VAR Models

Gloria González-RiveraUniversity of California, Riverside

and

Jesús Gonzalo U. Carlos III de Madrid

Some ReferencesSome 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

Multivariate ModelsMultivariate Models

• VARMAX Models as a multivariate generalization of the univariate ARMA models:

• Structural VAR Models:

• VAR Models (reduced form)

nn x 1k x k n x 1n x n n x

tjL

q

0j

jtXiL

r

0i

iGtYsL

p

0s

s

1 1 ...t t p t p tBY Y Y

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

Multivariate Models (cont)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 otherwiset sE a a

0

A Structural VAR(1)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:

From a Structural VAR to From a Structural VAR to a Standard VARa 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

From a Structural VAR to aFrom a Structural VAR to a Standard VAR Standard VAR (cont) (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 10 1 1

0 1 1

t t t

t t t

t t t

BY Y

Y B B Y B

Y Y a

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

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

Generalization of univariate analysis to an array of random variables

.....

income V rate,interest supply,money i.e.

2211

t

tptpttt

t

t

t

t

tt

aYYYcY

V

X

Z

Y

XZ

VAR(p)

t

taaEaE tt 0

)'(0)(

i are matrices)1(

333231

232221

131211

1

A typical equation of the system is

tptp

ptp

ptp

tttt

aVXZ

VXZcZ

1)(

13)(

12)(

11

113)1(

112)1(

111)1(

1 .....

Each equation has the same regressors

Stability ConditionsStability Conditions

ji

jiLLL

ij

L

acYL

acYLLLI

acYYYY

ijpp

ijijij

tt

ttp

p

tptpttt

0

1]....[

is (L) ofelement the

operator L lag thein polynomialmatrix nxna is )(

)(

)......(

......

)(2)2()1(ij

221

2211

A VAR(p) for is STABLE iftY

21 2 ..... 0

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

pn pI x x x

p n

cI pn1

21 ).....(

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

LLIL

aLaaaY

n

ttttt

Re-writing the system in deviations from its mean

tptpttt aYYYY )(...)()( 2211 Stack the vector as

0

0

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

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

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

...... 121

1

1

t

t

n

n

n

pp

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 tF v E v v

t

H

(nxp)x(nxp)STABLE:eigenvalues of F lie insideof the unit circle (WHY?).

VAR(p)VAR(p) VAR(1)VAR(1)

Estimation of Estimation of VAR VAR models models

Estimation: Conditional MLE

1 1 0 1 1 1 2 11

1 2 1 1

1 2

1 2

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

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

' [ ..... ]

[1 ...... ]'

'

( ) log

T

T T p t t t t pt

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 tt

f Y past

Tn TY X Y X

n x (np+1)

(np+1) x 1

Claim: OLS estimates equation by equation are good!!!1

1 1

ˆ ˆ ˆ ' ' 'T T

mle ols ols t t t tt t

Y X X X

Proof:

t t ttolsttolsolsttt

T

ttolsttolst

ttolstolst

T

tttolstolst

T

ttttt

XaXXaa

XaXa

XXXYXXXY

XYXY

)'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

tttolst

ttols

ttolst

ttolst

aXtraXtr

XatrXa

olsˆ when achieved is aluesmallest v the

definite positive is 1-matrix definite positive is because

t t

tX)'olsˆ(1)olsˆ(t'Xta1t'amin

T

1t

tX'tY1'tX'tYmin

Maximum Likelihood of Evaluate the log-likelihood at , then

T

tjtitij

T

titii

T

t

T

ttttt

T

ttt

aaT

aT

aaT

aaT

aaTTn

1

2

1

22

1 11

1

11

ˆˆ1

ˆ elements diagonal-off

ˆ1

ˆ elements diagonal

'ˆˆ1ˆ0'ˆˆ

2

1'

2

)ˆ,(

ˆ'ˆ2

1log

2)2log(

2)ˆ,(

Testing Hypotheses in a VAR modelTesting Hypotheses in a VAR model

Likelihood ratio test in VAR

: lags ofnumber theTesting2

ˆlog2

)2log(2

)ˆ,ˆ(

22

1ˆˆ2

1

'ˆˆˆ2

1ˆˆ'ˆ

2

1ˆˆ'ˆ

2

1

ˆˆ'ˆ2

1ˆlog2

)2log(2

)ˆ,ˆ(

01

1

1

1

1

1

1

1

1

1

11

pp

TnTTn

TnTItraceTtrace

aatraceaatraceaa

aaTTn

n

T

ttt

T

ttt

T

ttt

T

ttt

)(:

