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Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

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Page 1: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series

Page 2: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 1

The Contents

1. The main properties of Multivariate Time Series.

2. Estimation of the Mean and Covariance function.

3. Multivariate ARMA Models.

4. Modelling and Prediction with Multivariate Processes.

5. Cointegration.

Page 3: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 2

Introduction

Many time series arising in practice are best considered ascomponents of some vector-valued (multivariate) time series {Xt}having not only serial dependence within each component series{Xtj} but also independence between the different component series{Xtj} and {Xti}, i 6= j. Much of the theory of univariate time seriesextends in a natural way to the multivariate case.

Page 4: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 3

Introduction

Example 1 We consider the closing values D0, . . . , D250 of theDow-Jones Index of stocks on the New York Stock Exchange and theclosing values A0, . . . , A250 of the Australian All-ordinaries Index ofShare Prices, recorded at the termination of trading on 251 successivetrading days up to August 26th, 1994. The efficients markethypothesis suggests that these processes should resemble random walkswith uncorrelated increments.

Page 5: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 4

Page 6: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 5

In order to model the data as a stationary bivariate time series wereexpress the data as percentage relative price changes

Xt1 = 100Dt −Dt−1

Dt−1, t = 1, 2, . . . 250,

Xt2 = 100At −At−1

At−1, t = 1, 2, . . . 250.

Page 7: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 6

Page 8: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 7

Page 9: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 8

Second-Order Properties of Multivariate TimeSeries

Consider m time series {Xti, t = 0,−1,+1, . . . }, i = 1, 2, . . . ,m withEX2

ti < ∞ for all t and i. If all the finite dimensional distributions ofrandom variables {Xti} were multivariate normal, then thedistributional properties of {Xti} would be completely determined bythe means

µti = EXti

and the covariances

γij(t + h, t) = E[(Xt+h,i − µti) (Xtj − µtj)].

Page 10: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 9

Second-Order Properties of Multivariate TimeSeries

Even when the observations {Xti} do not have joint normaldistributions, the quantities µti and γij(t + h, t) specify thesecond-order properties, the covariance is a measure of dependence,not only between observations in the same series but also betweenthe observations in different series.

Page 11: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 10

Second-Order Properties of Multivariate TimeSeriesIt is more convenient when dealing with m interrelated series to usevector notation. Thus we define

Xt = [Xt1, . . . , Xtm]′, t = 0,−1,+1, . . . .

The second-order properties of the multivariate time series {Xt} arespecified by the mean vectors

µt = [µt1, . . . , µtm]′

and covariance matrices

Γ(t + h, t) =

γ11(t + h, t) . . . γ1m(t + h, t)

.... . .

...

γm1(t + h, t) . . . γmm(t + h, t)

.

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Multivariate Time Series 11

Second-Order Properties of Multivariate TimeSeries

As in the univariate case, a particularly important role is played bythe class of multivariate stationary time series, defined as follows

Definition 1 The m−variate series {Xt} is (weakly) stationary ifµt is independent of t and Γ(t + h, t) is independent of t for each h.

For a stationary time series we shall use the notation

µ = EXt Γ(h) = E[(Xt+h − µ) (Xt − µ)].

If {Xt} is stationary with covariance matrix function Γ(.), then foreach i, {Xti} is stationary with covariance function γii(.). Thefunction γij(.), i 6= j is called the cross-covariance function of the twoseries {Xti} and {Xtj}.

Page 13: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 12

Second-Order Properties of Multivariate TimeSeries

The correlation matrix function R(.) is defined by

R(h) =

ρ11(h) . . . ρ1m(h)

.... . .

...

ρm1(h) . . . ρmm(h)

,

where

ρij(h) =γij(h)

(γii(0)γjj(0))1/2.

The function R(.) is the covariance matrix function of the normalizedseries obtained by subtracting µ from {Xt} and then dividing eachcomponent by its standard deviation.

