CS代考 Financial Econometrics - Slides-01: RETURN PROPERTIES Part II

Financial Econometrics – Slides-01: RETURN PROPERTIES Part II

Stylized Facts of Returns

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School of Economics Financial Econometrics

Stylized Facts of Returns

Financial Econometrics
Slides-01: RETURN PROPERTIES Part II

School of Economics1

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removed from this material.

School 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

= var(Xt) = E[(Xt − µ)2] : spread

skewness(Xt) = S(X) = E

: symmetry

kustosis(Xt) = K(X) = E

: tail thickness

K(X)− 3 : Excess kurtosis

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

mk = E[(Xt − µ)k]

School 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.

School of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Examle: Normal Random Variable

Normal Distribution

 ∼ ( 2)

 −∞ ≤  ≤ ∞

var() = 2

skew() = 0

kurt() = 3

 = 0 for  odd

School 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 ̃

( − ̂)2 = ̂2

dskew = ̂3

 dkurt = ̂4

( − ̂)

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

School of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

Shape Characteristics: Visually

Topic 1. Features of Some Financial Time Series

UNSW Business School,

Slides-01, Financial Econometrics 20

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

Topic 1. Features of Some Financial Time Series

UNSW Business School,

Slides-01, Financial Econometrics 21

School 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

( ˆkurt− 3)2

Note: if rt is N(µ, σ

T ˆskew ∼ N(0, 6), and

T ( ˆkurt− 3) ∼ N(0, 24)

School 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 =

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

2 Kurtosis test: Zkt =

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

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

School 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

250 500 750 1000 1250 1500 1750

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:

+ 2Cov(*, @)]

UNSW Business School,

Slides-01, Financial Econometrics 23

School 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

-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

250 500 750 1000 1250 1500 1750

UNSW Business School,

Slides-01, Financial Econometrics 25

School 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)

-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

250 500 750 1000 1250 1500 1750

leptokurtic

Histogram of RC

-6 -4 -2 0 2 4

UNSW Business School,

Slides-01, Financial Econometrics 27

School 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 =

t=j+1(Xt −X)(Xt−j −X)

• Autocorrelation: ρj =

Sample Autocorrelation: ρ̂j =

• 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, · · ·

School 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)

School 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:

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:

= T (T + 2)

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

School 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.

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.

School 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,

Slides-01, Financial Econometrics 32

School 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?

5 ≈ �G�(��)

eg. NYSE Composite

return squared

UNSW Business School,

Slides-01, Financial Econometrics 33

School 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)

School of Economics Financial Econometrics

Stylized Facts of Returns Shape Characteristics Testing for Normality Autocorrelation

• 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)

School of Economics Financial Econometrics

Stylized Facts of Returns
Shape Characteristics
Testing for Normality
Autocorrelation

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