VARmodels

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VAR Models

Gloria González-Rivera

University of California, Riverside

and

Jesús Gonzalo U. Carlos III de Madrid

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Some References

• Hamilton, chapter 11

• Enders, chapter 5

• Palgrave Handbook of Econometrics, chapter 12 by Lutkepohl• Any of the books of Lutkepohl on Multiple Time Series

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Multivariate Models

• VARMAX Models as a multivariate generalization of the

univariate ARMA models:

• Structural VAR Models:

• VAR Models (reduced form)

nn x1k xk n x1n xnn x

t jL

q

0 j

jtXiL

r

0i

iGtYsL

p

0s

s

1 1 ...t t p t p t BY Y Y

1 1 ... at t p t p t Y Y Y

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Multivariate Models (cont)

where the error term is a vector white noise:

To avoid parameter redundancy among the parameters, we need

to assume certain structure on

and

This is similar to univariate models.

'( ) if s t

0 otherwise

t s E a a

0

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A Structural VAR(1)

t 10 12 t 11 t 1 12 t 1 yt

t 20 21 t 21 t 1 22 t 1 xt

y b b x y x

x b b y y x

• The error terms (structural shocks) yt and xt are white noise

innovations with standard deviations y

and x

and a zero covariance.

• The two variables y and x are endogenous (Why?)

• Note that shock yt affects y directly and x indirectly.

• There are 10 parameters to estimate.

Consider a bivariate Yt=(yt, xt), first-order VAR model:

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From a Structural VAR to a Standard VAR

• The structural VAR is not a reduced form.

• In a reduced form representation y and x are just functions of lagged y and x.

• To solve for a reduced form write the structural VAR in matrix form

as:

10 112 11 12

20 121 21 22

0 1 1

1

1

t t yt

t t xt

t t t

y b yb

x b xb

BY Y

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From a Structural VAR to a Standard VAR (cont)

• Premultipication by B-1 allow us to obtain a standard VAR(1):

• This is the reduced form we are going to estimate (by OLS equation

by equation)

• Before estimating it, we will present the stability conditions (the

roots of some characteristic polynomial have to be outside the unit

circle) for a VAR(p)

• After estimating the reduced form, we will discuss which information

do we get from the obtained estimates (Granger-causality, Impulse

Response Function) and also how can we recover the structural

parameters (notice that we have only 9 parameters now).

0 1 1

1 1 1

0 1 1

0 1 1

t t t

t t t

t t t

BY Y

Y B B Y B

Y Y a

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A bit of history ....Once Upon a Time

Sims(1980) “Macroeconomics and Reality” Econometrica, 48

Generalization of univariate analysis to an array of random variables

.....

incomeVrate,interestsupply,moneyi.e.

2211

t

t pt pt t t

t

t

t

t

t t

aY Y Y cY

V

X

Z

Y

X Z

VAR(p)

t t aa E a E t t

0)'(0)(

i are matrices)1(

333231

232221

131211

1

A typical equation of the system is

t pt

p

pt

p

pt

p

t t t t

aV X Z

V X Z c Z

1

)(

13

)(

12

)(

11

113)1(

112)1(

111)1(

1 .....

Each equation has the same regressors

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Stability Conditions

ji

ji L L L

ij

L

acY L

acY L L L I

acY Y Y Y

ij

p p

ijijij

t t

t t

p

p

t pt pt t t

0

1]....[

is(L)of elementthe

operator Llagthein polynomialmatrixnxnais)(

)(

)......(

......

)(2)2()1(

ij

2

21

2211

A VAR(p) for is STABLE if t Y

2

1 2..... 0

x roots of the characteristic polynomial are outside of the unit circle.

p

n p I x x x

p n

c I pn

1

21 ).....(

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If the VAR is stable then a representation exists.

This representation will be the “key” to study the impulse response

function of a given shock.

)( MA

......][)(

)(......

2

21

2211

L L I L

a LaaaY

n

t t t t t

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Re-writing the system in deviations from its mean

t pt pt t t aY Y Y Y )(...)()( 2211

Stack the vector as

0

0

0..........00

0...............0

0...............0

...... 121

1

1

t

t

n

n

n

p p

pt

t

t

t

a

v

I

I

I

F

Y

Y

Y

(nxp)x1 (nxp)x(nxp)(nxp)x1

1 ( ') 0

0.....0

0 0......0where

0 0......0

t t t t

H t

F v E v v t

H

(nxp)x(nxp)

STABLE:

eigenvalues of F lie inside

of the unit circle (WHY?).