Page 14: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 13

Second-Order Properties of Multivariate TimeSeries

Example 2 Let consider the bivariate stationary process {Xt}defined by

Xt1 = Zt, Xt2 = Zt + 0.75Zt−10,

where {Zt} ∼ WN(0, 1). In this case µ = 0 and

Γ(−10) = Γ(10)′=

0 0.75

0 0.75

,

Γ(0) =

1 1

1 1.5625

,

Page 15: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 14

and Γ(j) = 0 otherwise. The correlation matrix function is given by

R(10) = R(−10)′=

0 0.60

0 0.48

,

R(0) =

1 0.80

0.8 1

,

and R(j) = 0 otherwise.

Page 16: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 15

Second-Order Properties of Multivariate TimeSeries

Basic properties of Γ(.)

I Γ(h) = Γ′(−h),

I |γij(h)| ≤ [γii(0)γjj(0)]1/2, i, j = 1, 2, . . . m,

I γii(.) is an autocovariance function, i = 1, . . . m,

I∑n

j,k=1 a′

jΓ(j − k)ak ≥ 0 for all n ∈ {1, 2, . . . } anda1, . . . , an ∈ Rm.

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Multivariate Time Series 16

Second-Order Properties of Multivariate TimeSeries

Remark 1 The basic properties of the matrices Γ(h) are shared alsoby the corresponding matrices of correlations R(h) = [ρ(h)]mi,j=1,which have the additional property ρii(0) = 1 for all i.

The simplest multivariate time series is multivariate white noise

Definition 2 The m−variate series {Zt} is called white noise withmean 0 and covariance matrix Σ ({Zt} ∼ WN(0,Σ)) iff {Zt} isstationary with mean vector 0 and covariance matrix function

Γ(h) =

Σ h = 0,

0 otherwise

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Multivariate Time Series 17

Second-Order Properties of Multivariate TimeSeries

Definition 3 The m−variate series {Zt} is called IID noise withmean 0 and covariance matrix Σ ({Zt} ∼ IID(0,Σ)) if the randomvectors {Zt} are independent and identically distributed with mean 0and covariance matrix Σ.

Definition 4 The m−variate series {Xt} is a linear process if it hasthe representation

Xt =∞∑

j=−∞CjZt−j , {Zt} ∼ WN(0,Σ), (1)

where {Cj} is a sequence of m×m matrices whose components areabsolutely summable.

Page 19: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 18

Second-Order Properties of Multivariate TimeSeries

Remark 2 The linear process (1) is stationary with mean 0 andcovariance function

Γ(h) =∞∑

j=−∞Cj+hΣC

j , h = 0,−1,+1, . . . .

Definition 5 A MA(∞) process is a linear process with Cj = 0 forj < 0.

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Multivariate Time Series 19

Second-Order Properties in the frequencydomain

Provided the components of the covariance matrix function Γ(.) hasthe property

∑∞h=−∞ |γij(h)| < ∞, i, j = 1, 2, . . . m, then Γ has a

matrix-valued spectral density function

f(λ) =12π

∞∑h=−∞

e−iλhΓ(h), −π < λ < π,

and Γ can be expressed in terms of f as

Γ(h) =∫ π

−π

eiλhf(λ)dλ.

The second-order properties of the stationary process {Xt} cantherefore be described equivalently in terms of f(.) rather than Γ(.).

Page 21: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 20

Second-Order Properties in the frequencydomain

The series {Xt} has a spectral representation

Xt =∫ π

−π

eiλtdZ(λ),

where {Z(λ),−π < λ < π} is a process whose components arecomplex-valued processes satisfying

E(dZj(λ)dZ̄k(λ)

)=

fjk(λ)dλ λ = µ,

0 otherwise.

