Probit Model Inference in Logit and Probit Specification tests Intuitive illustration This illustration shows a sample of n independent observations, and two continuous distributions f1(x) and f2(x), Likelihood This illustration shows a sample of n independent observations, and two continuous distributions f1(x) and f2(x), with f2(x) being just f1(x) translated by a certain amount. Of these two distributions, which one is the most likely to have generated the sample ? Clearly, the answer is f1(x), and we would like to formalize this intuition. Although this is not strictly impossible, we don't believe that f2(x) generated the sample because all the observations are in regions where the values of f2(x) are small : the probability for an observation to appear in such a region is small, and it is even more unlikely that all the observations in the sample would appear in low density regions. On the other hand, the values taken by f1(x) are substantial for all the observations, which are then where one would expect them to be, would the sample be actually generated by f1(x). Definition of the likelihood Of the many ways to quantify this intuitive judgement, one turns out to be remarkably effective. For any probability distribution f(x), just multiply the values of f(x) for each of the observations of the sample, denote the result L, and call it the likelihood of the distribution f(x) for this particular sample : Clearly, the likelihood can have a large value only if all the observations are in regions where f(x) is not very small. This definition has the additional advantage that L receives a natural interpretation. The sample {xi} may be regarded as a single observation generated by the n-variate probability distribution f(x1, x2, ..., xn) = Πi f(xi) because of the independence of the individual observations. So the likelihood of the distribution is just the value of the n-variate probability density f (x1, x2, ..., xn ) for the set of observations in the sample considered as a unique n-variate observation. Likelihood and estimation, Maximum Likelihood estimators These considerations make us believe that "likelihood" might be a helpful concept for identifying the distribution that generated a given sample. First note, though, that as such, this approach is moot if we don't a priori restrict our search : the probability distribution leading to the largest possible value of the likelihood is obtained by assigning the probability 1/n to each of the points where there is an observation, and assigning the value 0 to f(x) for any other point of the x axis. This result is both trivial and useless. But consider the example given in the above illustration : f1(x) and f2(x) are assumed to belong to a family of distributions, all identical in shape and differing only by their position along the x axis (location family). It now makes sense to ask for which position of the generic distribution f(x) is the likelihood largest. If we denote θ the parameter adjusting the horizontal position of the distribution, one may consider the value of θ conducive to the largest likelihood as being probably fairly close to the true (and unknown) value θ 0 of the parameter of the distribution that actually generated the sample. It then appears that the concept of likelihood may lead to a method of parameter estimation. The method consists in retaining as an estimate of θ 0 the value of θ conducive to the largest possible value of the sample likelihood. This method is thus called Maximum Likelihood estimation, which is, in fact, the most powerful and widely used method of parameter estimation these days. An estimator θ* obtained by maximizing the likelihood of a probability distribution defined up to the value of a parameter θ is called a Maximum Likelihood estimator and is usually denoted "MLE". When we need to emphasize the fact that the likelihood depends on both the sample x = {xi} and the parameter θ, we'll denote it L(x, θ). ----- Interactive animation Likelihood = L =: Πi f(xi) i = 1, 2, ..., n Page 1 of 6Likelihood and method of Maximum Likelihood 21/03/2010http://www.aiaccess.net/English/Glossaries/GlosMod/e_gm_likelihood.htm Of these two distributions, which one is the most likely to have generated the sample ? Although it is not impossible, we don’t believe that f2(x) generated the sample. Why? On the other hand, the values taken by f1(x) are substantial for all the observations, which are then where one would expect them to be, would the sample be actually generated by f1(x). Dr. Rachida Ouysse ECON3208: Lecture 3 Maximum Likelihood Estimation Limited Dependent Variable Models Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 16 / 33 Maximum Likelihood Estimation ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Maximum Likelihood Estimation Binary dependent variable model The Probit Model Inference in Logit and Probit Specification tests Maximum Likelihood Estimation • Maximum Likelihood Estimation is a general method of estimation that can be used for many di↵erent types of data and economic models. It has very wide applicability. • The Maximum Likelihood Estimator (MLE) answers the following question: What are the parameter estimates that are most likely to have generated the observed data given the assumed model. • Begin by assuming a model for the outcome variable including a distribution function for the underlying population error term (and hence a distribution for the outcome variable in the population.) Dr. Rachida Ouysse ECON3208: Lecture 3 Maximum Likelihood Estimation Limited Dependent Variable Models Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 17 / 33 Estimation of ARMA: Maximum Likelihood ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Consider AR(1) model: yt = α0 + α1yt−1 + �t where �t ∼ i.i.d N(0, σ2). - it follows: yt|Ωt−1 ∼ N ( α0 + α1yt−1, σ 2 ) , t = 2, 3, · · · . y1 ∼ N ( [1− α1]−1α0, [1− α21] −1 σ 2 ) - Conditional pdf: f(yt|Ωt−1) = 1 √ 2πσ2 exp { − (yt − α0 − α1yt−1)2 2σ2 } - Information sets: Ω1 = {y1},Ω2 = {y2,Ω1}, · · · ,Ωt = {yt,Ωt−1}. - Joint pdf for a time series {y1, · · · , yT } can be factorised: f(yT , yT−1 · · · , y1) = = f(yT , yT−1 · · · , y2|Ω1)f(y1) = f(yT , yT−1 · · · , y3|Ω2)f(y2|Ω1)f(y1) = f(yT |ΩT−1)f(yT−1|ΩT−2) · · · f(y3|Ω2)f(y2|Ω1)f(y1) Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 18 / 33 Maximum Likelihood Estimation ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 3. Time Series Models • Maximum likelihood – Properties of ML estimators • When the pdf (likelihood) is correctly specified, the ML estimators have nice large-𝑇𝑇 sampling properties: – consistent, – asymptotically normally distributed, and – asymptotically efficient. Allow us to draw inference based on reported SEs. • When the pdf (likelihood) is incorrect, the “ML” procedure is called quasi (or pseudo) ML. – When the normal pdf is used, which may be incorrect, the quasi ML estimators are still consistent and asymptotically normal, as long as the model is defined by the conditional mean and variance that are correctly specified. School of Economics, UNSW Slides-05, Financial Econometrics 14 Must use “robust” SEs Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 19 / 33 ARMA Process: Identification ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Fitting ARMA models to the data The Box-Jenkins Approach The so-called Box-Jenkins approach toward fitting ARMA models comprises three stages: I Identification: determine tentative model(s) I Plot the time series to have a first idea on the DGP (stationary/non-stationary, structural break, ...) I Plot the ACF and the PACF to have a first idea on the order of the ARMA model I Estimation: estimate the various tentative models I Compare the estimated models using information criteria I Select parsimonious model I Diagnostic checking: check the selected model’s diagnostics Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 20 / 33 AR Process: Estimation ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Univariate Time Series Analysis: ARIMA models Fitting ARMA models to the data Estimating ARMA models Estimating ARMA models I Consider the AR(p) model yt = ↵0 + ↵1yt�1 + . . . + ↵pyt�p + "t ↵ (L) yt = "t with "t a zero-mean white noise process. As yt�1, . . . , yt�p are observed in the data, the model can be estimated using OLS. The OLS estimator is I biased, because: E (yt�j"t�j) 6= 0 I consistent, because: E (yt�j"t) = 0 8j > 0
I asymptotically normal

Intuition: the error terms and the explanatory variables are not
completely independent but contemporaneously uncorrelated.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 21 / 33

MA Process: Estimation

Univariate Time Series Analysis: ARIMA models

Fitting ARMA models to the data

Estimating ARMA models

I Consider the MA(q) model

yt = ↵0 + �1″t�1 + . . . + �q”t�q + “t
yt = ↵0 + � (L) “t

with “t a zero-mean white noise process.

As “t�1, . . . , “t�q are NOT observed in the data, the model
cannot be directly estimated using OLS.

A possible solution is to estimate the coe�cients in � (L) from
the AR representation of the MA model. For an invertible
MA(1) model, this is given by (cf. above):

yt = ↵0 /(1 + �1) �
1X

i=1

(��1)i yt�i + “t

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 22 / 33

Model Selection: order of the ARMA(p,q)

Univariate Time Series Analysis: ARIMA models

Fitting ARMA models to the data

Information criteria

A fundamental idea in the Box – Jenkins approach is the principle
of parsimony (meaning sparseness)

I A parsimonious model fits the data well without incorporating
any needless coe�cients

I In general, parsimonious models produce better forecasts than
over-parametrized models

Increasing the lag orders p and q will:

I Increase the goodness-of-fit of the model, i.e. reduce the RSS

I Reduce the degrees of freedom

! information criteria provide a trade-o↵ between the
goodness-of-fit of the model and the number of parameters used to
obtain that fit.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 23 / 33

Model Selection: order of the ARMA(p,q)

Univariate Time Series Analysis: ARIMA models

Fitting ARMA models to the data

Information criteria

The two most commonly used information criteria are:

I Akaike Information Criterion (AIC)

AIC = T ln (RSS) + 2k

I Schwarz Bayesian Criterion (SBC)

SIC = T ln (RSS) + k ln (T )

with k = p + q + 1 the number of estimated parameters.

The most appropriate model is the one that minimises AIC
and/or SBC.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 24 / 33

Model Selection: order of the ARMA(p,q)

Univariate Time Series Analysis: ARIMA models

Fitting ARMA models to the data

Information criteria

Note that:

I When you estimate models with lagged variables, some initial
observations are lost. In order to compare models using
information criteria, you should keep T fixed! Otherwise you
will be comparing the performance of the models over
di↵erent sample periods. Moreover, decreasing T has the
direct e↵ect of reducing the AIC and SBC.

I The SBC embodies a much sti↵er penalty for the loss of
degrees of freedom than the AIC. The main di↵erence between
the two in terms of performance is that SBC is consistent (i.e.
asymptotically it delivers the correct model) while the AIC is
biased toward selecting an over-parametrised model. However,
in small samples, the AIC can work better than the SBC.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 25 / 33

Information Criteria: Example

Topic 3. Time Series Models

• AIC & SIC
– If data are generated by an ARMA, the probability

that SIC selects the correct model converges to
one as 𝑇𝑇 → ∞.
• In finite samples, SIC may
select a too-small model.

eg. unanticipated
US monthly inflation:

• SIC selects AR(1)
• AIC selects ARMA(1,1)

School of Economics, UNSW Slides-05, Financial Econometrics 18

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 26 / 33

Forecasting

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Terminology

Forecasting using ARMA models

I In-sample versus out-of-sample forecasts
I In-sample forecasts are those generated for the same set of

data that was used to estimate the model’s parameters. Good
performance may be due to fitting a spurious model to the
noise in the sample, though!

I Out-of-sample forecasts are those generated for a set of data
that was not used to estimate the model, i.e. do not use all
observations in estimating the model and evaluate the model
from the forecasting accuracy in the holdout sample.

I Static versus dynamic forecasts
I Static forecasts are a sequence of one-step-ahead forecasts,

using actual, rather than forecasted values for lagged
dependent variables.

I Dynamic forecasts are a sequence of multi-step-ahead
forecasts starting from the first period in the forecast sample,
using forecasted values for lagged dependent variables.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 27 / 33

Forecasting

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

In addition to the prediction itself, it is important to know how
accurate this prediction is. To judge forecasting accuracy, define
the prediction error as

fet,s = yt+s � ft,s

and the variance of the forecasting error by

var (fet,s) = E (yt+s � ft,s)2

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 28 / 33

Forecasting MA(q)

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

For the MA(q) model we have

fet,1 = “t+1

fet,2 = “t+2 + �1″t+1

fet,3 = “t+3 + �1″t+2 + �2″t+1
…

fet,q = “t+q + �1″t+q�1 + . . . + �q�1″t+1
fet,q+1 = “t+q+1 + �1″t+q + . . . + �q”t+1

fet,q+2 = “t+q+2 + �1″t+q+1 + . . . + �q”t+2
…

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 29 / 33

Forecasting MA(q)

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

such that

var (fet,1) = E (“t+1)
2

= �2

var (fet,2) = E (“t+2 + �1″t+1)
2

=
�
1 + �21

�
�2

var (fet,3) = E (“t+3 + �1″t+2 + �2″t+1)
2

=
�
1 + �21 + �

2
2

�
�2

…

var (fet,q) = E (“t+q + �1″t+q�1 + . . . + �q�1″t+1)
2

=
�
1 + �21 + . . . + �

2
q�1

�
�2

var (fet,q+1) = E (“t+q+1 + �1″t+q + . . . + �q”t+1)
2

=
�
1 + �21 + . . . + �

2
q

�
�2

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 30 / 33

Forecasting MA(q)

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

The accuracy of the prediction

I decreases as we predict further into the future

I does not decrease any further from s = q + 1 onward as the
variance of the prediction error stabilises at the unconditional
variance. This is the upper bound on the inaccuracy of the
predictor.

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 31 / 33

Forecasting AR(p)

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

For a stationary AR(p) model, the prediction errors are most
easily obtained from the MA(1) representation

yt = ↵0 /(1 � ↵1 � . . .↵p) +
1X

i=0

�i”t�i

with �i undetermined coe�cients.

Consequently, the s-period-ahead prediction error is given by

fet,s =
s�1X

i=0

�i”t+s�i

with variance

var (fet,s) = �
2

s�1X

i=0

�2i

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 32 / 33

Forecasting AR(q)

Univariate Time Series Analysis: ARIMA models

Forecasting using ARMA models

Forecasting Accuracy

The accuracy of the prediction
I decreases as we predict further into the future as �2i > 0

I converges to the stable unconditional variance �2
1P
i=0

�2i as

t ! 1. This is the upper bound on the inaccuracy of the
predictor.

