chapitre 1 part 2 imea 1

8/2/2019 Chapitre 1 Part 2 IMEA 1

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Lecture 1: Fundamental concepts in Time Series Analysis(part 2)

Florian Pelgrin

University of Lausanne, Ecole des HECDepartment of mathematics (IMEA-Nice)

Sept. 2011 - Jan. 2012

Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 1 / 40
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Stationarity, stability, and invertibility

6. Stationarity, stability, and invertibility

Consider again a situation where the value of a time series at time t,Xt, is a linear function of a constant term, the last p values of Xt, thecontemporaneous and last q values of a white noise process, denotedby t :

Xt = +

pk=1

kXtk +

qj=0

jtj.

This process rewrites :

(L)Xt Autoregressive part

= + (L)t Moving average part

with (L) = 1 1L 2L2 pLp and(L) = 1 + 1L +

+ qL

q.

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Which conditions ?

Stationarity conditions regard the autoregressive part of the previous(linear) time series model (ARMA(p,q) model).

Stability conditions also regard the autoregressive part of the previous(linear) time series model (ARMA(p,q) model).

Stability conditions are generally required to avoid explosive solutions

of the stochastic difference equation : (L)Xt = + (L)t.

Invertibility conditions regard the moving average part

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Implications of the stability (stationarity) conditions

If stability conditions hold, stationarity conditions are satisfied (the

converse is not true) and (Xt) is weakly stationary.

If the stochastic process (Xt) is stable (and thus weakly stationary),then (Xt) has an infinite moving average representation (MA(

)

representation) :

Xt = 1(L) ( + (L)t)

= 1(1) + C(L)t

= 1 pk=1 k +

i=0

citi

where the coefficients ci satisfy

i=0|ci|


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This representation shows the impact of past shocks, ti, on the

current value of Xt :

dXtdti

= ci

This is an application of the Wolds decomposition

Omitting the constant term, Xt and t are linked by a linear filter inwhich C(L) = 1(L)(L) is called the transfer function.

The coefficients (ci) are also referred to as the impulse responsefunction coefficients.


St ti it t bilit d i tibilit
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If (Xt)tZ is weakly stationary but not stable, then (Xt) has thefollowing representation :

Xt = 1(L) ( + (L)t)

= 1(1) + C(L)t

=

1 pk=1 k +

i=

citi.

where (cj) is sequence of constants with :

i=

|ci|


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Determination of the stability and stationarity conditions

DefinitionConsider the stochastic process defined by :

(L)Xt = + (L)t

with (L) = 1 1L 2L2 pLp, (L) = 1 + 1L + + qLq,and (t) is a white noise process. This process is called stable if themodulus of all the roots, i, i = 1, , p, of the (reverse) characteristicequation :

() = 0 1 1 22 pp = 0

are greater than one, |i| > 1 for all i = 1, , p.


Stationarity stability and invertibility
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Implications of the stability (stationarity) conditions

Invertibility is the counterpart to stationarity for the moving averagepart of the process.

If the stochastic process (Xt) is invertible, then (Xt) has an infiniteautoregressive representation (AR() representation) :

1

(L)(L)Xt = 1

(L) + t

or

Xt = 1 q

k=1 k+

i=1

diXti + t

where the coefficients di satisfy

i=0|di|


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The AR(

) representation shows the dependence of the current value

Xt on the past values of Xti.

The coefficients are referred as the d-weights of an ARMA model.

If the stochastic process (Xt) is not invertible, it may exist a

representation of the following form (omitting the constant term) aslong as the magnitude of any (characteristic) root of () does notequal unity :

Xt = i=0

diXti + t


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Determination of the invertibility conditions

DefinitionConsider the stochastic process defined by :

(L)Xt = + (L)t

with (L) = 1 1L 2L2 pLp, (L) = 1 + 1L + + qLq,and (t) is a white noise process. This process is called invertible if themodulus of all the roots, i, i = 1, , q, of the (reverse) characteristicequation :

() = 0 1 + 1 + 22 + + qq = 0

are greater than one, |i| > 1 for all i = 1, , q.


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y, y, y

Three equivalent representations

DefinitionA stable (and thus weakly stationary), invertible stochastic process,(L)Xt = + (L)t has two other equivalent representations forms :

1 An infinite moving average representation :

Xt =

1 pk=1 k

+

i=0

citi

2 An infinite autoregressive representation :

Xt =

1 qk=1 k

+i=1

diXti + t


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y, y, y

Each of these representations can shed a light on the model from adifferent perspective.

For instance, the stationarity properties, the estimation of parameters,and the computing of forecasts can use different representations forms(see further).