Page 22: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 21

Estimation of the Mean and CovarianceFunction

As in the univariate case, the estimation of the mean vector andcovariances of a stationary time series plays an important role indescribing and modeling the dependence structure of the componentseries.A natural unbiased estimator of the mean vector µ based on theobservations X1, . . . ,Xn is the vector of sample means

X̄n =1n

n∑t=1

Xt.

The resulting estimate of the mean of the jth time series is then theunivariate sample mean (1/n)

∑nt=1 Xtj .

Page 23: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 22

Estimation of the Mean and CovarianceFunction

As in the univariate case , a natural estimator of the covarianceΓ(h) = E[(Xt+h − µ) (Xt − µ)] is

Γ̂(h) =

n−1∑n−h

t=1

(Xt+h − X̄n

) (Xt − X̄n

)′

0 ≤ h ≤ n− 1,

Γ̂(−h) −n + 1 ≤ h < 0.

Writing γ̂ij(h) for the (i, j)-component of ˆΓ(h), we estimate thecross-correlation

ρ̂ij(h) = γ̂ij(h)/[γ̂ii(0)γ̂jj(0)]1/2.

Page 24: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 23

Estimation of the Mean and CovarianceFunction

Example 3 We estimate mean and covariance function of the WestGerman fixed investment, disposable income and consumptionexpenditures in the years 1960-1982 (in Billions of DM).

µ = 103[0.4719 1.3551 1.1664]′

Γ(0) = 105

0.4441 1.4481 1.2235

1.4481 4.8851 4.1274

1.2235 4.1274 3.4901

.

Page 25: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 24

0 10 20 30 40 50 60 70 80 90 1000

500

1000

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

0 10 20 30 40 50 60 70 80 90 1000

1000

2000

3000

Investment

Income

Consumption

Page 26: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 25

Multivariate ARMA processes

As in the univariate case, we can define an extremely useful class ofmultivariate stationary processes {Xt} by requiring that {Xt} shouldsatisfy a set of linear difference equations with constant coefficients.Multivariate white noise {Zt} is a fundamental building block fromwhich these ARMA processes are constructed.

Page 27: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 26

Multivariate ARMA processes

Definition 6 The series {Xt} is an ARMA(p,q) process if {Xt} isstationary and if for every t

Xt − Φ1Xt−1 − · · · − ΦpXt−p = Zt + Θ1Zt−1 + · · ·+ ΘqZt−q, (2)

where {Zt} ∼ WN(0,Σ).

Equations (2) can be written in the more compact form

Φ(B)Xt = Θ(B)Zt, {Zt} ∼ WN(0,Σ),

where Φ(z) = I − Φ1z − · · · − Φpzp, Θ(z) = I + Θ1z + · · ·+ Θqz

q arematrix-valued polynomials, I is the m×m identity matrix.

Page 28: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 27

Multivariate ARMA processes

Example 4 Let consider a multivariate AR(1) model given by theequation

Xt −ΦXt−1 = Zt,

where Xt = [X1,t, X2,t]′, Φ = [φ1, φ2] = [0.3, 0.5] and

Zt = [Z1,t, Z2,t]′ ∼ WN(0,Σ). A realization of {X1, X2, . . . X100} and

the correlation and cross-correlation functions are shown in the nextfigures.

Page 29: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 28

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Multivariate Time Series 29

Page 31: Multivariate Time Series - prac.im.pwr.wroc.plprac.im.pwr.wroc.pl/~wyloman/Listy_ECMI/prez4a.pdf · Multivariate Time Series 1 The Contents 1. The main properties of Multivariate

Multivariate Time Series 30

Multivariate ARMA processesRemark 3 For the multivariate AR(1) series

Xt − ΦXt−1 = Zt, {Zt} ∼ WN(0,Σ) (3)

the solution is expressed as

Xt =∞∑

j=0

ΦjZt−j (4)

if all the eigenvalues of Φ(z) = I − Φz are less than 1 in absolutevalue, i.e.

det(I − Φz) 6= 0

for all z ∈ C such that |z| ≤ 1.If this condition is satisfied, then the coefficients Φj are absolutelysummable and hence the series AR(1) converges and series given in(4) is the unique stationary solution of (3).