As an illustration, consider an AR(1) model where �i = ↵
i
1. The

forecasting errors are given by

fet,1 = “t+1

fet,2 = “t+2 + ↵1″t+1
…

fet,s =
s�1X

i=0

↵i1″t+s�i

Dr. Rachida OuysseSchool of Economics (UNSW) Slides-06 ©UNSW 33 / 33

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

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Financial Econometrics
Slides-08: Nonstationary Processes

Identification, Testing and Estimation

Dr. Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this
material.

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Plan.

1 Stochastic and Deterministic Non-stationary Processes
• Properties of Deterministic Non-stationary Process
• Properties of Stochastic Non-stationary Process

1 Random Walk Process
2 Random Walk Process with a drift

• Transformation to achieve Stationarity
2 Unit Root Tests

1 Dickey Fuller Test: Basic
2 Dickey Fuller Test: Intercept
3 Dickey Fuller Test: Intercept and Trend
4 Augmented Dickey-Fuller Test

3 Power consideration

4 Selection of model for testing

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Stationarity versus Non-stationarity

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Deterministic Non-stationarity

Deterministic Non-stationarity Process

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Deterministic Non-stationarity

Deterministic Non-Stationary Process

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Deterministic Non-stationarity

Deterministic Non-Stationary Process

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Deterministic Non-stationarity

Deterministic Non-Stationary Process

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Stochastic Non-stationarity

Stochastic Non-Stationary Process

Univariate Time Series Analysis

Stationarity versus Non-stationarity

Stochastic non-stationarity

Consider the AR(1) process:

yt = α0 + α1yt−1 + εt

The MA representation is given by:

yt = α0
∑∞

i=0
αi1 + α

∞
1 y0 +

∑∞
i=0

αi1εt−i

Depending on the value for α1, two cases can be distinguished:

I Stationary case: |α1| < 1 ⇒ αi1 → 0 as i →∞ → shocks gradually die out. I Unit root case: |α1| = 1 ⇒ αi1 = 1 ∀i → shocks persist in the system. I Explosive case: |α1| > 1 ⇒ αi1 →∞ as i →∞
→ shocks have an increasingly large influence.

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Stochastic Non-stationarity

Stochastic Non-Stationary Process

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk Process

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk Process

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk Process

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk: Simulated example

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk: Simulated example

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk with a Drift

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk with a Drift: Properties

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Random Walk Model

Random Walk with a Drift: Properties

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Transformation to Stationarity

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Transformation to Stationarity

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Unit Root Tests

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

How do we test for a unit root?

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Basic Dickey-Fuller Test

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Basic Dickey-Fuller Test

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Basic Dickey-Fuller Test: Alternative τ statistic

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Basic Dickey-Fuller Test: Distribution

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Dickey-Fuller Test

Basic Dickey-Fuller Test: Example

Example of DF test: Australian GDP

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

DF Test with Intercept

Dickey-Fuller Test: Intercept

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

DF Test with Intercept

Dickey-Fuller Test with Intercept: Example

Example of DF test: Australian GDP

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

DF Test: Intercept and Trend

Dickey-Fuller Test: Intercept and Trend

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

DF Test: Intercept and Trend

Dickey-Fuller Test with Intercept: Example

Example of DF test: Australian GDP

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

The Augmented Dickey-Fuller test

Augmented DF Test

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

The Augmented Dickey-Fuller test

Augmented DF Test

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

The Augmented Dickey-Fuller test

Augmented Dickey-Fuller Test: Example

Example of ADF(1) test: Australian GDP

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

The Augmented Dickey-Fuller test

Augmented Dickey-Fuller Test: Example

Example of ADF(3) test: Australian GDP

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Model Selection

Model Selection for ADF

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Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Model Selection

Model Selection for ADF

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

Model Selection

Model Selection for ADF

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

A note on the power of ADF tests!

Slides-07 UNSW

Plan Stationarity versus Non-stationarity Unit Root Tests Power of ADF tests

A note on the power of ADF tests!

Slides-07 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

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offence under the law.
THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Financial Econometrics
Slides-09: Volatility Modelling

Dr. Rachida Ouysse
School of Economics1

1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Lecture Plan

• Motivation for modeling return volatility
• Measures of return volatility
• Conditional volatility via smoothing
• ARCH

• Conditional variance is a function of info set;
• It captures “clustering” in return series;
• It explains non-normality of return, to some extent;
• It can be used to improve interval forecasts and VaR (Value at Risk);
• Estimation and testing.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Introduction and Motivation

eg. Volatility in NYSE Composite index return

• Clustering.
• Squared returns are strongly autocorrelated.

Topic 5. Modelling Return Volatility: ARCH

• Motivation
eg. Volatility in NYSE Composite index return

• Clustering.
• Squared returns are strongly autocorrelated.

School of Economics, UNSW Slides-7, Financial Econometrics 3

-8

-6

-4

-2

250 500 750 1000 1250 1500 1750

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Motivation

eg. Volatility in NYSE Composite index return

• Monthly realised variance:
RV =sample mean of squared daily returns in a month

• RV is negatively correlated to lagged monthly return.
Corr(RV,Return(−1)) = −0.419.

Topic 5. Modelling Return Volatility: ARCH

• Motivation
eg. Volatility in NYSE Composite index return

• Monthly realised variance:
RV = sample mean of squared daily returns in a month

• RV is negatively correlated to lagged monthly return.
Corr(RV, Return(-1)) = −0.419

School of Economics, UNSW Slides-7, Financial Econometrics 4

1996 1998 2000 2002

0
1

2
3

4
5

R
V

Date

-1
5

-1
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-5
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et

ur
n(

-1
)

RV
Return(-1)

-15 -10 -5 0 5

0
1

2
3

4
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Comp. Return(-1)

C
om

p.
R

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Motivation

I Importance of return volatility
Asset pricing, risk management and portfolio selection

Substantial dependence structure in volatility

I Clustering:
– strong autocorrelations in squared returns,
– large variations tend to be followed by large variations

I Asymmetry:
– negative returns tend to cause more volatility than positives

I ARMA are unable to capture these features
Conditional variance is constant in ARMA.

Amend ARMA with a suitable conditional variance: ARCH and GARCH
models.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Volatility

Measures of return volatility (tendency of variation)

• Historical volatility: Sample variance or Stddev

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

(tendency of variation)
• Historical volatility: Sample variance or Stddev
eg. NYSE composite return: Sample Stddev

School of Economics, UNSW Slides-7, Financial Econometrics 7

250 500 750 1000 1250 1500 1750

0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

250 500 750 1000 1250 1500 1750

V1M
V3M

V6M
V12M

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Measures of return volatility

• Realised volatility: Realised variance = Sample mean of squared higher
frequency returns

(eg. daily RV = Sample mean of squared 5-min returns)

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Realised volatility: Realised variance =
Sample mean of squared higher frequency returns
(eg. daily RV = Sample mean of squared 5-min returns)

eg. NYSE composite return: Monthly realised variance
RV = Sample mean of squared daily returns in a month

School of Economics, UNSW Slides-7, Financial Econometrics 8
1996 1998 2000 2002

0
1

2
3

4
5

C
om

p.
R

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Measures of return volatility

• Range (high/low):
100× ln(high/low) in a time interval (eg, a day)

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Range (high/low) :
100∙ln(high/low) in a time interval (eg, a day)

eg. BHP
daily return and range

School of Economics, UNSW Slides-7, Financial Econometrics 9

-1
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2004 2006 2008 2010 2012

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an

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Implied Volatility

Implied volatility:

standard deviation derived from options prices
– Option of an asset: the right to buy/sell the asset at a future time

(maturity) at a fixed price (strike).
– Given theprice of an option, maturity, strike and risk-free interest rate, the

std deviation can be recovered from Black-Scholes formula, known as IV.
– IV represents market’s opinions on the return’s std deviation.

Black-Scholes formula:
price of an option =f(stdev, maturity,strike, rf−rate)

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Implied Volatility

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Implied volatility:
eg. VIX: index of IVs of a set of options on the SP500 index
SP500 daily return & VIX

School of Economics, UNSW Slides-7, Financial Econometrics 11

-1
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1990 1995 2000 2005 2010

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Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Conditional Volatility

Topic 5. Modelling Return Volatility: ARCH

• Conditional Volatility
– Conditional variance of return

• 𝜎𝜎𝑡𝑡+1|𝑡𝑡
2 = Var(𝑟𝑟𝑡𝑡+1|Ω𝑡𝑡) ,

where 𝑟𝑟𝑡𝑡+1 = 100ln (𝑃𝑃𝑡𝑡+1/𝑃𝑃𝑡𝑡) is the return and
Ω𝑡𝑡 is the information set at the end of period 𝑡𝑡 .
• It should capture “clustering” or autocorrelations in

squared returns, and facilitate predicting the return
volatility

• Knowing it helps to
– assess the risk of an asset via value-at-risk;
– price options;
– form mean-variance efficient portfolios.

School of Economics, UNSW Slides-7, Financial Econometrics 12

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Conditional Volatility

Exponentially weighted moving average (EWMA)

• The squared returns {r2t , r2t−1, · · · , r21} carry info about the volatility as
E(r2t ) ≡ variance.
• A weighted average of squared returns is an approximation to the

conditional variance. Recent observations should weigh more.

• EWMA: for 0 < λ < 1, σ 2 t+1|t = (1− λ)(r 2 t + λr 2 t−1 + λ 2 r 2 t−2 + · · · ) - weights decay exponentially; - weights sum up to 1. - RiskMetrics recommend λ = 0.94 Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model EWMA ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility – EWMA • EWMA: alternative formulation  𝜎𝜎1|0 2 = 𝑟𝑟1 2  𝜎𝜎𝑡𝑡+1|𝑡𝑡 2 = 1 − 𝜆𝜆 𝑟𝑟𝑡𝑡2 + 𝜆𝜆𝜎𝜎𝑡𝑡|𝑡𝑡−1 2 , for 𝑡𝑡 = 1,2,3, … – Quick and easy; – Can be used as 1-step ahead prediction. eg. NYSE Composite return: 𝜆𝜆 = 0.94 School of Economics, UNSW Slides-7, Financial Econometrics 14 0 4 8 12 16 20 24 28 1600 1650 1700 1750 1800 1850 1900 R2 EWMA Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ARCH (autoregressive conditional heteroskedasticity) Engle (1982) – Nobel price winner 1993 ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility ARCH models ARCH models Autoregressive conditional heteroscedasticity (ARCH) models are a class of models where the conditional variance evolves according to an autoregressive process. First define the conditional variance of the error term ut to be σ2t = var (µt |µt−1, µt−2, ...) = E ( (µt − E (µt))2 |µt−1, µt−2, ... ) As it is usually assumed that E (µt) = 0 σ2t = var (µt |µt−1, µt−2, ...) = E ( µ2t |µt−1, µt−2, ... ) = Et−1 ( µ2t ) The ARCH(1) model assumes σ2t = Et−1 ( µ2t ) = α0 + α1µ 2 t−1 The conditional variance captures ’clustering’: large past shock leads to large conditional variance. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ARCH (autoregressive conditional heteroskedasticity) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility ARCH models Extensions I An ARCH(q) model is given by σ2t = α0 + α1µ 2 t−1 + α2µ 2 t−2 + ...+ αqµ 2 t−q I Under ARCH, the conditional mean equation can take any form. An example of a full model would be yt = β1 + β2x2t + β3x3t + β4x4t + µt µt ∼ N ( 0, σ2t ) σ2t = α0 + α1µ 2 t−1 Alternative notation yt = β1 + β2x2t + β3x3t + β4x4t + µt µt = νtσt νt ∼ N (0, 1) σ2t = α0 + α1µ 2 t−1 Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Properties of ARCH(1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • ARCH(1): µt|Ωt−1 ∼ N(0, σ2t ), Ωt−1 = {yt−1, µt−1, yt−2, µt−2 · · · } is the info set at the end of period t− 1: σ2t = α0 + α1µ 2 t−1, α0 > 0, 0 ≤ α1 < 1 • Its conditional variance is time varying: Var(µt|Ωt−1) = σ2t , CI(95%) =?• It is WN:(Use LIE) E(µt) = 0, Var(µt) = α01−α1 , Cov(µt, µt−j) = 0 But it is NOT independent WN or iid WN. Why? Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Proof of properties ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Definition (Law of Iterated Expectations) For a random variable Y and information sets Ω1 and Ω2, the the LIE states that E (Y |Ω1) = E (E (Y |Ω2) |Ω1) , where information set Ω1 is included in information set Ω2. Example: E (Yt|Ωt−2) = E (E (Yt|Ωt−1) |Ωt−2) Special Case: If Ω1 is empty set, then E (Y ) = E (E (Y |Ω2)) . µt = νtσt = νt √ α0 + α1µ 2 t−1, where νt is N(0, 1) 1 Unconditional Expectation of µt. We have that µt|Ωt−1 ∼ N(0, σ2t ): E (µt) = E [E [µt|Ωt−1]] (1) E [µt|Ωt−1] = 0 (2) E (µt) = 0. (3) Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Proof of properties ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material µt = νtσt = νt √ α0 + α1µ 2 t−1, where νt is N(0, 1) 2 Unconditional variance of µt. We have that E ( µ 2 t ) = E [ E [ µ 2 t |Ωt−1 ]] (4) = E [ E [ ν 2 t ( α0 + α1µ 2 t−1 ) |Ωt−1 ]] (5) = E [( α0 + α1µ 2 t−1 ) E [ ν 2 t |Ωt−1 ]] (6) = E [ α0 + α1µ 2 t−1 ] = E [ α0 + α1E [ µ 2 t−1|Ωt−2 ]] (7) = α0 + α1E [ α0 + α1µ 2 t−2 ] (8) = · · · = α0 ( 1 + α1 + α 2 1 + · · ·+ α t−1 1 ) + α t 1E [ µ 2 0 ] (9) As t→∞, the unconditional variance converges if α1 < 1 to: E ( µ2t ) = α0 1−α1 . −→ Unconditionally, the process µt is homoskedastic. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Properties of ARCH(1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • It can be alternatively expressed as: µt = σtvt, vt ∼ iidN(0, 1), where vt = µt/σt is the standardised shock. • When model is correct,v2t should have no autocorrelation • The unconditional distribution of µt is NOT normal, with heavy tails (kurtosis > 3).