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y y y

Examples

1. An autoregressive process of order 1 :

(1 L)Xt = twhere t WN(0, 2 ) for all t.

The stability and (weak) stationarity of (Xt) depends on the (reverse)

characteristic root of () = 1 :

() = 0 = 1.

The stability condition writes :

|| > 1 || < 1.The stationarity condition writes :

|

| = 1

|

| = 1.


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Therefore,

1 If || < 1, then (Xt) is stable and weakly stationary :

(1 L)1 =k=0

kLk and Xt =k=0

ktk

2 If || > 1, then a non-causal stationary solution (Xt) exists :

(1 L)1 = k=1

kFk and Xt = k=1

kt+k

3 If || = 1, then (Xt) is not stable and stationary : (1 L) cannot beinverted !


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2. A moving average process of order 1 :

Xt = t t1where t WN(0, 2 ) for all t.

(Xt) is weakly stationary irrespective of the characteristic root of() = 1 (why ?).The invertibility of (Xt) depends on the (reverse) characteristic root of() = 1

:

() = 0 = 1.

The invertibility condition writes :

||>

1 || q) : MA(q) process has no memory beyond q

periods.

The autocorrelation (respectively, autocovariance) function of astationary AR(p) process exhibits exponential decay towards zero

(but does not vanish for lags greater than p).

The autocorrelation (respectively, autocovariance) function of astationary ARMA(p, q) process exhibits exponential decay towardszero : it does not cut off but gradually dies out as h increases.

The autocorrelation function of a nonstationary process decreasesvery slowly even at very high lags, long after the autocorrelationsfrom stationary processes have declined to (almost) zero.


Identification tools Autocorrelations


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The sample autocorrelation function

DefinitionGiven a sample of T observations, x1, , xT, the sample autocorrelationfunction, denoted by (X(h)), is computed by :

X(h) =

T

t=h+1(xt )(xth )T

t=1(xt )2

where is the sample mean :

=1

T

Tt=1

xt.


Identification tools Autocorrelations


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Proposition

If (Xt)tZ is a second order stationary process with :

Xt = m +jZ

ajtj

where the ts are i.i.d. with mean zero and variance2 , jZ|aj|


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The partial autocorrelation function

The partial autocorrelation function is another tool for identifying theproperties of an ARMA process.

It is particularly useful to identify pure autoregressive process(AR(p)).

A partial correlation coefficient measures the correlation between tworandom variables at different lags after adjusting for the correlationthis pair may have with the intervening lags : the PACF thusrepresents the sequence of conditional correlations.

A correlation coefficient between two random variables at differentlags does not adjust for the influence of the intervening lags : theACF thus represents the sequence of unconditional correlations.


Identification tools Partial autocorrelation


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Definition

The partial autocorrelation function of a stationary stochastic process(Xt)tZ is defined to be :

aX(h) = Corr(Xt,Xth | Xt1, ,Xth+1)

h > 0.




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Definition

The partial autocorrelation function of order h is defined to be :

aX(h) = Corr

Xt, Xth

=Cov

Xt, Xth

VXtVXth1/2

where

Xt = Xt

EL(Xt

|Xt1,

,Xth+1)

Xth = Xth EL(Xth | Xt1, ,Xth+1).




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Definition

The partial autocorrelation function of a second order stationary stochastic

process (Xt)tZ satisfies :

aX(h) =|R(h)||R(h)|

where R(k) is the autocorrelation matrix of order h definie par :

R(h) =

X(0) X(1) X(h 1)X(1) X(0) X(h 2)

......

. . ....

X(h 1) X(h 2) X(0)

and R(h) is the matrix that is obtained after raplacing the last columnofR(h) by = (X(1), , X(k))t.




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Definition

The partial correlation function can be viewed as the sequence of the h-th

autoregressive coefficients in a h-th order autoregression. Let ah denotethe -th autoregressive coefficient of an AR(h) process :

Xt = ah1Xt1 + ah2Xt2 + + ahhXth + t.

Then

aX(h) = ahh

for h = 1, 2, Remark : The theoretical partial autocorrelation function of an AR(p)model will be different from zero for the first p terms and exactly zero forhigher order terms (why ?).



Th l i l l i f i


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The sample partial autocorrelation function

The sample partial autocorrelation function can be obtained bydifferent methods :

1

Find the ordinary least squares (or maximum likelihood) estimates ofahh2 Use the recursive equations of the autocorrelation function

(Yule-Walker equations) after replacing the autocorrelation coefficientsby their estimates (see further).

3 Etc.




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Proposition

If (Xt)tZ is a second order stationary process with :

Xt = m +jZ

ajtj

where the ts are i.i.d. with mean zero and variance2 , jZ|

aj

|

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