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Multivariate Time Series 31

Multivariate ARMA processes

Definition 7 An ARMA(p,q) process {Xt} is causal or a causalfunction of {Zt}, if there exist matrices {Ψj} with absolutelysummable components such that

Xt =∞∑

j=0

ΨjZt−j .

Causality is equivalent to the condition detΦ(z) 6= 0 for all z ∈ C

such that |z| ≤ 1. The matrices Ψj are found recursively from theequations

Ψj = Θj +∞∑

k=1

ΦkΨj−k, j = 0, 1, . . . ,

where Θ0 = I, Θj = 0 for j > q, Φj = 0 for j > p, and Ψj = 0 forj < 0.

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Multivariate Time Series 32

Multivariate ARMA processes

Definition 8 An ARMA(p,q) process {Xt} is invertible if there existmatrices {Πj} with absolutely summable components such that

Zt =∞∑

j=0

ΠjXt−j .

Invertibility is equivalent to the condition detΘ(z) 6= 0 for all z ∈ C

such that |z| ≤ 1. The matrices Πj are found recursively from theequations

Πj = −Φj −∞∑

k=1

ΘkΠj−k, j = 0, 1, . . . ,

where Φ0 = −I, Φj = 0 for j > p, Θj = 0 for j > q, and Πj = 0 forj < 0.

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Multivariate Time Series 33

Multivariate ARMA processes

Example 5 For the multivariate AR(1) process (3) the recursionsgiven in definition of causality give

Ψ0 = I,

Ψ1 = ΦΨ0 = Φ,

Ψ2 = ΦΨ1 = Φ2,

...

Ψj = ΦΨj−1 = Φj , j ≥ 3.

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Multivariate Time Series 34

Multivariate ARMA processes

Example 6 For the bivariate AR(1) process (3) with

Φ =

0 0.5

0 0

It is easy to check that Ψj = Φj = 0 for j > 1 and hence {Xt} hasthe alternative representation

Xt = Zt + ΦZt−1,

as MA(1) process. This example shows that it is not always possibleto distinguish between multivariate ARMA models of different orderswithout imposing further restrictions.

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Multivariate Time Series 35

The covariance matrix function of a causalARMA processWe can express the covariance matrix Γ(h) = EXtX

t+h of the causalmultivariate ARMA(p,q) process:

Γ(h) =∞∑

j=0

Ψh+jΣΨ′

j , h = 0,−1,+1, . . . ,

where the matrices are found from the recursive equations given inDefinition 7. The covariance matrices Γ(h) can also be found bysolving the Yule-Walker equations

Γ(j)−p∑

r=1

ΦrΓ(j − r) =∑

j≤r≤q

ΘrΣΨr−j , j = 0, 1, 2, . . .

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Multivariate Time Series 36

Modeling with multivariate AR processes

If {Xt} is the causal multivariate AR(p) process defined by thedifference equations

Xt = Φ1Xt−1 + · · ·+ ΦpXt−p + Zt, {Zt} ∼ WN(0,Σ),

then post-multiplying by X′

t−j , j = 0, . . . , p and taking expectationsgives the equations

Σ = Γ(0)−p∑

j=1

ΦjΓ(j)

and

Γ(i) =p∑

j=1

ΦjΓ(i− j), i = 1, . . . p.

Given the matrices Γ(0), . . . ,Γ(p) the above equations can be used todetermine the coefficient matrices Φ1, . . . ,Φp.