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

ML Estimation

MLE of ARCH(1)

• An example: AR(1)−ARCH(1)
yt = c+ φ1yt−1 + µt, µt|Ωt−1 ∼ N(0, σ2t ), (10)
σ2t = α0 + α1µ

2
t−1, (11)

α0 > 0, 0 ≤ α1 < 1. (12) • Likelihood of {y1, y2, · · · , yT−1, yT }: L(Θ) = f (yT |ΩT−1) f (yT−1|ΩT−2) · · · f (y2|Ω1) f(y1) f (yt|Ωt−1) = (2πσ2t )−1/2exp{− (yt − c− φ1yt−1)2 2σ2t }. (13) • ML Estimator maximises the Log likelihood function lnL(Θ) = −T 2 ln(2π)− 1 2 T∑ t=2 [ ln(σ2t ) + (yt − c− φ1yt−1)2 σ2t ] . Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation MLE of ARCH(1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • ML estimators are generally consistent with an asymptotic normal distribution. • The above holds even when the conditional normality µt|ωt−1 ∼ N(0, σ2t ) is incorrectly assumed, as long as the conditional mean and conditional variance are correctly specified. • With robust quasi ML standard errors, inference is standard. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation Example ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material eg. NYSE composite return: AR(1)-ARCH(5) Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility – ML Estimation of ARCH(1) eg. NYSE composite return: AR(1)-ARCH(5) School of Economics, UNSW Slides-7, Financial Econometrics 19 0 100 200 300 400 500 600 -6 -4 -2 0 2 4 Series: E Sample 1 1931 Observations 1929 Mean -0.033709 Median -0.037415 Maximum 5.392314 Minimum -6.773783 Std. Dev. 1.004962 Skewness -0.198864 Kurtosis 7.131158 Jarque-Bera 1384.431 Probability 0.000000 0 40 80 120 160 200 240 280 -5.00 -3.75 -2.50 -1.25 0.00 1.25 2.50 Series: V Sample 1 1931 Observations 1929 Mean -0.048030 Median -0.043219 Maximum 3.427925 Minimum -5.188789 Std. Dev. 0.999109 Skewness -0.413633 Kurtosis 4.558816 Jarque-Bera 250.3099 Probability 0.000000 Type in Eviews upper panel: arch(5,0,h) rc c ar(1) Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation Example ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material eg. NYSE composite return: AR(1)-ARCH(5) • Squared residuals (E2) of AR(1) have strong autocorrelation. Squared standardised residuals (V2) are not autocorrelated • Residuals (E) of AR(1) have larger kurtosis. Standardised residuals (V) larger negative skewness. • Normality is rejected for both E and V. Two essential checks for the ’adequacy’ of a model I Adequate mean equation: E (residuals) has no autocorrelation; I Adequate variance equation: V2 has no autocorrelation Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Comments and limitations of ARCH ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Advantages of ARCH • It is able to capture ’clustering’ in return series or the autocorrelation in squared returns. • It facilitates volatility forecasting. • It explains, partially, non-normality in return series. Limitations of ARCH I In ARCH(q), the q may be selected by AIC, SIC or LR test. The correct value of q might be very large. The model might not be parsimonious. (eg. ARCH(1) would not work for the composite return) I The conditional variance σ2t cannot be negative: Requires non-negativity constraints on the coefficients. Sufficient (but not necessary) condition is: αi ≥ 0 for all i = 0, 1, 2, · · · q. Especially for large values of q this might be violated Slides-09 UNSW Slides-08 Modeling Long Run relationship Dr. Rachida Ouysse School of Economics UNSW ECON3206 Lecture Plan • Long-run relationship: co-movement in trending time series • Cointegration and common trend • Interest rate and inflation • Long and short term interest rates • Regression with I(1) series under cointegration and dynamic OLS • Spurious regression • Test for cointegration • Error correction models • Information & price discovery ECON3206 Long-run relationships Long-run relationships • Co-movement among time series eg. US zero coupon rates: 3-month vs 9-month Topic 4. Modelling Long-run Relationships • Long-run relationships – Co-movement among time series eg. US zero coupon rates: 3-month vs 9-month (1946:12-1987:2, 483 monthly observations) Both appear non-stationary but move together. School of Economics, UNSW Slides-06, Financial Econometrics 3 0 4 8 12 16 20 1950 1955 1960 1965 1970 1975 1980 1985 3-month coupon rate 9-month coupon rate -4 -3 -2 -1 0 1 2 3 4 1950 1955 1960 1965 1970 1975 1980 1985 (3 Month Rate - 9 Month Rate) (1946:12-1987:2, 483 monthly observations) Both appear non-stationary but move together. ECON3206 Long-run relationships Long-run relationships • Co-movement among time series eg. US zero coupon rates: 3-month vs 9-month Topic 4. Modelling Long-run Relationships • Long-run relationships – Co-movement among time series eg. NYSE log Composite & Industrial indices Both are non-stationary but move together. – How to characterise such “co-movement”? School of Economics, UNSW Slides-06, Financial Econometrics 4 5.4 5.6 5.8 6.0 6.2 6.4 6.6 6.8 250 500 750 1000 1250 1500 1750 LCOMP LINDU .10 .15 .20 .25 .30 .35 .40 250 500 750 1000 1250 1500 1750 (LINDU - LCOMP) ECON3206 Long-run relationships Long-run relationships • Co-movement among time series • Two (or more) time series move together over time and never depart for long. • The time series are individually I(1) and vary a great deal. But their long-run relationship appears stable over time. • There must be a common trend that drives both time series. • Important to exploit long-run relationships in finance eg. pairs-trading; rational bubbles; bi-listed stocks • We introduce basic facts on modelling long-run relationships, mainly with bi-variate cases. ECON3206 Long-run relationships pairs=trading 12/09/2018 1:13 pmPairs Trading: Introduction | Investopedia Page 1 of 3https://www.investopedia.com/university/guide-pairs-trading/ Guide to Pairs Trading The origin of Pairs Trading 1. Pairs Trading: Introduction 2. Pairs Trading: Market Neutral Investing 3. Pairs Trading: Correlation 4. Arbitrage and Pairs Trading 5. Fundamental and Technical Analysis for Pairs Trading Pairs trading is a market-neutral trading strategy that matches a long position with a short position in a pair of highly correlated instruments such as two stocks, exchange-traded funds (ETFs), currencies, commodities or options. Pairs traders wait for weakness in the correlation and then go long the under-performer while simultaneously short selling the over-performer, closing the positions as the relationship returns to statistical norms. The strategy’s profit is derived from the di!erence in price change between the two instruments, rather than from the direction each moves. Therefore, a profit can be realized if the long position goes up more than the short, or the short position goes down more than the long (in a perfect situation, the long position rises and the short position falls, but that’s not a requirement for making a profit). It’s possible for pairs traders to profit during a variety of market conditions, including periods when the market goes up, down or sideways – and during periods of either low or high volatility. (See also: 4 Factors That shape Market Trends.) By Jean Folger | Updated February 21, 2018 — 8:30 AM EST SHARE ECON3206 Spurious Regression The spurious regression problem I General result: a linear combination zt of a set of variables xit, with order xit ∼ I(1), will have an order of integration equal to 1, if there exists a linear combination, zt = ∑k i=1 αixit ∼ I(0) I Example: consider two series yt and xt, with yt ∼ I(1); xt ∼ I(1) and a linear combination zt thereof, i.e. zt = α0 + α1yt + α2xt ∼ I(0) ECON3206 Spurious Regression The spurious regression problem • Example: NYSE log Composite index vs Simulated RW Symptom: the residual looks like RW Topic 4. Modelling Long-run Relationships • Spurious regression – What if 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated? • If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated, the linear regression 𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡 is cal School of Economics, UNSW Slides-06, Financial Econometrics 10 -.6 -.4 -.2 .0 .2 .4 5.2 5.6 6.0 6.4 6.8 250 500 750 1000 1250 1500 1750 Residual Actual Fitted 5.4 5.6 5.8 6.0 6.2 6.4 6.6 -2 0 2 4 6 8 250 500 750 1000 1250 1500 1750 LCOMP SIMUL Topic 4. Modelling Long-run Relationships • Spurious regression – What if 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated? • If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are not cointegrated, the linear regression 𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡 is called “spurious”. eg. NYSE log Composite index vs Simulated RW School of Economics, UNSW Slides-06, Financial Econometrics 10 -.6 -.4 -.2 .0 .2 .4 5.2 5.6 6.0 6.4 6.8 250 500 750 1000 1250 1500 1750 Residual Actual Fitted 5.4 5.6 5.8 6.0 6.2 6.4 6.6 -2 0 2 4 6 8 250 500 750 1000 1250 1500 1750 LCOMP SIMUL ECON3206 Spurious Regression Examples of Spurious Regression Multivariate Time Series Analysis: Cointegration analysis Basic concepts Spurious regression Examples of spurious regression I Egyptian infant mortality rate (Yt), 1971-1990, annual data, on gross aggregate income of American farmers (It) and total Honduran money supply (Mt) Ŷt = 179.9 (16.63) − 0.30It (−2.32) − 0.04Mt (−4.26) R2 = 0.918; F = 95.17; DW = 0.475 I US export index (Yt), 1960-1990, annual data, on Australian males life expectancy (Xt) Ŷt = −2943 (16.70) + 45.80Xt (17.76) R2 = 0.916; F = 315.2; DW = 0.360 ECON3206 Spurious Regression The spurious regression problem yt = β1 + β2xt + �t • The spurious regression problem is characterized by • Highly significant value for β2 • Fairly high R2 • Reason: distribution of the conventional test statistics are very different from conventional case (stationary data) • OLS estimator does not converge in probability as T →∞ • t−stats do not have well-defined asymptotic distributions • Estimated stdv strongly underestimates true stdv (b/c autocorrelation) • Sign something is wrong: • Highly autocorrelated residuals ECON3206 Spurious Regression Implication The spurious regression problem implies that when regressing non-stationary variables, the estimation results should not be taken too seriously!!! I Take first-differences of I(1) variables (GLS correction for autocorrelation) An important exception arises when the non-stationary series have a common stochastic trend: cointegration. I Don’t take first-differences - specification error! - advantage of I(1) variables (superconsistency) ECON3206 Cointegration Definition cointegration The k variables of the k × 1 vector xt = (x1t, x2t, · · · , xkt)′ are said to be cointegrated of order one, denoted as x1 ∼ CI(1) if 1 All variables in xt are integrated of the same order one, i.e. xit ∼ I(1), for all i 2 There exists at least one vector β = (β1, β2, · · · , βk)′ of coefficients, called the cointegrating vector, such that the linear combination x′tβ = (β1x1t + β2x2t + · · · + βkxkt) is integrated of a order zero, i.e. xt ∼ I(0) ECON3206 Cointegration Example In practice, xt ∼ CI(1) is most common. Consider for instance two variables, yt and xt , which are both I(1). If the residuals �t of the regression yt = β1 + β2xt + �t are I(0), i.e. �t ∼ I(0), then yt and xt are said to be cointegrated of order CI(1) with cointegrating vector β = (1,−β1,−β2) as yt − β1 − β2xt = �t ∼ I(0) • eg. When (9monthRate − 3monthRate) is stationary, they are cointegrated with cointegrating vector β = [1,−1]. • eg. When (logIndustrial − 0.98 logComposite) is stationary, they are cointegrated with cointegrating vector β = [1,−0.98]. ECON3206 Cointegration Cointegration & common trend • Common trend eg. A model of interest rates (Fisher equation) • Short & long term interest rates (rst , rlt) are directly influenced by the inflation πt), subject to stationary shocks (� s t , � l t) : r s t = a s + πt + � s t , r l t = a l + πt + � l t • Both will be I(1) when the πt is I(1). Here πt acts as the common trend that represents the trend (non-stationary part) in both rst and r l t. • (rst , rlt) are cointegrated with β = [1,−1]′ because rst − rlt = as − al + �st − �lt is I(0). ECON3206 Cointegration Economic Interpretation If two (or more) series are linked to form an equilibrium relation yt = β1 + β2xt then even though the series themselves are non-stationary they will nevertheless move closely together over time, i.e. they have a common trend, such that deviations from the equilibrium �t = yt − (β1 + β2xt) are stationary. I The concept of cointegration indicates the existence of a long-run equilibrium to which an economic system converges over time and �t can be interpreted as the equilibrium error, i.e. the distance the system is away from the equilibrium at time t. As equilibrium errors should be temporary, �t should be stationary. ECON3206 Cointegration Economic Interpretation I The concept of spurious regression indicates that there is no long-run equilibrium relation between yt and xt as the error term �t is non-stationary, implying that deviations from the presumed relation between yt and xt are permanent such that this relation is not a long-run equilibrium relation. ECON3206 Cointegration Econometric implication I If non-stationary variables are cointegrated, regression analysis imparts meaningful information about the long-run relationship between the variables. In fact, it can be shown that in this case, the OLS estimator β̂ is even a super consistent estimator for β, i.e. β̂ converges to β at a much faster rate than with conventional asymptotics (i.e. for stationary variables). I If non-stationary variables are not cointegrated, regression results are not meaningful, i.e. spurious regression problem. ECON3206 Error-Correction Mechanism Cointegration and Error-Correction Mechanisms The existence of a long-run equilibrium relationship also has its implications for the short-run behaviour of the I(1) variables • The Granger representation theorem states that if a set of variables is cointegrated, there has to be a mechanism that drives the variables back to their long-run equilibrium relationship after the equilibrium has been disturbed by a shock • This mechanism is called an error-correction model ECON3206 Error-Correction Mechanism Example of an error-correction model Consider two variables yt and xt which are cointegrated with cointegrating vector β = (1,−β1,−β2). A simple error-correction model (ECM) is given by ∆yt = γ1∆xt − α(yt−1 − β1 − β2xt−1) + µt (1) = γ1∆xt − α�t−1 + µt (2) The ECM incorporates both short-run and long-run effects I The long-run equilibrium is obtained by imposing the ’no change’ condition ∆yt = ∆xt = µt = 0 and solve for yt yt = β1 + β2xt Thus, the long-run impact of xt on yt is given by β2. I The contemporaneous impact of xt on yt is given by γ1. ECON3206 Error-Correction Mechanism Error correction mechanism I The term −α�t−1 captures the error-correction mechanism. If yt and xt are cointegrated, the Granger representation theorem implies that α > 0.

