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Financial Econometrics – Slides-05: Time Series Analysis using ARMA models Part 2

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

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

School of Economics1

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

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

Time Series Models (Mainly Theoretical Aspects)

• MA process
• AR process

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

Dr. School 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,

Θ(L) = 1 + θ1L+ θ2L

+ · · ·+ θqLq.

Dr. School 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),

yt = µ+ Θ(L)�t,

where Θ(L) = 1 + θ1L.

Dr. School 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

γj = Cov(yt, yt−j) =

(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. School 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

V ar (yt+h|Ωt) =
(1 + θ21)σ

• Conditional variance ≤unconditional variance (why?)

Dr. School 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:

σθ1, h = 1

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

MA(1) model: Invertibility

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

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. School of Economics (UNSW) Slides-05 ©UNSW 9 / 27 MA(1) model: Invertibility ©Copyright University of 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: −1(yt − µ). For MA(1), when invertible, = (1 + θ1L) = 1 + (−θ1)L+ (−θ1)2L2 + · · · , (1) �t = yt − µ+ (−θ1)i(yt−i − µ) (2) (−θ1)iyt−i − µ/(1 + θ1). (3) Hint. Use expansion: 1/(1− x) = 1 + x+ x2 + · · · - Parameters can be estimated by minimizing - The alternative expression: yt = µ/(1 + θ1)− (−θ1)iyt−i + �t indicates that the PAC function of invertible MA(1) has no cutoffs and decays exponentially. Dr. School of Economics (UNSW) Slides-05 ©UNSW 10 / 27 MA(1) model: Example ©Copyright University of 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 eg. NYSE comp return School of Economics, UNSW Slides-04, Financial Econometrics 19 25 50 75 100 125 150 175 200 MA(1): theta = -0.9 25 50 75 100 125 150 175 200 MA(1): theta = 0, White Noise 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. School of Economics (UNSW) Slides-05 ©UNSW 11 / 27 MA(q) model: Dynamic Behaviour ©Copyright University of 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 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. School 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. School 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!

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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)

↵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

iyt + “t with L
iyt = yt�i

↵ (L) yt = ↵0 + “t (13)

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

Dr. School 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. School 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 + ↵
1yt�2 + ↵1″t�1 + “t

Next substitute

yt�2 = ↵0 + ↵1yt�3 + “t�2

in the equation for yt to obtain

1 + ↵1 + ↵

1″t�2 + ↵1″t�1 + “t

AR Process: Impulse response function

Univariate Time Series Analysis: ARIMA models

Building ARIMA models

Autoregressive Process

After repeating this t � 1 times, we obtain

1 + ↵1 + . . . + ↵

1 “1 + . . . + ↵1″t�1 + “t

↵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 = ↵

dyt+1/d”t = ↵1

dyt+2/d”t = ↵

dyt+3/d”t = ↵

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.

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):

↵i1″t�i (15)

I The expected value of yt is given by

E (yt) = E

1 + ↵1 + ↵

! if |↵1| < 1 : E (yt) converges to ! if |↵1| � 1 : E (yt) is time-dependent Dr. School of Economics (UNSW) Slides-05 ©UNSW 20 / 27 Properties of AR(1) Process: Unconditional Variance ©Copyright University of 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 t�2 + . . . + cross-products 1 + ↵21 + ↵ ! if |↵1| < 1 : V (yt) converges to ! if |↵1| � 1 : V (yt) is time-dependent Dr. School of Economics (UNSW) Slides-05 ©UNSW 21 / 27 Properties of AR(1) Process: ACF ©Copyright University of 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))) "t + ↵1"t�1 + ↵ 1"t�2 + . . . "t�1 + ↵1"t�2 + ↵ 1"t�3 + . . . t�3 + . . . + cross-products 1 + ↵21 + ↵ ! if |↵1| < 1 : �1 converges to ↵1 ! if |↵1| � 1 : �1 is time-dependent Dr. School of Economics (UNSW) Slides-05 ©UNSW 22 / 27 Properties of AR(1) Process: ACF ©Copyright University of 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))) "t + ↵1"t�1 + ↵ 1"t�2 + . . . "t�2 + ↵1"t�3 + ↵ 1"t�4 + . . . t�4 + . . . + cross-products 1 + ↵21 + ↵ ! if |↵1| < 1 : �2 converges to ↵21 ! if |↵1| � 1 : �2 is time-dependent Dr. School of Economics (UNSW) Slides-05 ©UNSW 23 / 27 Properties of AR(1) Process: ACF ©Copyright University of 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 ! if |↵1| � 1 : �k is time-dependent I The ACF (for stationary series!) is given by ⇢1 = �1 /�0 = ↵1 ⇢2 = �2 /�0 = ↵ ⇢k = �k /�0 = ↵ Dr. School of Economics (UNSW) Slides-05 ©UNSW 24 / 27 AR Process: Stationary Conditions ©Copyright University of 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 1 + ↵1 + ↵ 1 + ↵21 + ↵ 1 + ↵21 + ↵ ! an AR(1) process is stationary is |↵1| < 1. Dr. School of Economics (UNSW) Slides-05 ©UNSW 25 / 27 AR Process: Conclusions ©Copyright University of 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): 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. School of Economics (UNSW) Slides-05 ©UNSW 26 / 27 AR(1) Example: Simulated and Fitted ©Copyright University of 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 eg. NYSE comp return: ‘c’ below is in fact 𝜇𝜇 = 𝑐𝑐/(1 − 𝜙𝜙1) School of Economics, UNSW Slides-04, Financial Econometrics 25 25 50 75 100 125 150 175 200 AR(1): phi = 0, White Noise 25 50 75 100 125 150 175 200 AR(1): phi = 0.5 25 50 75 100 125 150 175 200 AR(1): phi = 0.9 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. School of Economics (UNSW) Slides-05 ©UNSW 27 / 27 Moving average process MA(q) Defining MA(q) Dynamic behaviour: Impulse response function MA(q) Dynamic behaviour Autoregressive Process AR(p) Characteristics of AR(p) Properties of an AR(1) Process AR(1) Example 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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