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Multivariate Time Series 37

Modeling with multivariate AR processes

The Yule-Walker estimators Φ̂1, . . . , Φ̂p and Σ̂ for the modelARMA(p,q) fitted to the data X1, . . .Xn are obtained by replacingΓ(j) by Γ̂(j) in the above equations and solving the resultingequations for Φ1, . . . ,Φp and Σ. Order selection criterion form-variate autoregressive models can be made by minimizing amultivariate analogue of the univariate AICC statistic

AICC = −2lnL(Φ1, . . . ,Φp,Σ) +2(pm2 + 1)nm

nm− pm2 − 2,

where L(Φ1, . . . ,Φp,Σ) is the likelihood of observations X1, . . . ,Xn.

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Multivariate Time Series 38

Modeling with multivariate AR processes

Example 7 To illustrate the results of the Yule-Walker estimationmethod we use the West German investment, income andconsumption data.

p 1 2 3 4 5 6

AICC 2617.23 2631.87 2644.77 2663.12 2671.82 2686.04

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Multivariate Time Series 39

Φ̂1 =

0.981507 0.113112 −0.138330

0.357728 1.285401 −0.497579

0.229938 0.494436 .305645

Σ̂ =

0.265464 0.794858 0.679089

0.794858 2.73250 2.30377

0.679089 2.30377 1.95470

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Multivariate Time Series 40

Modeling with multivariate AR processes

Example 8 We use the data from Example 1: the closing valuesD0, . . . , D250 of the Dow-Jones Index of stocks on the New YorkStock Exchange and the closing values A0, . . . , A250 of the AustralianAll-ordinaries Index of Share Prices, recorded at the termination oftrading on 251 successive trading days up to August 26th, 1994. Tothe estimation we use the data as percentage relative price changes.

p 1 2 3 4 5 6

AICC 1048.68 1050.63 1057.64 1062.56 1065.85 1070.22

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Multivariate Time Series 41

µ̂ = [0.0295 0.0309]′

Φ̂1 =

−0.0148 0.0357

0.6584 0.0998

Σ̂ =

0.3653 0.0224

0.0224 0.6016

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Multivariate Time Series 42

Best linear predictors of second-order randomvectors

Let {Xt} = {Xt1, . . . , Xtm} be a m-variate time series with meansEXt = µt and covariance matrix function given by the m×m

matricesK(i, j) = EXiX

j − µiµ′

j

If Y = (Y1, . . . , Ym)′is a random vector with finite second moment

and EY = µ we define PnYj as the best linear predictor of thecomponent Yj of Y in therms of all of the components of the vectorsXt, t = 1, . . . n and the constant 1.

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Multivariate Time Series 43

Best linear predictors of second-order randomvectors

It follows from the properties of the prediction operator that

Pn(Y) = µ + A1(Xn − µn) + · · ·+ An(X1 − µ1),

for some matrices A1, A2, . . . , Am and that

Y − Pn(Y)⊥Xn+1−i, i = 1, . . . , n.

The vector of best predictors is uniquely determined by the aboveequations, although it is possible that there may be more than onepossible choice for A1, . . . Am.

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Multivariate Time Series 44

Best linear predictors of second-order randomvectors

Example 9 Let consider a zero-mean time series {Xt}. The bestlinear predictor Pn(Xn+1) of Xn+1 in terms of X1, . . .Xn is obtainedon replacing Y by Xn+1 in the above equations. We can write

Pn(Xn+1) = Φn1Xn + . . .ΦnnX1, n = 1, 2, . . . ,

where the coefficients Φnj, j = 1, 2, . . . n are such that

EPn(Xn+1)X′

n+1−i = EXn+1X′

n+1−i, i = 1, . . . , n,

i.e.n∑

j=1

ΦnjK(n + 1− j, n + 1− i) = K(n + 1, n + 1− i), i = 1, . . . n.

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Multivariate Time Series 45

In case when {Xt} is stationary with K(i, j) = Γ(i− j), theprediction equations simplify to the following

n∑j=1

ΦnjΓ(i− j) = Γ(i), i = 1, . . . n.