I When yt is below its equilibrium value implied by xt, �t < 0 such that yt increases back the equilibrium I When yt is above its equilibrium value implied by xt , �t > 0 such
that yt decreases back to the equilibrium

Note that α measures the speed of adjustment towards the
equilibrium. The smaller α (i.e. the closer to zero), the lower this
speed of adjustment.
• When yt and xt are cointegrated, �t is the deviation from their

long-run equilibrium.
• yt+1 and xt+1 must move toward eliminating the deviation, or

correcting the cointegation error �t.
• Hence, �t is useful for predicting ∆yt+1 and ∆xt+1 and the models

for ∆yt+1 and ∆xt+1 should include �t as an explanatory variable.

ECON3206

Vector Error Correction Model

Vector Error correction VEC

• Vector error correction (VEC) model:

�t−1 = yt−1 − β0 − β1xt−1 (3)
∆xt = c1 + α1�t−1 + φ11∆xt−1 + φ12∆yt−1 + u1t (4)

∆yt = c2 + α2�t−1 + φ21∆xt−1 + φ22∆yt−1 + u2t (5)

• Eg, when α1 = 0, the adjustment toward equilibrium is all done by
yt and the common trend is xt.

Call α1 and α2 adjustment coefficients.

What happens when both α1 and α2 are zero?

ECON3206

Vector Error Correction Model

Price discovery in parallel markets

How information is incorporated into prices?

• Examples (usually require intraday price series)
• Bi-listed stock: which market sets the price?
• Spot & futures prices: does spot follows futures?

• For two log prices, yt and xt, on the same asset, the rule-of-one-price
dictates that �t = yt − xt can only fluctuate around zero.

• Hence, yt and xt are cointegrated with [1,−1] being the
cointegrating vector. The error correction model is applicable.

• The relative magnitudes of α1 and α2 can tell us to what extent xt
acts as price setter, sx =

|α1|
|α1|+|α2|

ECON3206

Vector Error Correction Model

Example: Price discovery in parallel marketsTopic 4. Modelling Long-run Relationships
• Price discovery in parallel markets

eg. SP500 spot & futures indices: VEC
(20100104-20120810, 656 obs.)
– The adjustment coefficients:
𝛼𝛼futures is insignificant (t-stat = 0.46).
𝛼𝛼spot is significant (t-stat = -2.34).
– Futures appears to be the price-setter.

School of Economics, UNSW Slides-06, Financial Econometrics 21

690

700

710

720

730

-1.0

-0.5

0.0

0.5

1.0

1.5

100 200 300 400 500 600

LSPT LFUT DLSF

ECON3206

Vector Error Correction Model

Example: US and Canadian 10-years bond yeilds
Topic 4. Modelling Long-run Relationships

• Error correction & cointegration
eg. US and Canadian 10-year bond yields
Error correction model:
dca = ca – ca(-1), dus = us – us(-1),
e = ca – b0 – b1∙us .

School of Economics, UNSW Slides-06, Financial Econometrics 18

Correction is
done by CA, not US.

US acts as
the common trend.

ECON3206

Super consistency

Properties of OLS : Super consistency

Consider two time series yt and xt which are both I(1). Estimating the
static equation

yt = β1 + β2xt + �t

using OLS yields super consistent estimates of the long-run parameters
β1 and β2 when �t is I(0).

I Super consistency means that the OLS estimator converges to the true
population parameters at a much faster rate than with stationary variables

I This result arises as OLS picks the coefficients β̂ such that the variance of
the estimated residuals �̂t is as small as possible. As setting β̂ 6= β implies
that �t ∼ I(1) such that its variance becomes infinitely large when
T →∞, OLS is very efficient in picking the correct β

I The super consistency property of the OLS estimator implies that in
estimating the long-run relation between cointegrated variables, dynamics
and endogeneity issues can be ignored asymptotically

ECON3206

Super consistency

Properties of OLS : Super consistency

Topic 4. Modelling Long-run Relationships

• Cointegration & common trend
– Cointegration regression

• If two I(1) series 𝑦𝑦𝑡𝑡 and 𝑥𝑥𝑡𝑡 are cointegrated, they may
be fitted in the linear regression

𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + 𝜀𝜀𝑡𝑡 , 𝜀𝜀𝑡𝑡 being stationary
where [1,−𝛽𝛽1] is the cointegrating vector.
• As long as 𝜀𝜀𝑡𝑡 is stationary, the OLS estimator of 𝛽𝛽1 is

consistent, but generally has a non-standard asymptotic
distribution.

• To make valid inference about 𝛽𝛽1, the “dynamic” OLS
estimator of 𝛽𝛽1 from

𝑦𝑦𝑡𝑡 = 𝛽𝛽0 + 𝛽𝛽1𝑥𝑥𝑡𝑡 + ∑ 𝜓𝜓𝑗𝑗Δ𝑥𝑥𝑡𝑡−𝑗𝑗
𝑞𝑞
𝑗𝑗=−𝑞𝑞 + 𝜀𝜀𝑡𝑡 .

See Saikkonen (1992, ET) or Stock & Watson (1993, Etrca).

School of Economics, UNSW Slides-06, Financial Econometrics 8 ECON3206

Super consistency

Properties of OLS : Super consistency

• The addition of leads and lags removes the deleterious effects that
short-run dynamics of the equilibrium process �t have on the
estimate of the cointegrating vector

• The DOLS estimator is consistent, asymptotically normally
distributed, and efficient.

• Asymptotically valid standard errors for the individual elements of
the estimated cointegration vector are given by their corresponding
HAC (e.g., Newey-West) standard errors.

ECON3206

Testing for cointegration

Consider two time series yt and xt.
Suppose we want to estimate the following equation:

yt = β1 + β2xt + �t

Prior to estimation, test the variables for their order of integration

1 If both are I(0): standard regression analysis is valid

2 If they are integrated of a different order, e.g. yt is I(1) and xt is
I(0): there can be no (long-run) relation between these two variables

3 If both are I(1): use cointegration analysis

Note however that there is almost never certainty about the true order of
integration

ECON3206

The Engle-Granger two-step approach

A popular methodology to test for cointegration and to analyse
cointegrating relationships is the so-called Engle-Granger two-step
approach:

1 Estimate the static model and test for cointegration

2 Estimate an ECM to analyse the short-run dynamics

ECON3206

The Engle-Granger two-step approach

Multivariate Time Series Analysis: Cointegration analysis

Testing for cointegration

The Engle-Granger two-step approach

Step 1: estimate static model and test for cointegration

Estimate the model in levels using OLS. Two cases can be
distinguished

1. The regression results are spurious if εt ∼ I (1)
2. OLS is super consistent if εt ∼ I (0)

After estimating a model including non-stationary variables, it is
therefore very important to test the order of integration of the
estimated residuals ε̂t . We consider two alternative tests:

1. The cointegrating regression Durbin-Watson (CRDW) test

2. ADF cointegration test

ECON3206

The Engle-Granger two-step approach

Multivariate Time Series Analysis: Cointegration analysis

Testing for cointegration

The Engle-Granger two-step approach

1. Cointegrating Regression Durbin-Watson (CRDW) test
Tests whether the residuals ε̂t are generated by a unit root
process:

ε̂t = ε̂t−1 + υt

against the alternative that ε̂t is generated by a stationary
AR(1) process:

ε̂t = ρε̂t + υt with |ρ| < 1 using the Durbin-Watson (DW) statistic. As DW ≈ 2(1− ρ̂) this test boils down to testing whether DW is significantly larger than zero. ECON3206 The Engle-Granger two-step approach Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration The Engle-Granger two-step approach I Formally: H0 : ε̂t ∼ I (1) corresponds to ρ = 1 or d = 0 H1 : ε̂t ∼ I (0) corresponds to ρ < 1 or d > 0

I The 5% critical values for the CRDW test are given by

Number of variables Number of observations
(incl. yt) 50 100 250

2 0.72 0.38 0.20
3 0.89 0.48 0.25
4 1.05 0.58 0.30
5 1.19 0.68 0.35

I Drawback: the CRDW test is only valid when εt follows an
AR(1) process as the DW statistic only checks for an AR(1)
pattern in the data.

ECON3206

The Engle-Granger two-step approach
Multivariate Time Series Analysis: Cointegration analysis

