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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
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