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### Transcript of Lesson 17: Vector AutoRegressive · PDF fileLesson 17: Vector AutoRegressive Models Umberto...

• Lesson 17: Vector AutoRegressive Models

Umberto Triacca

Dipartimento di Ingegneria e Scienze dellInformazione e MatematicaUniversita dellAquila,

[email protected]

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Vector AutoRegressive models

The extension of ARMA models into a multivariate frameworkleads to

Vector AutoRegressive (VAR) models

Vector Moving Average (VMA) models

Vector AutoregRegressive Moving Average (VARMA)models

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Vector AutoRegressive models

The Vector AutoRegressive (VAR) models ,

made famous in Chris Simss paper Macroeconomics and Reality,Econometrica, 1980,

are one of the most applied models in the empirical economics.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Vector AutoRegressive models

Let {yt = (y1t , ..., yKt)

; t Z}

be a K -variate random process.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Vector AutoRegressive models

We say that the process {yt ; t Z} follows a vector autoregressivemodel of order p, denoted VAR(p) if

yt = + A1yt1 + ...+ Apytp + ut , t Z

p is a positive integer,

Ai are fixed (K K ) coefficient matrices, = (1, ..., K )

is a fixed (K 1) vector of intercept terms,ut = (u1t , ..., uKt)

is aK -dimensional white noise withcovariance matrix u.

The covariance matrix u is assumed to be nonsingular.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Example: Bivariate VAR(1)

[y1ty2t

]=

[12

]+

[a11 a12a21 a22

] [y1t1y2t1

]+

[u1tu2t

]or

y1t = 1 + a11y1t1 + a12y2t1 + u1t

y2t = 1 + a21y1t1 + a22y2t1 + u2t

where cov(u1t , u2s) = 12 for t = s, 0 otherwise

Umberto Triacca Lesson 17: Vector AutoRegressive Models

1 They are easy to estimate.

2 They have good forecasting capabilities.

3 The researcher does not need to specify which variables areendogenous or exogeneous. All are endogenous.

4 In a VAR system is very easy to test for Granger non-causality.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Problems with VARs

1 So many parameters. If there are K equations, one for each Kvariables and p lags of each of the variables in each equation,(K + pK 2) parameters will have to be estimated.

2 VARs are a-theoretical, since they use little economic theory.Thus VARs cannot used to obtain economic policyprescriptions.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

In this lesson, the estimation of a vector autoregressive model isdiscussed.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

Consider the VAR(1)

y1t = 1 + a11y1t1 + a12y2t1 + u1t

y2t = 1 + a21y1t1 + a22y2t1 + u2t

wherecov(u1t , u2s) = 12 for t = s, 0 otherwise.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

The model corresponds to 2 regressions with different dependentvariables and identical explanatory variables.

We could estimate this model using the ordinary least squares(OLS) estimator computed separately from each equations.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

It is assumed that a time series

y1 = [y11, y21], ..., yT = [y1T , y2T ]

of the y variables is available.In addition, a presample value

y0 = [y10, y20]

is assumed to be available.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

Consider the first equation

y1t = 1 + a11y10 + a12y2t1 + u1t ; t = 1, ...,T

y11 = 1 + a11y10 + a12y20 + u11

y12 = 1 + a11y11 + a12y21 + u12

...

y1T = 1 + a11y1T1 + a12y2T1 + u1T

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

We definey1 = [y11, ..., y1T ]

,

X1 =

1 y1,0 y2,01 y1,1 y2,1...

......