Testing for cointegration

The Engle-Granger two-step approach

2. ADF cointegration test
Tests for a unit root in the estimated residuals using the
standard DF specification

∆ε̂t = γε̂t−1 +
∑p−1

i=1
αi∆ε̂t−i + ωt

with H0 : γ = 0 → no cointegration
H1 : γ < 0 → cointegration Important notes: I Deterministic components (i.e. intercept and trend) can be included either in the cointegrating regression or in the ADF test (but not in both!) I The standard DF critical values are not valid! Reason: the OLS estimator ‘picks’β such that the residuals ε̂t have the lowest possible variance, i.e. making the residuals appear as stationary as possible even if there is no cointegration (i.e. εt is non-stationary). ECON3206 The Engle-Granger two-step approach Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration The Engle-Granger two-step approach Step 2: Estimate an ECM to analyse the short-run dynamics Upon finding cointegration, estimate an ECM A (L) ∆yt = δ + B (L) ∆xt + αε̂t−1 + C (L)µt where ε̂t−1 = yt−1 − β̂1 − β̂2xt−1. Since all variables are I (0), this can be done using OLS and statistical inference using standard t- and F -tests is possible. ECON3206 The Engle-Granger two-step approach Topic 4. Modelling Long-run Relationships • Test for cointegration – Engle-Granger cointegration test eg. US zero coupon rates: 3-month vs 9-month, Cointegrated (H0 rejected). eg. NYSE logFinance & logUtility, Not cointegrated (H0 not rejected). School of Economics, UNSW Slides-06, Financial Econometrics 13 ECON3206 Example Example: consumption, income and wealth in the US Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Example: consumption, income and wealth in the US I All data in natural logs, sample 1951:Q4-2005:Q4. I ADF tests show that all series are I(1) I Unit root in first differences is rejected I Unit root in levels is not rejected I The null hypothesis of no cointegration can be rejected at the 5% level of significance I The CRDW equals 0.31, which is just above the 5% critical value of ≈ 0.30. I The ADF test on the residuals of the static regression equals −4.19, which is below the 5% critical value −3.78( = −3.7429− 8.352 /217 − 13.41 / 2172 ) . I The error-correction term is significant and shows that consumption is only slowly converting to the long-run equilibrium implied by income and wealth, i.e. every quarter 5.7% of the equilibrium gap is closed. ECON3206 Example Example: consumption, income and wealth in the US Perform unit root tests on levels and first differences Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 19 : ADF Unit root test on first difference of consumption Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 20 : ADF Unit root test on consumption Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 21 : ADF Unit root test on first difference of income Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 22 : ADF Unit root test on income ECON3206 Example Example: consumption, income and wealth in the US Perform Static Regression Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 25 : Results static regression ECON3206 Example Example: consumption, income and wealth in the US Perform unit root test on residualsMultivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 27 : ADF Unit root test on the estimated residuals Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 26 : Residuals static regression ECON3206 Example Example: consumption, income and wealth in the US Estimate the Error Correction Model Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 28 : ECM model Multivariate Time Series Analysis: Cointegration analysis Testing for cointegration Example Figure 29 : Residuals ECM model ECON3206 ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Copyright©Copyright University of New South Wales 2020. All rights reserved. Course materials subject to Copyright UNSW Sydney owns copyright in these materials (unless stated otherwise). The material is subject to copyright under Australian law and overseas under international treaties. The mater ials are provided for use by enrolled UNSW students. The materials, or any part, may not be copied, shared or distributed, in print or digitally, outside the course without permission. Students may only copy a reasonable portion of the material for personal research or study or for criticism or review. Under no circumstances may these materials be copied or reproduced for sale or commercial purposes without prior written permission of UNSW Sydney. Statement on class recording To ensure the free and open discussion of ideas, students may not record, by any means, classroom lectures, discussion and/or activities without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the student?s own private use. WARNING: Your failure to comply with these conditions may lead to disciplinary action, and may give rise to a civil action or a criminal offence under the law. THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL. Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Financial Econometrics Slides-10: Modeling Return Volatility: Testing/Estimating/Forecasting ARCH and Introduction to GARCH Dr. Rachida Ouysse School of Economics1 1©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Lecture Plan ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • ARCH LM-test • Forecasting with ARCH • Generalised ARCH: why and how Formulation of GARCH: parameter restrictions • Properties of GARCH(1,1) • Mean, variance, ARMA(1,1) representation • ML estimation of GARCH • Forecasting with GARCH Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary ARCH-LM TEST LM test for ARCH effect ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary ARCH-LM TEST LM test for ARCH effect: Example ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material eg. NYSE composite return: 1 Estimate the model for mean (eg. AR(1)) and save the residual series µ̂t. 2 OLS auxilary regression: µ̂2t = γ0 + γ1µ̂ 2 t−1 + · · ·+ γqµ̂2t−q + errort Save the R2. (q depends on T and data frequency) 3 T ′ = T − q, with q = 5 reject when T ′R2 exceeds χ2(5) Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility – LM test for ARCH effect • Estimate the model for mean (eg. AR(1)) and save the residual series 𝑒𝑒𝑡𝑡. • OLS auxiliary regression 𝑒𝑒𝑡𝑡2 = 𝑐𝑐0 + 𝑐𝑐1𝑒𝑒𝑡𝑡−1 2 + ⋯+ +𝑐𝑐𝑞𝑞𝑒𝑒𝑡𝑡−𝑞𝑞2 + error𝑡𝑡 and save 𝑅𝑅𝑎𝑎2. (𝑞𝑞 depends on 𝑇𝑇 and data frequency) • Reject “H0: no ARCH” if 𝑇𝑇 − 𝑞𝑞 𝑅𝑅𝑎𝑎 2 exceeds 𝜒𝜒(𝑞𝑞) 2 cv. eg. NYSE composite return: LM test with q = 5 E V Performed on “V” to check the adequacy of variance equation School of Economics, UNSW Slides-7, Financial Econometrics 21 Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Forecasting with ARCH Models Forecasting with ARCH Models ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Using repeated substitutions, we can make multi-step forecasts for the return and its volatility • Example. AR(1)-ARCH(2) yt = c+ φ1yt−1 + µt, µt|Ωt−1 ∼ N(0, σ2t ) σ 2 t = α0 + α1µ 2 t−1 + α2µ 2 t−2 yt+1|t = c+ φ1yt, yt+2|t = c+ φ1yt+1|t, · · · σ 2 t+1|t = α0 + α1µ 2 t + α2µ 2 t−1, σ 2 t+2|t = α0 + α1σ 2 t+1|t + α2µ 2 t , σ 2 t+3|t = α0 + α1σ 2 t+2|t + α2σ 2 t+1|t, · · · Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Forecasting with ARCH Models Forecasting with ARCH models: Example ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility – Forecasting with ARCH models eg. NYSE composite return: AR(1)-ARCH(5) forecasts revert to unconditionals (mean reverting) School of Economics, UNSW Slides-7, Financial Econometrics 23 -8 -6 -4 -2 0 2 4 6 250 500 750 1000 1250 1500 1750 RC SIGMA -6 -4 -2 0 2 4 6 1870 1880 1890 1900 1910 1920 1930 R RF RF_LO RF_UP 0 2 4 6 8 10 12 1870 1880 1890 1900 1910 1920 1930 VF SIGMA2 Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Forecasting with ARCH Models Remember the limitations of ARCH! ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Advantages of ARCH • It is able to capture ’clustering’ in return series or the autocorrelation in squared returns. • It facilitates volatility forecasting. • It explains, partially, non-normality in return series. Limitations of ARCH I In ARCH(q), the q may be selected by AIC, SIC or LR test. The correct value of q might be very large. The model might not be parsimonious. (eg. ARCH(1) would not work for the composite return) I The conditional variance σ2t cannot be negative: Requires non-negativity constraints on the coefficients. Sufficient (but not necessary) condition is: αi ≥ 0 for all i = 0, 1, 2, · · · q. Especially for large values of q this might be violated Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary GARCH Models: Introduction ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Generalised ARCH (GARCH) models allow the conditional variance to depend upon previous own lags. • Let µt be the error term or shock in a model. ARCH(q): Var(µt|Ωt−1) = σ2t , σ 2 t = α0 + α1µ 2 t−1 + α2µ 2 t−2 + · · ·+ αqµ 2 t−q, is not parsimonious as a large q is often required. • If σ2t−1 is a summary of volatility info in Ωt−2, then Ωt−1 = {µt−1, µt−2, µt−3, · · · } = {µt−1,Ωt−2} ≈ {µt−1, σ2t−1} (volatility wise!) • This leads to the GARCH(1,1) model: σ 2 t = α0 + α1µ 2 t−1 + β1σ 2 t−1 Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary GARCH: Introduction ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • More generally, GARCH(p, q) model Var(µt|Ωt−1) = σ2t , σ 2 t = α0 + α1µ 2 t−1 + · · ·+ αqµ 2 t−q + β1σ 2 t−1 + · · ·+ βpσ 2 t−p, where the parameters should satisfy: (1) Positivity constraint: α0 > 0, αi ≥ 0, βj ≥ 0 for all i = 1, · · · , q and
j = 1, · · · , p

(2) Finite Variance
∑q
i=1

αi +
∑p
j=1

βj < 1 • In practice, the models for asset returns rarely go beyond GARCH(1,1). Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material The generalisation implied by GARCH can be seen from backward iterating the GARCH(1,1) model: σ 2 t = α0 1− β1 + α1 ∞∑ j=1 β j−1 1 µ 2 t−j . This shows that the GARCH model is an ARCH(∞) with geometrically declining coefficients (for |β1| < 1). Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility GARCH models Alternatively, if we define the surprise in the squared innovations as ωt = µ 2 t − σ2t , the GARCH(1,1) model can be rewritten as µ2t − ωt = α0 + α1µ2t−1 + β1 ( µ2t−1 − ωt−1 ) µ2t = α0 + (α1 + β1)µ 2 t−1 + ωt − β1ωt−1 which shows that the squared errors follow an ARMA(1,1) model. As the root of the autoregressive part is α1 + β1, the squared residuals are stationary provided |α1 + β1| < 1. Under stationarity, E ( µ2t ) = E ( µ2t−1 ) = E ( σ2t−1 ) = σ2, the unconditional variance of µt is given by σ2 = α0 + α1σ 2 + β1σ 2 = α0 1− (α1 + β1) Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility GARCH models Two general cases can be distinguished I α1 + β1 < 1 → the unconditional variance is defined, i.e. finite I α1 + β1 ≥ 1 → the unconditional variance is not defined, i.e. infinite The latter case is denoted non-stationarity in variance I Variance does not converge to an unconditional mean I The special case where α1 + β1 = 1 is known as a unit root in variance or integrated GARCH (IGARCH) Slides-10 UNSW ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary Properties of GARCH(1,1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • GARCH(1,1): µt|Ωt−1 ∼ N(0, σ2t ), σ 2 t = α0 + α1µ 2 t−1 + β1σ 2 t−1 α0 > 0, α1 ≥ 0, β1 ≥ 0, α1 + β1 < 1 • Its conditional variance is time varying: E(µt|Ωt−1) = 0, Var(µt|Ωt−1) = σ2t , CI(95%) = E(yt+1|Ωt−1) + 2σt • µt is a White Noise: E(µt) = 0, Var(µt) = α01−(α1+β1) , Cov(µt, µt−j) = 0 • But it is NOT an independent WN or iid WN. It is NOT unconditionally Normally distributed: kurt(µt) > 3

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Properties of GARCH(1,1)

• GARCH(1,1) can be expressed in terms of standardised shocks νt:
µt = σtνt and νt ∼ iid N(0, 1)
• When model is correct, ν2t should have no autocorrelation.

Advantages of the GARCH model (compared to ARCH)

I Avoids overfitting, i.e. a higher order ARCH model may have a more
parsimonious GARCH representation

I Due to less estimated parameters, violations of the non-negativity
constraint are less likely

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

For instance, estimate the following AR(1)-GARCH(1,1) model

yt = µ+ φyt−1 + µt
µt = νtσt νt ∼ N (0, 1)
σ2t = α0 + α1µ

2
t−1 + β1σ

2
t−1

OLS is inappropriate

I OLS minimises the RSS,
∑
µ̂2t =

∑(
yt − µ̂− φ̂yt−1

)2
,

which is a function of the parameters in the conditional mean
equation only and not in the conditional variance equation

I In fact, OLS assumes that the residuals are homoscedastic,
i.e. all slope coefficients in the conditional variance equation
are set to zero

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

Maximum Likelihood

I Make assumptions about conditional distribution of µt , e.g.

νt ∼ N (0, 1) such that µt ∼ N
(
0, σ2t

)

This means that conditional on information available at t − 1,
µt is normally distributed with mean zero and variance σt
with the latter being known at time t − 1. Note that this does
not imply that the unconditional distribution of µt is
normal, as σt becomes a random variable if we do not
condition on all information available on t − 1.

I The conditional distribution of yt is then also normal, given by

f (yt |yt−1, . . . , µt−1, . . .) =
1√

2πσ2t
exp

(
−1

µ2t
σ2t

)

with µt = yt − µ− φyt−1 and σ2t = α0 + α1µ2t−1 + β1σ2t−1.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

I The loglikelihood function is given by the sum over all t of
the log of the conditional distribution of yt

L = −T
2

log (2π)− T
2

T∑

t=1

log
(
σ2t
)
− 1

T∑

t=1

µ2t
σ2t

I The ML estimator is obtained by maximising the loglikelihood
with respect to the unknown parameters (µ, φ, α0, α1, β1).

I Analytical solution not possible: use numerical procedures
I These algorithms ‘search’ over the parameter space, from an

initial guess, until a maximum for the loglikelihood function is
found

I Potential problem: the loglikelihood function may have several
local maxima such that alternative initial guesses may yield
different results.

I In practice: use linear regression to get initial estimates of the
parameters in the conditional mean equation and choose some
(alternative) parameter value for the parameters in the
conditional variance equation 6= 0.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

Figure 6: The problem of local maxima

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

GARCH(1,1) Estimation

Extra topics MBF: Modelling volatility

Estimating GARCH models

I Fortunately, first order conditions are, under some weak
assumptions, valid even when νt is not normally distributed.

I The parameter estimates are still consistent
I Adjustments have to be made to the standard errors, i.e. use

Bollerslev-Wooldridge variance-covariance matrix, also known
as Quasi Maximum Likelihood Estimation, which is robust
for non-normality.

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

• GARCH
– ML Estimation of GARCH(1,1)

eg. NYSE composite return

School of Economics, UNSW Slides-08, Financial Econometrics 8

100

150

200

250

-5.0 -2.5 0.0 2.5

Series: Standardized Residuals
Sample 3 1931
Observations 1929

Mean -0.048341
Median -0.039867
Maximum 2.850528
Minimum -6.601836
Std. Dev. 0.996820
Skewness -0.547486
Kurtosis 4.973199

Jarque-Bera 409.3080
Probability 0.000000

q = 5

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

• GARCH
– ML Estimation of GARCH(1,1)

eg. NYSE composite return (continued)
Large 𝛽𝛽1 estimate: about 0.9
Small 𝛼𝛼1 estimate: about 0.1
𝛼𝛼1 + 𝛽𝛽1 estimate: very close to 1

GARCH(1,1) is preferred by AIC and SIC.

School of Economics, UNSW Slides-08, Financial Econometrics 9

AIC SIC

AR(1)-ARCH(5) 2.664 2.687

AR(1)-GARCH(1,1) 2.622 2.636

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Example 1

Example 1: GARCH(1,1) Estimation

Topic 5. Modelling Return Volatility: GARCH

• GARCH
– ML Estimation of GARCH(1,1)

eg. NYSE composite return (continued)
GARCH(1,1) 𝜎𝜎𝑡𝑡 plot is smoother than ARCH(5).
Large 𝛽𝛽1 estimate implies persistence:
𝜎𝜎𝑡𝑡 tends to continue at the current level.
𝜎𝜎𝑡𝑡2 = 𝛼𝛼0 + 𝛼𝛼1𝜀𝜀𝑡𝑡−1

2 + 𝛽𝛽1𝜎𝜎𝑡𝑡−1
2

School of Economics, UNSW Slides-08, Financial Econometrics 10

250 500 750 1000 1250 1500 1750

EA SGM

GARCH(1,1)

250 500 750 1000 1250 1500 1750

EA SGM

ARCH(5)

Slides-10 UNSW

ARCH: Test and Forecasting GARCH Models Properties of GARCH GARCH Estimation Summary

Summary facts about GARCH models

• GARCH(1,1) is usually preferred to ARCH or higher order GARCH,
because of its parsimony.

• Usually, GARCH β1 estimate is about 0.9 or more and α1 + β1 estimate is
very close to 1, for daily returns.

• Standardised residuals are usually non-normal, with negative skewness and
excessive kurtosis.

• GARCH(1,1) is able to capture clustering in returns but unable to account
for

Asymmetry: negative returns tend to cause more volatility;

Non-normality; Structural change

• Coefficient restrictions are hard to impose in MLE

Slides-10 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

ials are provided for use by enrolled UNSW students. The materials, or any part, may not be copied, shared or distributed, in print or
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Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

ECON3206/5206 Financial Econometrics
Slides-11: GARCH, VaR and Extensions

Dr. Rachida Ouysse
School of Economics1

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Lecture Plan

• Forecasting Volatility with GARCH
• Volatility and Risk: VaR
• Typical estimates of GARCH parameters

A measure of volatility persistence

• Integrated GARCH and EWMA

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

At first sight, forecasting the volatility in the error terms may not
seem very useful.
However, keep in mind that

var (yt |yt−1, yt−2, …) = var (µt |µt−1, µt−2, …)

Therefore, these models are very useful as they can add a model
for the volatility of a time series to traditional ARMA models.

I forecasting the volatility of stock returns is useful e.g. in
option pricing as this requires the expected volatility of the
underlying asset over de lifetime of the option as an input

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

Consider the following GARCH(1,1) model

yt = µ+ µt µt ∼ N
(
0, σ2t

)

σ2t = α0 + α1µ
2
t−1 + β1σ

2
t−1

Generate one-, two- and three-step-ahead forecasts for the
conditional variance of yt at time T .