)(:

11

00

pVARH

pVARH

0under H

2ˆlog

2)2log(

2

2

ˆlog2

)2log(2

ˆ,ˆlags p andconstant a

on variableeach of sregression OLS n perform

11

*1

10

*0

000

TnTTn

TnTTn

1under H

)(nsrestrictio ofnumber

ˆlogˆlogˆlogˆlog)(2

0122

101

01

1*

01*

ppnmLR

TTLR

m

equation each in )(

variableeach on nrestrictio has equation each

01

01

ppn

pp

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

estimated 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 y t t t itT iT

t=1 t=1n.T

ˆAsymptotic distribution of is

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

2 1 ˆˆ( ) (0, ) with lim(1 / )

T N M

T N M M p TiT i i

'X Xt tt

In general, linear hypotheses can be tested directly as usual and their A.D follows from the next asymptotic result:

Information Criterion in a Standard VAR(p)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 is that 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)

Granger CausalityGranger Causality

Granger (1969) : “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods”, Econometrica, 37

Consider two random variablestt YX ,

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 Yt t s t t t s t tt t t

MSE X s E X X st t s t

MSE X s MSE X s Yt t t

ause 0

is not linearly informative to forecast

X st

Y Xt t

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 tX c X X X Y Y Y a

Estimate by OLS and test for the following hypothesis

0any :

) cause-Grangernot does ( 0......:

1

210

i

ttp

H

XYH

Unrestricted sum of squared residuals

Restricted sum of squared residuals

t

taRSS 21 ˆ

t

taRSS 22

ˆ

2 1

1

( )

/( 2 1)

RSS RSSF

RSS T p

• Under general conditions ( )F p

Impulse Response Function (IRF)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

)(,

)(

'

sij

jt

sti

sijs

t

st

a

y

a

Y

n x n

Reaction of the i-variable to a unit changein innovation j

(multipliers)

Impluse Response Function (cont)Impluse Response Function (cont)

Impulse-response function: response of to one-time impulse in with all other variables dated t or earlier held constant.

stiy ,

jty

ijjt

sti

a

y

,

s

ij

1 2 3

Example: IRF for a VAR(1)Example: IRF for a VAR(1)

2212

122

1

2

1

2

11

2221

1211

2

1 ;1

at

tt

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

2

12 11 12 11 11 12

22 21 22 21 21 22

1 11 121

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)

If you work with the MA representation:

1

212

11

1)()(

ss

LL

In this example, the variance-covariance matrix of the innovationsis 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 bediagonalized (the shocks have to be orthogonal) and here is wherea serious problems appear.

This is not very realistic

Reminder: is positive definite (symmetric) matrix.

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

Then, the MA representation:

00

1

0

1

0

Let us call ;

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

has components that are all uncorrelated and unit variance

t i t i ni

t i t ii

i i t t t i t ii

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 ss s

t

YM Q

w

Orthogonalized impulse-responseFunction.

Problem: Q is not unique

Variance decompositionVariance decomposition

Contribution of the j-th orthogonalized innovation to the MSE of the s-period ahead forecast

1 1 1 1

1 1 1 1

1 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

10 0

recall that

and ,

i iM Q

M Q I

contribution of the first orthogonalizedinnovation to the MSE (do it for a two variables VAR model)

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

• 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

2y

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)

Identification in a Standard VAR(1)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) and only 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 model letting b21=0.

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

211 11 12 21 21 21 2 x

212 12 12 22 22 22 1 2 12 x

b b b b var(e ) b

b var(e )

b co v(e ,e ) b

• b21=0 implies

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

•Note both structural shocks can now be identified from the residuals 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).

Critics on VARCritics on VAR

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

• To calculate the IRF, the order matters: remember that “Q” is not unique.

• Sensitive to the lag selection

• Dimensionality problem.

•THINK on TWO MORE weak points of VAR modelling