VAR(p) VAR(1)

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Estimation of VAR models

Estimation: Conditional MLE

1 1 0 1 1 1 2 1

1

1 2 1 1

1 2

1 2

( , ..... | , .... ; ) ( | , .... ; )

| , .... ( .... , )

' [ ..... ]

[1 ...... ]'

'

( ) log

T

T T p t t t t p

t

t t t t p t p

p

t t t t p

t t t

t

f Y Y Y Y Y Y f Y Y Y Y

Y Y Y N c Y Y

c

X Y Y Y

Y X a

1

1 1

1

( | ; )

1log(2 ) log ' ' '2 2 2

T

t

T

t t t t

t

f Y past

Tn T Y X Y X

n x (np+1)

(np+1) x 1

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Claim: OLS estimates equation by equation are good!!!

1

1 1

ˆ ˆ ˆ ' ' 'T T

mle ols ols t t t t

t t

Y X X X

Proof:

t t t

t olst t olsolst t t

T

t

t olst t olst

t t olst olst

T

t

t t olst olst

T

t

t t t t

X a X X aa

X a X a

X X X Y X X X Y

X Y X Y

)'ˆ('ˆ2)'ˆ()ˆ('ˆ'ˆ

)'ˆ(ˆ')'ˆ(ˆ

''ˆ'ˆ'''ˆ'ˆ

'''

111

1

1

1

1

1

1

0'ˆ)ˆ('ˆ)'ˆ(

)'ˆ('ˆ)'ˆ('ˆ(*)

11

11

t

t t olst

t

t ols

t

t olst

t

t olst

a X tr a X tr

X atr X a

T

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olsˆwhenachievedisaluesmallest vthe

definite positiveis1-matrixdefinite positiveis because

t t

tX)'olsˆ(1)ols

ˆ(t'Xta1t'amin

T

1t

tX'tY1'tX'tYmin

Maximum Likelihood of Evaluate the log-likelihood at , thenˆ

T

t jt it ij

T

t

it ii

T

t

T

t

t t t t

T

t

t t

aaT

aT

aa

T

aaT

aaT Tn

1

2

1

22

1 1

1

1

11

ˆˆ

1ˆ

elementsdiagonal-off

ˆ

1ˆelementsdiagonal

'ˆˆ

1ˆ0'ˆˆ

2

1'

2

)ˆ,(

ˆ'ˆ

2

1log

2)2log(

2)ˆ,(

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Testing Hypotheses in a VAR model

Likelihood ratio test in VAR

:lagsof number theTesting

2ˆlog

2)2log(

2)ˆ,ˆ(

221ˆˆ

21

'ˆˆˆ

2

1ˆ

ˆ'ˆ

2

1ˆ

ˆ'ˆ

2

1

ˆˆ

'ˆ

2

1ˆ

log2)2log(2)ˆ

,ˆ

(

01

1

1

1

1

1

1

1

1

1

11

p p

TnT Tn

TnTI traceT trace

aatraceaatraceaa

aa

T Tn

n

T

t

t t

T

t

t t

T

t

t t

T

t t t

)(:

)(:

11

00

pVAR H

pVAR H

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0under H

2ˆlog

2)2log(

2

2ˆlog

2)2log(

2

ˆ,ˆlags pandconstanta

onvariableeachof sregressionOLSn perform

1

1

*

1

1

0

*

0

000

TnT Tn

TnT Tn

1under H

)(nsrestrictioof number

ˆ

logˆ

logˆ

logˆ

log)(2

01

22

10

1

0

1

1

*

01

*

p pnm LR

T T LR

m

equationeachin)(

variableeachonnrestrictiohasequationeach

01

01

p pn

p p

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Let ( ) denote the (nk 1) (with k=1+np number of parameters

Testimated per equation) vector of coef. resulting from OLS regressions of each

of the elements of y on x for a sample of size T:t t

vec

T

-11.T T T'

. , where = x x x yt t t itT iTt=1 t=1

n.T

ˆ

Asymptotic distribution of is

1( ) (0, ( )), and the coef of regression iT

2 1ˆˆ( ) (0, ) with lim(1 / )