I First update the equations for the conditional variance:

σ2T+1 = α0 + α1µ
2
T + β1σ

2
T

σ2T+2 = α0 + α1µ
2
T+1 + β1σ

2
T+1

σ2T+3 = α0 + α1µ
2
T+2 + β1σ

2
T+2

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

I Then let σ2f1,T be the one-step-ahead forecast for σ
2 at time T

σ2f1,T = ET
(
σ2T+1

)
= α0 + α1µ

2
T + β1σ

2
T

σ2f2,T = ET
(
σ2T+2

)
= α0 + α1ET

(
µ2T+1

)
+ β1σ

2f
1,T

= α0 + α1ET
(
σ2T+1

)
+ β1σ

2f
1,T

= α0 + (α1 + β1)σ
2f
1,T

= α0 + (α1 + β1)
(
α0 + α1µ

2
T + β1σ

2
T

)

σ2f3,T = ET
(
σ2T+3

)
= α0 + (α1 + β1)σ

2f
2,T

= α0 + α0 (α1 + β1) + (α1 + β1)
2
σ2f1,T

σ2fs,T = ET
(
σ2T+s

)
= α0

s−1∑

i=1

(α1 + β1)
i−1

+ (α1 + β1)
s−1

σ2f1,T

I For s →∞ σ2fs,T = α0 /(1− (α1 + β1)) if |α1 + β1| < 1 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Example 1:Forecasting with GARCH(1,1) Example: Forecasting volatility with GARCH(1,1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Forecasting volatility Example: forecasting mean and variance from an AR(1)-GARCH(1,1) for returns on the S&P500 index in a hold-out sample of 100 observations. EViews: in the Equation Window select Forecast Note that volatility is highly persistent! I forecasted volatility converges only slowly to the unconditional mean, which is equal to σ2 = 0.000000792 1− 0.068012− 0.923437 = 0.000093 I there is a great deal of predictability in volatility! Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Example 1:Forecasting with GARCH(1,1) Forecasting volatility with GARCH(1,1) ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Figure 20: Forecasting mean and volatility Extra topics MBF: Modelling volatility Forecasting volatility Figure 20: Forecasting mean and volatility Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Volatility and Risk: Risks of Large Losses ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Amaranth h/f ($6.5 billion in one week in September 2006) • Credit Lyonnais ($5.0 billion in 1990) • Long-Term Capital Management h/f ($4.6 billion in 1998) • Lehman Brothers ($3.9 billion in September 2008) • Orange County ($2 billion in 1994) • Barings ($1.4 billion in 1995) • Daiwa Bank ($1.1 billion in 1995) • Allied Irish Bank ($0.7 billion in 2002) • China Aviation Oil ($0.6 billion in 2004) Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Value at Risk VaR ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Risk managers/regulators are often interested in the following statement: ”I am 99% certain that my portfolio of assets will not lose more than $V over the next period and have sufficient reserves to cover losses lower than this level. ” period is often one day, but can be a month, quarter, year (1− α)100% VaR : VaR1−α = F−1(α)× Value of Investment VaR is the maximum portfolio loss in a given period (eg, 1 day) with a given probability (eg, 0.99). 99% Value at Risk VaR0.99 $ PORTFOLIO RETURNS 1% 1 ( ) ( ) 0.01 (0.01) VaR f y F VaR VaR F −∞ − = = = ∫ Pdf , f(x), of the next period portfolio returns School of Economics, UNSW Slides-08, Financial Econometrics 15 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Conditional Value at Risk Conditional Value at Risk ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Consider AR(1)−GARCH(1, 1) for the portfolio return yt yt = c+ φ1yt−1 + µt, where µt|Ωt−1 ∼ N(0, σ2t ) σ2t = α0 + α1µ 2 t−1 + β1σ 2 t−1 � νt = µtσt = yt−yt|t−1 σt ∼ N(0, 1), where yt|t−1 = E(yt|Ωt−1) � P (νt < −2.326) = 0.01 = 1− 0.99 implies: P (yt < yt|t−1 − 2.326σt) = 0.01 � VaR0.99 = 1100 (yt|t−1 − 2.326σt)× Portfolio Value Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Conditional Value at Risk Example 1 ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: GARCH – Conditional value at risk (VaR) eg. NYSE composite return (continued) Portfolio valued at $1m at T = 2002-08-29. AR(1)-GARCH(1,1): 𝜎𝜎𝑇𝑇+1 =1.64196, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05132. VaR = 1 100 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.326𝜎𝜎𝑇𝑇+1 ×$1m = −$37,678 If using the sample mean, sample variance and normality, we find VaR = 1 100 [.0353 – 2.326(1.0062)] ×$1m = − $23,051. • But, normality is strongly rejected! School of Economics, UNSW Slides-08, Financial Econometrics 17 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Empirical Quantile VaR using Empirical Quantile ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material BUT normality is often rejected? � GARCH is able to account for clustering, such that the standardised shock (νt) can be viewed as iid. � To compute VaR, we only need the lower quantile of νt, which can be estimated by the empirical quantile of the standardised residuals. • Instead of using the N(0, 1) to find F−1(α), we need to use the distrubution of the the estimated standardised residuals νt • νt = µtσt = yt−yt|t−1 σt ∼ iid(0, 1) • P (νt < Q0.01) = 0.01 = 1− 0.99 implies P (yt < yt|t−1 −Q0.01σt) = 0.01 = 1− 0.99 VaR0.99 = 1 100 (yt|t−1 −Q0.01σt)× Portfolio Value Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Empirical Quantile Example 2 ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: GARCH – Conditional value at risk (VaR) eg. NYSE composite return (continued) Portfolio valued at $1m at T = 2002-08-29. AR(1)-GARCH(1,1): 𝜎𝜎𝑇𝑇+1 =1.64196, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05132. The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.873 VaR = 1 100 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.873𝜎𝜎𝑇𝑇+1 ×$1m = −$46,660 For ARCH(5): 𝜎𝜎𝑇𝑇+1 = 1.25322, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05037, 𝑞𝑞0.01 = −2.774, VaR = −$34,260 School of Economics, UNSW Slides-08, Financial Econometrics 20 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH A measure of persistence: half-life time ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Let ωt = µ2t − σ2t , then µ2t has an ARMA(1,1) representation:µ2t = α0 + (α1 + β1)µ 2 t−1 + ωt − β1ωt−1 • When the shocks are zero, ie, ω = 0 for all t, by substitution: µ 2 t = α0 [ 1 + · · ·+ (α1 + β1)t−1 ] + (α1 + β1) t µ 2 0 The impact of µ20 on µ 2 t is (α1 + β1) t, ceteris paribus. I Half-life time, tH , is defined as the number of periods required for the impact to be halved (α1 + β1) tHµ 2 0 = 1 2 µ 2 0, or tH = ln(1/2) ln(α1 + β1) eg. Composite return: α1 + β1 = 0.996, tH = 172.9 (days). Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Integrated GARCH: iGARCH ©Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material I What happens if α1 + β1 = 1? (known as iGARCH) I When α0 > 0, the unconditional variance is NOT finite and grows with t:
E(σ2t ) = α0t+ E(σ

2
0).

True because E(σ2t ) = α0 + (α1 + β1)E(σ
2
t−1) = α0 + E(σ

2
t−1)

We may write α0 = (1− α1 − β1)ω, where ω is the unconditional variance
of µt for α1 + β1 = 1.

I When α1 + β1 = 1 and α0 = 0, the conditional variance is an EWMA of
µ2t :

σ
2
t = (1− β1)µ

2
t−1 + β1σ

2
t−1

which, as an EWMA, is not mean-reverting.

eg. NYSE composite return: The above explains why GARCH is very slow to
revert to the average level

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Summary

• Forecasting with the GARCH follows similar recurssive structure as an
autoregressive model.

• The long run forcast of volatility converges to the unconditional variance
of the process.
• Application to VaR: measures the risk exposure and the maximum amount

of loss in dollar value forecast for the next period:
• The VaR involves the mean, and variance of the distribution of

returns/payoffs of investment,
• GARCH/ARCH models allow us to compute Conditional VaR,
• The unconditional mean and variance underestimates the VaR: conditional

VaR bigger in absolute value than Unconditional VaR (based on the mean
and sample variance)

• The normal distribution quantile leads to underestimating the VaR
compared to using the empirical quantile.

• The Half-life time measures the amount of persistence in the GARCH.

Slides-11 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Copyright©Copyright University of New South Wales 2020. All rights reserved.
Course materials subject to Copyright
UNSW Sydney owns copyright in these materials (unless stated otherwise). The material is subject to copyright under Australian law and
overseas under international treaties. The materials are provided for use by enrolled UNSW students. The materials, or any part, may
not be copied, shared or distributed, in print or digitally, outside the course without permission. Students may only copy a reasonable
portion of the material for personal research or study or for criticism or review. Under no circumstances may these materials be copied or
reproduced for sale or commercial purposes without prior written permission of UNSW Sydney.
Statement on class recording
To ensure the free and open discussion of ideas, students may not record, by any means, classroom lectures, discussion and/or activities
without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the
student?s own private use.
WARNING: Your failure to comply with these conditions may lead to disciplinary action, and may give rise to a civil action or a criminal
offence under the law.
THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL.

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Financial Econometrics
Slides-12: Further Issues for GARCH & Realized Volatility

Dr. Rachida Ouysse
School of Economics1

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Lecture Plan

• Asymmetric GARCH: Leverage effect
• Quantify the effect of standardised shock and avoid positivity restrictions:

EGARCH

• Measure the risk premium effect: GARCH-M model

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

GARCH Extensions

Asymmetric GARCH models

I Motivation: a negative shock to financial time series is likely to cause volatility to
rise by more than a positive shock of the same magnitude

I This is due to leverage effects, i.e. a fall in the value of a firm’s stock causes the
firm’s debt to equity ratio to rise, which makes the future stream of dividends
more volatile

I Standard GARCH models assume a symmetric response of volatility to positive
and negative shocks since by squaring the lagged error term the sign is lost:

In GARCH(1,1): σ2t = α0 + α1µ
2
t−1 + β1σ

2
t−1, the impact µt−1 on σ

2
t is

symmetric.

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Asymmetric GARCH: Motivation

• In equity markets, however, bad news (-ve shock) tends to cause more
volatility than good news (+ve shock), aka “asymmetric effect” or
“leverage effect”.

• Desirable to allow for asymmetric effect in GARCH
Topic 6. GARCH Extensions

• Asymmetric GARCH
– Introduction

• In GARCH(1,1):
𝜎𝜎𝑡𝑡2 = 𝛼𝛼0 + 𝛼𝛼1𝜀𝜀𝑡𝑡−1

2 + 𝛽𝛽1𝜎𝜎𝑡𝑡−1
2

the impact of 𝜀𝜀𝑡𝑡−1 on 𝜎𝜎𝑡𝑡2
is symmetric.

• In equity markets, however, bad news (-ve shock) tends

to cause more volatility than good news (+ve shock),
aka “asymmetric effect” or “leverage effect”.

• Desirable to allow for asymmetric effect in GARCH.

School of Economics, UNSW Slides-09, Financial Econometrics 3

-2 -1 0 1 2

0.
0

0.
1

0.
2

0.
3

0.
4

0.
5

News Impact Curve

t1


t2

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Asymmetric GARCH

I The Threshold GARCH (TGARCH) model. Glosten, Jagannathan and Runkle
[JF, 1993, 48(5), p1779-1801] propose a so-called TGARCH model (GJR) in
which the conditional variance equation is given by

σ2t = α0 + α1µ
2
t−1 + γµ

2
t−1It−1 + β1σ

2
t−1,

where It−1 is a dummy variable: It−1 = 1 if µt−1 < 0 and It−1 = 0 otherwise. If leverage effects are present γ > 0

– If µt−1 < 0, its effect on σ 2 t is α1 + γ If µt−1 ≥ 0, its effect on σ2t is α1 - The asymmetric effect exists if and only if γ > 0. Reduced back to GARCH if
γ = 0.

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: GJR/TGARCH

Example: estimates TGARCH/GJR model for returns for S&P500 index with
robust standard errors

Extra topics MBF: Modelling volatility

Extensions of GARCH models

Asymmetric GARCH models

Figure 17: Estimates AR(1)-TGARCH(1,1) model with robust standard
errors

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

News impact curve

I Graphical representation of the degree of asymmetry of volatility to positive and
negative shocks: the curves are drawn by using the estimated conditional
variance equation of the model under consideration.

I Calculate the values of the conditional variance σt over a range of past error
terms. Set the lagged conditional variance at the unconditional variance

I Example: News impact curve from estimates TGARCH model for returns for
S&P500 index

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: GJR/TGARCH

News impact curve from estimates TGARCH model

Extra topics MBF: Modelling volatility

Extensions of GARCH models

Asymmetric GARCH models

Figure 18: News impact curve from estimates TGARCH model

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Properties of the TGARCH/GJR model

I Unconditional variance:

µt|Ωt−1 ∼ N(0, σ2t ), σ
2
t = α0 + α1µ

2
t−1 + γµ

2
t−1It−1 + β1σ

2
t−1

• E(µ2t−1) = E
(
σ2t−1

)
•

E(It−1µ
2
t−1) = E

[
E
(
It−1µ

2
t−1|Ωt−2

)]
= E

[
1

2
E
(
µ2t−1|Ωt−2

)]
=

2
E
(
σ2t−1

)
• E(σ2t ) = α0 +

(
α1 + β1 +

1
2
γ
)
E
(
σ2t−1

)
• Stationarity: E

(
σ2t
)

= E
(
σ2t−1

)
= α0/

[
1− (α1 + β1 + 12γ)

]
• The above is valid when the conditional distribution of µt|Ωt−1 is

symmetric.