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Multivariate Time Series 46

Forecasting multivariate autoregressiveprocessesWe can compute the minimum mean squared error one-step linearpredictors Pn(Xn+1) for any multivariate stationary time series fromthe autocovariance matrices Γ(h) by recursively determining thecoefficients Φni, i = 1, . . . n and evaluating

Pn(Xn+1) = Φn1Xn + · · ·+ ΦnnX1.

The situation is simplified when {Xt} is the AR(p) process (the casein practice) since for n > p

Pn(Xn+1) = Φ1Xn + · · ·+ ΦpXn+1−p.

Because Xn+1 − Φ1Xn + · · ·+ ΦpXn+1−p = Zn+1, then thecovariance matrix of the one-step prediction error is EZn+1Z

n+1 = Σ.

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Multivariate Time Series 47

Forecasting multivariate autoregressiveprocesses

To compute the best h−step linear predictor Pn(Xn+1) based on allthe component of X1, . . . ,Xn we apply the linear operator Pn to (3)to obtain the recursions

Pn(Xn+h) = Φ1Pn(Xn+h−1) + · · ·+ ΦpPn(Xn+h−p).

These equations are easily solved recursively, first for Pn(Xn+1), thenfor Pn(Xn+2),etc.

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Multivariate Time Series 48

Forecasting multivariate autoregressiveprocesses

To compute the h−step error covariance matrices we use that for thecausal AR(p) process we have

Xn+h =∞∑

j=0

ΨjZn+h−j ,

where

Ψj =∞∑

k=1

ΦkΨj−k, j = 0, 1, . . . ,

with the convention Φj = 0 for j > p, and Ψj = 0 for j < 0.

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Multivariate Time Series 49

Therefore for n > p we obtain

Pn(Xn+h) =∞∑

j=h

ΨjZn+h−j .

The h−step prediction error is therefore given by

Xn+h − Pn(Xn+h) =h−1∑j=0

ΨjZn+h−j ,

with covariance matrix

E (Xn+h − Pn(Xn+h)) (Xn+h − Pn(Xn+h))′=

h−1∑j=0

ΨjΣΨ′

j , n > p.

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Multivariate Time Series 50

Cointegration

Nonstationary univariate time series can frequently be madestationary by applying the differencing operator ∇ = 1−B

repeatedly. If {∇dXt} is stationary for some positive integer d but{∇d−1Xt} is nonstationary, we say that {Xt} is integrated of order d

({Xt} ∼ I(d)). Many macroeconomic time series are found to beintegrated of order 1.If {Xt} is a k−variate time series, we define {∇dXt} to be the serieswhose j−th component is obtained by applying the operator(1−B)d to the j−th component of {Xt}, j = 1, 2, . . . , k.

Definition 9 The I(d) process {Xt} is said to be cointegratedCI(d, b) with cointegration vector α if α is k × 1 vector such that{αXt} is I(d− b).

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Multivariate Time Series 51

Cointegration

Example 10 A simple example is provided by the bivariate processwhose first component is the random walkXt =

∑tj=1 Zj , t = 1, 2, . . . , {Zt} ∼ IID(0, σ2) and whose second

component consists of noisy observations of the same random walkYt = Xt + Wt, {Wt} ∼ IID(0, τ2), where {Wt} is independent of{Zt}. Then {(Xt, Yt)

′} is integrated of order 1 and conitegrated withcointegration vector α = (1,−1).

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Multivariate Time Series 52

0 100 200 300 400 500 600 700 800 900 1000−60

−50

−40

−30

−20

−10

0

10

20X

tY

t

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Multivariate Time Series 53

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Multivariate Time Series 54

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Multivariate Time Series 55

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Multivariate Time Series 56

Cointegration

The notion of cointegration captures the idea of univariatenonstationary time series ”‘moving together”’. Thus even though{Xt} and {Yt} in the previous example are both nonstationary, theyare linked in the sense that they differ only by the stationarysequence {Wt}. Series that behave in a cointegrated manner areoften encountered in economics.