T N M

T N M M p T iT i i

' X X t t

t

In general, linear hypotheses can be tested directly as usual and

their A.D follows from the next asymptotic result:

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Information Criterion in a Standard VAR(p)

2

2

2(n p n)AIC ln T

(n p n) ln(T)SBC ln

T

• In the same way as in the univariate AR(p) models,

Information Criteria (IC) can be used to choose the “right”number of lags in a VAR(p): that minimizes IC(p) for

p=1, ..., P.

p

• Similar consistency results to the ones obtained in the univariate

world are obtained in the multivariate world.The only difference isthat as the number of variables gets bigger, it is more unlikely that

the AIC ends up overparametrizing (see Gonzalo and Pitarakis

(2002), Journal of Time Series Analysis)

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Granger Causality

Granger (1969) :

“Investigating Causal Relations by Econometric Models and Cross-

Spectral Methods”, Econometrica, 37

Consider two random variablest t Y X ,

Two Forecast of , periods ahead:t

(1) (2)ˆ ˆ( ) ( | , , ....) ( ) ( | , , .... , , ....)1 1 1

2ˆ ˆ

( ( )) ( ( ))(1) (2)

ˆ ˆIf ( ( ) ) ( ( ) ) then does not Granger-c

X s

X s E X X X X s E X X X Y Y t t s t t t s t t t t t

MSE X s E X X st t s t

MSE X s MSE X s Y t t t

ause 0

is not linearly informative to forecast

X st

Y X t t

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Test for Granger-causality

Assume a lag length of p

1 1 1 2 2 1 1 2 2..... ....t t t p t p t t p t p t X c X X X Y Y Y a

Estimate by OLS and test for the following hypothesis

0any:

)cause-Granger notdoes( 0......:

1

210

i

t t p

H

X Y H

Unrestricted sum of squared residuals

Restricted sum of squared residuals

t

t a RSS 2

1ˆ

t

t a RSS 2

2ˆˆ

2 1

1

( )

/( 2 1)

RSS RSS F

RSS T p

• Under general conditions ( ) F p

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Impulse Response Function (IRF)

Objective: the reaction of the system to a shock

1 1 2 2

1 1 2 2

1

1 1 2 2 1 1

....

If the system is stable,

( ) ....

( ) [ ( )]

Redating at time :

.... ....

t t t p t p t

t t t t t

t s t s t s t s s t s t

Y c Y Y Y a

Y L a a a a

L L

t s

Y a a a a a

)(,

)(

'

s

ij

jt

st i

s

ij s

t

st

a

y

a

Y

n x n

Reaction of the i-variable to a unit change

in innovation j

(multipliers)

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Impluse Response Function (cont)

Impulse-response function: response of to one-time impulse in

with all other variables dated t or earlier held constant. st i y ,

jt y

ij

jt

st i

a y

,

s

ij

1 2 3

f A (1)

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Example: IRF for a VAR(1)

2

212

12

2

1

2

1

2

11

2221

1211

2

1;

1

a

t

t t

t

t

a

a

y

y

y

y

t

1 2

10 20 2

0 0

0 0, 1 ( increases by 1 unit)

(no more shocks occur)

t t

t

t y y

t a a y

Reaction of the system 10

20

11 11 12 12

21 21 22 22

212 11 12 11 11 12

22 21 22 21 21 22

1 11 12

1

2 21 22

0

1

0

1

0

1

0 0

1 1

s

s s

s

y

y

y

y

y y

y y

y

y

(impulse)

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If you work with the MA representation:

1

212

11

1)()(

s

s

L L

In this example, the variance-covariance matrix of the innovations

is not diagonal, i.e. 012

There is contemporaneous correlation between shocks, then

1

0

20

10

y

y

To avoid this problem, the variance-covariance matrix has to be

diagonalized (the shocks have to be orthogonal) and here is where

a serious problems appear.

This is not very realistic

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Reminder:

is positive definite (symmetric) matrix.

(non-singular) such that Q 'Q Q I

Then, the MA representation:

0

0

1

0

1

0

Let us call ;

[ ' ] [ ' '] [ ' ] ' '

has components that are all uncorrelated and unit variance

t i t i n

i

t i t ii

i i t t t i t i

i

t t t t t t n

t

Y a I

Y Q Qa

M Q w Qa Y M w

E w w E Qa a Q QE a a Q Q Q I

w

1t s s s

t

Y M Q

w

Orthogonalized impulse-response

Function.