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Properties of the TGARCH/GJR model

Properties of TGARCH/GJR: persistence

• Let ωt = µ2t − σ
2
t , then µ

2
t has a representation:

µ2t = α0 + (α1 + β1 + γIt−1)µ
2
t−1 + ωt − β1ωt−1

• When the shocks are zero, ie, ωt = 0 for all t, by substitution,

µ2t ≈ Π
t−1
τ=0(α1 + β1 + γIτ )µ

2
0.

• E (Iτ |Ωτ−1) = 12 by symmetry.
• On average, the impact of µ20 on µ

2
t is

E
{

Πt−1τ=0(α1 + β1 + γIτ )
}

= E
{

(α1 + β1 + γE [It−1|Ωt−2]) Πt−2τ=0(α1 + β1 + γIτ )
}

= (α1 + β1 + γ/2)E
{

Πt−2τ=0(α1 + β1 + γIτ )
}

= · · · = (α1 + β1 + γ/2)t .

• Half-life time, tH , is defined as tH =
ln(1/2)

ln(α1+β1+γ/2)

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: TGARCH/GJR

Example.

eg. NYSE composite return: γ̂ = 0.1977, significant α̂1 negative, insignificant

Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR

eg. NYSE composite return:
𝛾𝛾� = 0.1977, significant
𝛼𝛼�1 negative, insignificant

School of Economics, UNSW Slides-09, Financial Econometrics 7

100

150

200

250

-5.0 -2.5 0.0 2.5

Series: Standardized Residuals
Sample 3 1931
Observations 1929

Mean -0.015283
Median -0.007326
Maximum 3.437926
Minimum -6.279817
Std. Dev. 0.999823
Skewness -0.465783
Kurtosis 4.617748

Jarque-Bera 280.1008
Probability 0.000000

Type in Eviews upper panel:
arch(1,1,h,thrsh=1) rc c ar(1)

Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR

eg. NYSE composite return:
𝛾𝛾� = 0.1977, significant
𝛼𝛼�1 negative, insignificant

School of Economics, UNSW Slides-09, Financial Econometrics 7

100

150

200

250

-5.0 -2.5 0.0 2.5

Series: Standardized Residuals
Sample 3 1931
Observations 1929

Mean -0.015283
Median -0.007326
Maximum 3.437926
Minimum -6.279817
Std. Dev. 0.999823
Skewness -0.465783
Kurtosis 4.617748

Jarque-Bera 280.1008
Probability 0.000000

Type in Eviews upper panel:
arch(1,1,h,thrsh=1) rc c ar(1)

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: TGARCH/GJR

Example: Test for asymmetry

eg. NYSE composite return: Asymmetric news impact. GJR is preferred by AIC/SIC.
Test for asymmetry,

LR = 2 (logLU − logLR) = 2 [(−2472.7)− (−2523.6)] = 97.8

Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR model

eg. NYSE composite return:
Asymmetric news impact.
GJR is preferred by AIC/SIC.

Test for asymmetry,
LR = 2(logLU − logLR) = 2[(−2474.7)−(−2523.6)] = 97.8
“H0: symmetry” is rejected.

School of Economics, UNSW Slides-09, Financial Econometrics 8

-2 -1 0 1 2

0.
0

0.
1

0.
2

0.
3

0.
4

0.
5

News Impact Curve

t1


t2

GARCH(1,1)
GJR

log Likelihood AIC SIC

AR(1)-GARCH(1,1) -2523.6 2.622 2.636

AR(1)-GJR -2474.7 2.572 2.589 Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR model

eg. NYSE composite return:
Asymmetric news impact.
GJR is preferred by AIC/SIC.

Test for asymmetry,
LR = 2(logLU − logLR) = 2[(−2474.7)−(−2523.6)] = 97.8
“H0: symmetry” is rejected.

School of Economics, UNSW Slides-09, Financial Econometrics 8

-2 -1 0 1 2

0.
0

0.
1

0.
2

0.
3

0.
4

0.
5

News Impact Curve

t1


t2

GARCH(1,1)
GJR

log Likelihood AIC SIC

AR(1)-GARCH(1,1) -2523.6 2.622 2.636

AR(1)-GJR -2474.7 2.572 2.589

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: TGARCH/GJR

Example: Forecasts

eg. NYSE composite return: forecasts
σ2t is still persistent, but less than GARCH(1,1).

α1 + β1 +
1
2
γ = 0.985, tH = 45.9 (days)

Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR model

eg. NYSE composite return: forecasts
𝜎𝜎𝑡𝑡2 is still persistent, but less than GARCH(1,1).

𝛼𝛼1 + 𝛽𝛽1 +
1
2
𝛾𝛾 = 0.985, 𝑡𝑡𝐻𝐻 = 45.9 (days)

School of Economics, UNSW Slides-09, Financial Econometrics 9

-6

-4

-2

1870 1880 1890 1900 1910 1920 1930

RF
FITTED
RF_LO

R
RF_UP

1870 1880 1890 1900 1910 1920 1930

VF SIGMA2

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Example: TGARCH/GJR

Example: VaR

eg. NYSE composite return: VaR
Portfolio valued at $1m at T = 2002− 08− 29.
AR(1)-GJR : σT+1 = 1.577, yT+1|T = 0.0185.
The 1% quantile of νt: Q0.01 = −2.678

V aR =
1

100

(
yT+1|T − 2.678σT+1

)
× $1m

Topic 6. GARCH Extensions

• Asymmetric GARCH
– GJR model

eg. NYSE composite return: VaR
Portfolio valued at $1m at T = 2002-08-29.
AR(1)-GJR: 𝜎𝜎𝑇𝑇+1 =1.577, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.0185.
The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.678

VaR =
1
100

𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.678𝜎𝜎𝑇𝑇+1 ×$1m = −$42,048

School of Economics, UNSW Slides-09, Financial Econometrics 10

𝜎𝜎𝑇𝑇+1 𝑦𝑦𝑇𝑇+1|𝑇𝑇 𝑞𝑞0.01 VaR

AR(1)-ARCH(5) 1.253 0.050 −2.774 −34260

AR(1)-GARCH(1,1) 1.642 0.051 −2.873 −46660

AR(1)-GJR 1.577 0.019 −2.678 −42048

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Exponential GARCH

I In GARCH, positivity restrictions on parameters make the ML estimation
difficult. Why not exponential?

I In GARCH, new info is incorporated via the term

α1µ
2
t−1 = α1ν

2
t−1σ

2
t−1

Why not separate the news ν2t−1 from non-news σ
2
t−1?

I EGARCH (Nelson, 1991, Econometrica, 59(2), p347-370)

• Exponential functional form: no need to worry about positivity;
• Separation of the effect of pure news;
• Incorporation of asymmetric effect.

Slides-12 UNSW

Asymmetric GARCH Exponential GARCH GARCH in mean

Exponential GARCH

• Model: µt|Ωt−1 ∼ N(0, σ2t ),

ln(σ2t ) = α0 + α1|νt−1|+ γνt−1 + β1ln(σ
2
t−1),

−1 < β1 < 1, νt−1 = µt−1/σt−1 - if νt−1 < 0, its effect on ln(σ 2 t ) is (α1 − γ)|νt−1|. if νt−1 ≥ 0, its effect on ln(σ2t ) is (α1 + γ)|νt−1|. - Negative shocks cause more volatility if and only if γ < 0. Reduced to symmetry if γ = 0. - σ2t = (σ 2 t−1) β1exp {α0 + α1|νt−1|+ γνt−1} Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Exponential GARCH: persistence • µt|Ωt−1 ∼ N(0, σ2t ), ln(σ2t ) = α0 + α1|νt−1|+ γνt−1 + β1ln(σ 2 t−1), −1 < β1 < 1, νt−1 = µt−1/σt−1 - By substitution, ln(σ2t ) ≈ β t−1 1 (α1|ν0|+ γν0) . Initial impact of the shock ν0 on ln(σ 2 1) : (α1|ν0|+ γν0) . - The time for the initial impact to halve: β tH−1 1 (α1|ν0|+ γν0) = 1 2 (α1|ν0|+ γν0) - Half-life time: tH = ln(1/2) ln(β1) + 1. Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH eg. NYSE composite return:AR(1)-EGARCH γ̂ = −0.1573, significant Topic 6. GARCH Extensions • Asymmetric GARCH – EGARCH eg. NYSE composite return: AR(1)-EGARCH 𝛾𝛾� = −0.1573, significant School of Economics, UNSW Slides-09, Financial Econometrics 14 Type in Eviews upper panel: arch(1,1,h,egarch) rc c ar(1) 0 40 80 120 160 200 240 -5.0 -2.5 0.0 2.5 Series: Standardized Residuals Sample 3 1931 Observations 1929 Mean -0.005748 Median 0.000146 Maximum 3.521650 Minimum -6.035894 Std. Dev. 1.002586 Skewness -0.377514 Kurtosis 4.385068 Jarque-Bera 200.0118 Probability 0.000000 Topic 6. GARCH Extensions • Asymmetric GARCH – EGARCH eg. NYSE composite return: AR(1)-EGARCH 𝛾𝛾� = −0.1573, significant School of Economics, UNSW Slides-09, Financial Econometrics 14 Type in Eviews upper panel: arch(1,1,h,egarch) rc c ar(1) 0 40 80 120 160 200 240 -5.0 -2.5 0.0 2.5 Series: Standardized Residuals Sample 3 1931 Observations 1929 Mean -0.005748 Median 0.000146 Maximum 3.521650 Minimum -6.035894 Std. Dev. 1.002586 Skewness -0.377514 Kurtosis 4.385068 Jarque-Bera 200.0118 Probability 0.000000 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH Topic 6. GARCH Extensions eg. NYSE composite return: Asymmetric news impact. �̂�𝛽1=0.9645, 𝑡𝑡𝐻𝐻 = 20.2 (days). Revert to mean quickly. School of Economics, UNSW Slides-09, Financial Econometrics 15 -2 -1 0 1 2 1. 0 1. 2 1. 4 1. 6 News Impact Curve: EGA vt1  t2 -4 -2 0 2 4 6 1870 1880 1890 1900 1910 1920 1930 RF FITTED RF_LO R RF_UP 1 2 3 4 5 6 7 8 1870 1880 1890 1900 1910 1920 1930 VF SIGMA2 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example: EGARCH eg. NYSE composite return:VaR. Portfolio valued at $1m at T = 2002− 08− 29. AR(1)-EGARCH:σT+1 = 1.482, yT+1|T = 0.0124 The 1% quantile of νt : Q0.01 = −2.678 V aR = 1 T ( yT+1|T − 2.678σT+1 ) × $1m = −39, 565 Topic 6. GARCH Extensions • Asymmetric GARCH – EGARCH eg. NYSE composite return: VaR Portfolio valued at $1m at T = 2002-08-29. AR(1)-EGARCH: 𝜎𝜎𝑇𝑇+1 =1.482, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.0124. The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.678 VaR = 1 100 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.678𝜎𝜎𝑇𝑇+1 ×$1m = −$39,565 School of Economics, UNSW Slides-09, Financial Econometrics 16 𝜎𝜎𝑇𝑇+1 𝑦𝑦𝑇𝑇+1|𝑇𝑇 𝑞𝑞0.01 VaR AR(1)-ARCH(5) 1.253 0.050 −2.774 −34260 AR(1)-GARCH(1,1) 1.642 0.051 −2.873 −46660 AR(1)-GJR 1.577 0.019 −2.678 −42048 AR(1)-EGARCH 1.482 0.012 −2.678 −39565 Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean GARCH in mean I Risk premium effect: investing in a riskier asset should be rewarded by a higher expected return. I In the context of a market index: investing in a riskier (more volatile) period should be rewarded by a higher expected return. � In AR(1)-GARCH, the mean equation yt = c+ φyt−1 + µt: implies the expected return = yt = c+ φyt−1, which is unrelated to the volatility or risk measure σt. � Motivation: investors should be rewarded for taking additional risk by obtaining a higher return I GARCH-M is used to account for the risk premium yt = c+ δσt−1 + µt µt|Ωt−1 ∼ N(0, σ2t ) σ2t = α0 + α1µ 2 t−1 + β1σ 2 t−1, where δ measures the risk premium effect. (See Lundblad (2007, JFE, p123-150) among others.) Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Example. eg. NYSE composite return No evidence for the “risk premium” effect in any of GARCH(1,1), TGARCH/GJR and EGARCH. Slides-12 UNSW Asymmetric GARCH Exponential GARCH GARCH in mean Summary • We completed the ARCH/GARCH extensions that capture: • Leverage effect/Asymmetry in the returns volatility • Positivity of the volatility and the impossibility constraints • Next... how about structural change in volatility? Slides-12 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Copyright c�Copyright University of New South Wales 2020. All rights reserved. Course materials subject to Copyright UNSW Sydney owns copyright in these materials (unless stated otherwise). The material is subject to copyright under Australian law and overseas under international treaties. The materials are provided for use by enrolled UNSW students. The materials, or any part, may not be copied, shared or distributed, in print or digitally, outside the course without permission. Students may only copy a reasonable portion of the material for personal research or study or for criticism or review. Under no circumstances may these materials be copied or reproduced for sale or commercial purposes without prior written permission of UNSW Sydney. Statement on class recording To ensure the free and open discussion of ideas, students may not record, by any means, classroom lectures, discussion and/or activities without the advance written permission of the instructor, and any such recording properly approved in advance can be used solely for the student?s own private use. WARNING: Your failure to comply with these conditions may lead to disciplinary action, and may give rise to a civil action or a criminal o↵ence under the law. THE ABOVE INFORMATION MUST NOT BE REMOVED FROM THIS MATERIAL. Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Financial Econometrics Slides-13: Remainaing Issues for GARCH and Alternative Models Dr. Rachida Ouysse School of Economics1 1 c�Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material. Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Lecture Plan c�Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Measure the risk premium e↵ect: GARCH-M model • Deal with structural break in volatility • Seasonality and distributional assumptions • Inclusion of other volatility measures • SV models Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary GARCH in mean I Risk premium e↵ect: investing in a riskier asset should be rewarded by a higher expected return. I In the context of a market index: investing in a riskier (more volatile) period should be rewarded by a higher expected return. ⌅ In AR(1)-GARCH, the mean equation yt = c+ �yt�1 + µt: implies the expected return = yt = c+ �yt�1, which is unrelated to the volatility or risk measure �t. ⌅ Motivation: investors should be rewarded for taking additional risk by obtaining a higher return I GARCH-M is used to account for the risk premium yt = c+ ��t�1 + µt µt|⌦t�1 ⇠ N(0,�2t ) � 2 t = ↵0 + ↵1µ 2 t�1 + �1� 2 t�1, where � measures the risk premium e↵ect. (See Lundblad (2007, JFE, p123-150) among others.) Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary eg. NYSE composite return No evidence for the “risk premium” e↵ect in any of GARCH(1,1), TGARCH/GJR and EGARCH. Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Structural break in Volatility • The composite return series appears to have a change in its volatility level. • The change is permanent. • If ignored, it can result in - over-estimating the persistence measure (↵1 + �1); - making the unconditional variance estimate inconsistent; - reducing the quality of forecasts, and VaR. • Important to detect and account for the structural break. Topic 6. GARCH Extensions • Structural break in volatility – Break in volatility • The composite return series appears to have a change in its volatility level. • The change is permanent. If ignored, it can result in – over-estimating the persistence measure (𝛼𝛼1 + 𝛽𝛽1); – making the unconditional variance estimate inconsistent; – reducing the quality of forecasts, and VaR. • Important to detect and account for the structural break. School of Economics, UNSW Slides-09, Financial Econometrics 19 1996 1998 2000 2002 -6 -4 -2 0 2 4 C om p R et ur n Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Test for structural break I As the variance is closely related to squared returns, we may check the break in an AR model for the squared returns, using the CUSUM test. I Model stability: Its structure changes over time?: • Recursive parameter estimates. Monitor changes in parameter estimates over time. {y1, · · · , y⌧}, {y1, · · · , y⌧+1}, · · · , {y1, · · · , yT } �̂(⌧), �̂(⌧ + 1), �̂(T ) • Recursive residuals: e⌧+1|⌧ = y⌧+1 �X⌧+1�̂(⌧) {y1, · · · , y⌧}, {y1, · · · , y⌧+1}, · · · , {y1, · · · , y⌧} e⌧+1|⌧ , e⌧+2|⌧+1, eT |T�1 • If the model is stable/correct: w⌧+1|⌧ = e⌧+1|⌧ se(e⌧+1|⌧ ) ⇠ N(0, 1) Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Model stability test: CUSUM CUSUM test (cumulative sum of standardised recursive residuals) CUSUMt = tX ⌧=K+1 w⌧+1|⌧ , t = K + 1,K + 2, · · · , T � 1 Reject stability if it goes outside the 95% bands. Eviews: View/Stability Tests/Recursive Estimates after a linear regression is estimated Test in volatility break: eg. AR(5) for the composite return squared: r 2 t = a0 + a1r 2 t�1 + · · ·+ a5r 2 t�5 + errort CUSUM test rejects the null hypothesis of no break. Topic 6. GARCH Extensions • Structural break in volatility – Test for a break in volatility • As the variance is closely related to squared returns, we may check the break in an AR model for the squared returns, using the CUSUM test. eg. AR(5) for the composite return squared: 𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1 2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5 2 + error𝑡𝑡 CUSUM test rejects the null hypothesis of no break. School of Economics, UNSW Slides-09, Financial Econometrics 20 -150 -100 -50 0 50 100 150 200 250 500 750 1000 1250 1500 1750 CUSUM 5% Significance Slides-13 UNSW GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary Structural break in volatility c�Copyright University of New South Wales 2020. All rights reserved. This copyright notice must not be removed from this material Find the break point Topic 6. GARCH Extensions • Structural break in volatility – Find the break point 1) Run the restricted regression (no break) and save the log likelihood as ℓ0. 2) Set 𝜏𝜏 = .15𝑇𝑇 (15% trim). Define the break dummy as 𝐵𝐵𝑡𝑡,𝜏𝜏, which is 0 for 𝑡𝑡 < 𝜏𝜏 and 1 for 𝑡𝑡 ≥ 𝜏𝜏. 3) Run the unrestricted regression 𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1 2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5 2 + 𝜓𝜓𝐵𝐵𝑡𝑡,𝜏𝜏 + error𝑡𝑡 and save the log likelihood ℓ𝜏𝜏 and 𝐿𝐿𝑅𝑅𝜏𝜏 = 2(ℓ𝜏𝜏 − ℓ0). 4) Set 𝜏𝜏 = 𝜏𝜏 + 1. If 𝜏𝜏 ≤ .85𝑇𝑇 (15% trim), go to 3). Otherwise go to 5). 5) The break point is estimated as the 𝜏𝜏 associated with the greatest 𝐿𝐿𝑅𝑅𝜏𝜏. It could be used as a test: the null of no break is rejected if max LR > cv.
The cv for 15% trim is 8.85, see Andrews (1993, Etrca, p821-856).