1 y1,T1 y2,T1

,

1 = [1, a11, a12],

u1 = [u11, ..., u1T ],

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Thusy1 = X1 + u,

The OLS estimator 1 is given by

1 = (XX)1Xy1

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

Consider the second equation

y2t = 2 + a21y10 + a22y2t1 + u2t ; t = 1, ...,T

y21 = 2 + a21y10 + a22y20 + u21

y22 = 2 + a21y11 + a22y21 + u22

...

y2T = 2 + a21y1T1 + a22y2T1 + u2T

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

We definey2 = [y21, ..., y2T ]

,

2 = [2, a21, a22],

u2 = [u21, ..., u2T ],

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation

Thusy2 = X2 + u2,

The OLS estimator 2 is given by

2 = (XX)1Xy2

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• OLS estimators

1 = (XX)1Xy1 1 = [1, a11, a12]

2 = (XX)1Xy2 2 = [2, a21, a22]

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• OLS Estimation

Are the OLS estimators for 2 and 2 efficient estimators?

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Estimation

We remember that

cov(u1t , u2t) = 12 6= 0

When contemporaneous correlation exists, it may be more efficientto estimate all equations jointly, rather than to estimate each oneseparately using least squares.

The appropriate joint estimation technique is the GLS estimation

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

Let as consider the following set of two equations

y1t = 10 + 11x1t + 12z1t + u1t

y2t = 10 + 11x2t + 12z2t + u2t

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

The two equation can be written in the usual matrix algebranotation as

y1 = X11 + u1

y2 = X22 + u2

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

whereyi = [yi1, ..., yiT ]

i = 1, 2

Xi =

1 xi ,1 zi ,11 xi ,2 zi ,2...

......

1 xi ,T zi ,T

i = 1, 2,i = [i0, i1, i2]

, i = 1, 2,

andui = [ui1, ..., uiT ]

, i = 1, 2,

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

Further, we assume that

1 E [uit ] = 0 i = 1, 2 t = 1, 2...T

2 var(uit) = 2i i = 1, 2 t = 1, 2...T ;

3 E [uitujt ] = ij i , j = 1, 2;

4 E [uitujs ] = 0 for t 6= s and i , j = 1, 2;

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

Now we write the two equations as the folllowing super model[y1y2

]=

[X1 00 X2

] [12

]+

[u1u2

]or

y = X + u

The first point to note is that it can be shown that least squaresapplied to this system is identical to applying least squaresseparately to each of the two equations.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

Note that the covariance matrix of the joint disturbace vector u isgiven by

=

[21I 12I12I 22I

]=

[21 1212

22

] IT = IT

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

The disturbance covariance matrix is of dimension(2T 2T ).An important point to note is that cannot be writtenas scalar multiplied by a 2T -dimensional identity matrix. Thus thebest linear unbiased estimator for is given by the generalizedleast squares estimator

GLS = (X1X)1X1y = [X(1 IT )X]1X(1 IT )y

It has lower variance than the least squares estimators for because it takes into account the contemporaneous correlationbetween the disturbances in different equation.

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Seemingly Unrelated Regressions Equations

There are two conditions under the which least squares is identicalto generalized least squares.

1 The first is when all contemporaneus covariances are zero,12 = 0.

2 The second is when the explanatory variables in each equationare identical, X1 = X2

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation of A VAR model

The use of generalized least squares estimator does not lead to again in efficiency when each equation contains the sameexplanatory variables.

Thus the autoregressive cooefficients of our VAR(1) model can beestimated, without loss of estimation efficiency, by ordinary leastsquares. This in turn is equivalent to estimating each equationseparately by OLS

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation of A VAR model

The (2 2) unknown covariance matrix may be consistentestimated by whose elements are

ij =ui uj

T 2p 1for i , j = 1, 2

where ui = yi Xi

Umberto Triacca Lesson 17: Vector AutoRegressive Models

• Estimation of A VAR model

In general, the autoregressive cooefficients of a K -dimensionalVAR(p) model

yt = + A1yt1 + ...+ Apytp + ut

can be estimated, without loss of estimation efficiency, by ordinaryleast squares.In the first equation, we have to run the regression

y1t on y1t1, ..., yKt1, ..., y1tp, ..., yKtp,

in the second equation, we regress

y12 on y1t1, ..., yKt1, ..