Problem: Q is not unique

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Variance decomposition

Contribution of the j-th orthogonalized innovation to the MSE of

the s-period ahead forecast

1 1 1 1

1 1 1 11 1' 1 1'

1 1

1 1'

1 1

1 1'

ˆ ˆ ˆ( ( )) ( ( ))( ( )) '

ˆ( ) ( ) .....

[ ( ) ( ) '] ' .... '

( ) ' ' ' ....

' '

t t s t t s t

t t s t t s t s s t

t t a a s a s

a a

s a s

MSE Y s E Y Y s Y Y s

e s Y Y s a a a

E e s e s

MSE s Q Q Q Q Q Q Q Q

Q Q Q Q

Q Q

1 1' 1 1'

1 1 1 1

0 0 1 1 1 1

' ....... '' ' ......... '

s s

s s

Q Q Q Q M M M M M M

1

1

0 0

recall that

and ,

i i M Q

M Q I

contribution of the first orthogonalized

innovation to the MSE (do it for a two variables VAR model)

E l V i d i i i

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Example: Variance decomposition in a two

variables (y, x) VAR

• The s-step ahead forecast error for variable y is:

y E y M (1,1) M (1,1) ... M (1, 1)t s t t s yt s0 1 yt s 1 s 1 yt 1

M (1, 2) M (1, 2) ... M (1, 2)xt s0 1 xt s 1 s 1 xt 1

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• Denote the variance of the s-step ahead forecast

error variance of yt+s as for y(s)2:

2 2 2 2 2(s) [M (1, 1) M (1, 1) ... M (1, 1) ]y y 0 1 s 1

2 2 2 2[M (1, 2) M (1, 2) ... M (1, 2) ]x 0 1 s 1

• The forecast error variance decompositions are

proportions of y(s)2.

2y

2

y

2 2 2 2[M (1, 1) M (1, 1) ... M (1, 1) ]y 0 1 s 1

2 2 2 2[M (1, 2) M (1, 2) ... M (1, 2) ]x 0 1 s 1

due to shocks to y / (s)

due to shocks to x / (s)

Id tifi ti i St d d VAR(1)

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Identification in a Standard VAR(1)

ytt 10 t 111 1212

t 20 21 22 t 1 xt

y b y1 b

x b x0 1

• Remember that we started with a structural VAR model, and jumped into the reduced form or standard VAR for estimation

purposes.

•Is it possible to recover the parameters in the structural VAR from the estimated parameters in the standard VAR? No!!

•There are 10 parameters in the bivariate structural VAR(1) andonly 9 estimated parameters in the standard VAR(1).

•The VAR is underidentified.

•If one parameter in the structural VAR is restricted the

standard VAR is exactly identified.•Sims (1980) suggests a recursive system to identify the modelletting b21=0.

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Identification in a Standard VAR(1) (cont.)

ytt 10 t 111 1212 12 12

t 20 21 22 t 1 xt

t 10 t 1 1t11 12

t 20 21 22 t 1 2t

y b y1 b 1 b 1 b

x b x0 1 0 1 0 1

y y e

x x e

• The parameters of the structural VAR can now be identified from the

following 9 equations

2 2 210 10 12 20 20 20 1 y 12 x

2

11 11 12 21 21 21 2 x

2

12 12 12 22 22 22 1 2 12 x

b b b b var(e ) b

b var(e )

b cov(e ,e ) b

• b21=0 implies

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Identification in a Standard VAR(1) (cont.)

• Note both structural shocks can now be identified from theresiduals of the standard VAR.

• b21=0 implies y does not have a contemporaneous effect on x.

•This restriction manifests itself such that both

yt and

xt affect y contemporaneously but only xt affects x contemporaneously.

•The residuals of e2t are due to pure shocks to x.

•Decomposing the residuals of the standard VAR in this triangular fashion is called the Choleski decomposition.

•There are other methods used to identify models, like Blanchard

and Quah (1989) decomposition (it will be covered on the

blackboard).

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Critics on VAR

• A VAR model can be a good forecasting model, but in a sense it isan atheoretical model (as all the reduced form models are).

• To calculate the IRF, the order matters: remember that “Q” is notunique.

• Sensitive to the lag selection

• Dimensionality problem.

•THINK on TWO MORE weak points of VAR modelling

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