School of Economics, UNSW Slides-09, Financial Econometrics 21

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Structural break in volatility

Find the break point
eg. AR(5) for the composite return squared:

r2t = a0 + a1r
2
t�1 + · · ·+ a5r

2
t�5 + Bt,⌧ + errort

Topic 6. GARCH Extensions

• Structural break in volatility
– Find the break point

eg. AR(5) for the composite return squared:
𝑟𝑟𝑡𝑡2 = 𝑎𝑎0 + 𝑎𝑎1𝑟𝑟𝑡𝑡−1

2 +⋯+ 𝑎𝑎5𝑟𝑟𝑡𝑡−5
2 + 𝜓𝜓𝐵𝐵𝑡𝑡,𝜏𝜏 + error𝑡𝑡

The break point = 566.
AR(5) with the break passes the CUSUM test

School of Economics, UNSW Slides-09, Financial Econometrics 22

250 500 750 1000 1250 1500 1750

-120

-80

-40

120

750 1000 1250 1500 1750

CUSUM 5% Significance

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Break in volatility models

Incorporating breaks in volatility models

Incorporate a break in GARCH

I Once the break point is known and the break dummy Bt,⌧ is defined, the
break should be included in the conditional variance.

I GARCH(1,1) :
�2t = ↵0 + ↵1µ

2
t�1 + �1�

2
t�1 + Bt,⌧

I TGARCH/GJR :
�2t = ↵0 + ↵1µ

2
t�1 + �µ

2
t�1It�1 + �1�

2
t�1 + Bt,⌧

I TGARCH/GJR :
ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + Bt,⌧

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Seasonality: January E↵ect

⌅ Including a dummy in the variance equation,
• GARCH(1,1):
�2t = ↵0 + ↵1µ

2
t�1 + �1�

2
t�1 + �Jt

• GJR:
�2t = ↵0 + ↵1µ

2
t�1 + �µ

2
t�1It�1 + �1�

2
t�1 + �Jt

• EGARCH: ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + �Jt

where Jt is 1 if t is in January and 0 otherwise.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Non-normality

⌅ Normality alternatives
• In our examples, normality is usually rejected owing to

– heavy tails (Kurtosis> 3) and
– negative skewness

in the distribution of the standardised shock ⌫t.
• Alternative distributions may be assumed

– Student’s t: t(n)
with heavy tails but symmetry.
t(n) ⇡ N(0, 1) when the df n ! 1

– Mixture distributions: heavy tails and asymmetry.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Student-t

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Mixture of two normals

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Mixture of gaussians

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Incorporate other volatility measures

⌅ Range and implied volatility
• In addition to µt�1 or ⌫t�1, other volatility measures may have predictive

power for conditional variance.

Typically, the range (100ln(high/low)) and implied volatility (IV) are
informative measures of volatility.

• For EGARCH, we may specify
ln(�2t ) = ↵0 + ↵1|⌫t�1|+ �⌫t�1 + �1ln(�2t�1) + a1rngt�1 + a2ivt�1,
where the range (rng) and IV (iv) are included.

! It is good for 1-step ahead forecast. However, we need models for the
range and IV to do multi-step ahead forecasts.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Example: Range and Implied Volatility in EGARCH

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Stochastic volatility (SV) model: Latent Volatility

• In GARCH type models, the shock µt�1 or ⌫t�1 can be recovered
from the mean equation. The conditional variance, as a function of
µt�1 is ”observable”.

• In SV,

yt = µ+ �t⌫t,

ln(�2t ) = ↵0 + �1ln(�
2
t�1) + ⌘t, ⌘t ⇠ iid N(0,!

the conditional variance �2t is latent (unobservable):
• there are two shocks: ⌫t and ⌘t. Often used in theoretical options

pricing literature;
• it is di�cult to estimate (likelihood evaluation is challenging)
• it is awkward for forecasting, as �2t is conditional on an unobservable

information set.

Slides-13 UNSW

GARCH in mean Structural Break in volatility Seasonality Non-normality Control for Volatility Proxies Stochastic Volatility Summary

Summary

• We have seen a variety of models for conditional volatility for
niveriate returns models

• Next… Multivariate Volatility models: Portfolio management,
hedging strategies…

Slides-13 UNSW

Financial Econometrics
Slides-14: Multivariate Volatility Models

Dr. Rachida Ouysse
School of Economics1

Dr. Rachida OuysseSchool of Economics 1

Multivariate GARCH Models
• Multivariate GARCH models are used to estimate and to

forecast covariances and correlations.

• The basic formulation is similar to that of the GARCH model,
but where the covariances as well as the variances are
permitted to be time-varying.

• There are 3 main classes of multivariate GARCH formulation
that are widely used: VECH, diagonal VECH and BEKK.

VECH and Diagonal VECH

• e.g. suppose that there are two variables used in the model.
The conditional covariance matrix is denoted H t, and would
be 2× 2. Ht and VECH(Ht) are

Ht =

[
h11t h12t
h21t h22t

]
, VEC (Ht) =


 h11th22t

h12t




Dr. Rachida OuysseSchool of Economics 2

VECH and Diagonal VECH

• In the case of the VECH, the conditional variances and
covariances would each depend upon lagged values of all of
the variances and covariances and on lags of the squares of
both error terms and their cross products.

• In matrix form, it would be written

VECH(Ht) = C + AVECH(Ξt−1Ξ
′
t−1) + BVECH(Ht−1)

Ξt |ψt−1 ∼ N(0,Ht)

Dr. Rachida OuysseSchool of Economics 3

VECH and Diagonal VECH (Cont’d)

• Writing out all of the elements gives the 3 equations as

h11t = c11 + a11u
2
1t−1 + a12u

2
2t−1 + a13u1t−1u2t−1 + b11h11t−1

+ b12h22t−1 + b13h12t−1

h22t = c21 + a21u
2
1t−1 + a22u

2
2t−1 + a23u1t−1u2t−1 + b21h11t−1

+ b22h22t−1 + b23h12t−1

h12t = c31 + a31u
2
1t−1 + a32u

2
2t−1 + a33u1t−1u2t−1 + b31h11t−1

+ b32h22t−1 + b33h12t−1

Dr. Rachida OuysseSchool of Economics 4

VECH and Diagonal VECH (Cont’d)

• Such a model would be hard to estimate. The diagonal VECH
is much simpler and is specified, in the 2 variable case, as
follows:

h11t = α0 + α1u
2
1t−1 + α2h11t−1

h22t = β0 + β1u
2
2t−1 + β2h22t−1

h12t = γ0 + γ1u1t−1u2t−1 + γ2h12t−1

Dr. Rachida OuysseSchool of Economics 5

BEKK and Model Estimation for M-GARCH

• Neither the VECH nor the diagonal VECH ensure a positive
definite variance-covariance matrix.

• An alternative approach is the BEKK model (Engle & Kroner,
1995).

• The BEKK Model uses a Quadratic form for the parameter
matrices to ensure a positive definite variance / covariance
matrix H t.

• In matrix form, the BEKK model is

Ht = W
′W + A′Ht−1A + B

′Ξt−1Ξ
′
t−1B

Dr. Rachida OuysseSchool of Economics 6

BEKK and Model Estimation for M-GARCH
(Cont’d)

• Model estimation for all classes of multivariate GARCH model
is again performed using maximum likelihood with the
following LLF:

`(θ) = −
TN

2
log 2π −

T∑
t=1

(
log |Ht |+ Ξ′tH

−1
t Ξt

)
where N is the number of variables in the system (assumed 2
above), θ is a vector containing all of the parameters, and T is
the number of obs.

Dr. Rachida OuysseSchool of Economics 7

Correlation Models and the CCC
• The correlations between a pair of series at each point in time

can be constructed by dividing the conditional covariances by
the product of the conditional standard deviations from a
VECH or BEKK model

• A subtly different approach would be to model the dynamics
for the correlations directly

• In the constant conditional correlation (CCC) model, the
correlations between the disturbances to be fixed through time

• Thus, although the conditional covariances are not fixed, they
are tied to the variances

• The conditional variances in the fixed correlation model are
identical to those of a set of univariate GARCH specifications
(although they are estimated jointly):

hii ,t = ci + ai�
2
i ,t−i + bihii ,t−1, i = 1, . . . ,N

Dr. Rachida OuysseSchool of Economics 8

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