VAR Models and Applications Laurent Ferrara1 1 University

of Paris West

M2 EIPMC 2015

Overview of the presentation

1. Vector Auto-Regressions I I I

Definition Estimation Tests

2. Impulse responses functions (IRF) I I

Concept General IRF

3. Applications

Vector Auto-Regressions: Short introduction

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The VAR models are widely used in economic analysis.

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While simple and easy to estimate, they make it possible to conveniently capture the dynamics of multivariate systems.

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VAR popularity is mainly due to Sims (1980) influential work.

Vector Auto-Regressions: Notations

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Let yt denote an (n × 1) vector of random variables. yt follows a p th order Gaussian VAR if, for all t, we have yt = c + Φ1 yt−1 + . . . Φp yt−p + εt where εt ∼ N(0, Ω).

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Consequently yt | yt−1 , yt−2 , . . . , y−p+1 ∼ N(c + Φ1 yt−1 + . . . Φp yt−p , Ω).

Vector Auto-Regressions: Exemple n = 2

VAR(1)for yt = (y1,t , y2,t ):

y1,t

= c1 + φ11 y1,t−1 + φ12 y2,t−1 + ε1,t

y2,t

= c2 + φ21 y1,t−1 + φ22 y2,t−1 + ε2,t .

where ε1,t ∼ GWN(σε21 ), ε2,t ∼ GWN(σε22 ) and ρ(ε1,t , ε2,t ) = 0 φ11 and φ22 are autoregressive coefficients, φ21 and φ12 are exogeneous coefficients.

Vector Auto-Regressions: MLE I

0  and Denoting with Π the matrix c Φ1 Φ2 . . . Φp  0 0 0 0 with xt the vector 1 yt−1 yt−2 . . . yt−p , the log-likelihood is given by L(YT ; θ) = −(Tn/2) log(2π) + (T /2) log Ω−1 T
0
i 1 Xh yt − Π0 xt Ω−1 yt − Π0 xt . − 2 t=1

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ˆ is given by The MLE of Π, denoted with Π " Πˆ0 =

T X t=1

yt xt0

#" T X t=1

#−1 xt xt0

.

(1)

Vector Auto-Regressions: MLE Proof of equation (1) Let’s rewrite the last term of the log-likelihood T h X


0
i yt − Π0 xt Ω−1 yt − Π0 xt =

t=1 T  X

ˆ 0 xt + Π ˆ 0 xt − Π0 xt yt − Π

0

Ω

−1



ˆ 0 xt + Π ˆ 0 xt − Π0 xt yt − Π



t=1 T  X

0   −1 0 ˆ − Π) xt − Ω ˆ − Π) xt εˆt + (Π εˆt + (Π 0

t=1

where the j th element of the (n × 1) vector εˆt is the sample residual for observation t from an OLS regression of yjt on xt .

=

Vector Auto-Regressions: MLE

T h X


0
i yt − Π0 xt Ω−1 yt − Π0 xt =

t=1 T X

εˆ0 t Ω−1 εˆt + 2

t=1

T X

ˆ − Π)0 xt εˆ0 t Ω−1 (Π

t=1

+

T X t=1

ˆ − Π)Ω−1 (Π ˆ − Π)0 xt xt0 (Π

Vector Auto-Regressions: MLE

Let’s apply the trace operator on the second term (that is a scalar): ! T T X X ˆ − Π)0 xt = trace ˆ − Π)0 xt εˆ0 t Ω−1 (Π εˆ0 t Ω−1 (Π t=1

= trace

t=1 T X

! Ω

−1

0

ˆ − Π) xt εˆ0 t (Π

t=1

= trace

ˆ − Π)0 Ω−1 (Π

T X t=1

! xt εˆ0 t

Vector Auto-Regressions: MLE Given that, by construction, the sample residuals are orthogonal to the explanatory variables, this term is equal to zero. ˆ − Π)0 xt , we have If x˜t = (Π T h X


0
i yt − Π0 xt Ω−1 yt − Π0 xt =

t=1 T X t=1

εˆ0 t Ω−1 εˆt +

T X

x˜t0 Ω−1 x˜t

t=1

Since Ω is a positive definite matrix, Ω−1 is as well. Consequently, the smallest value that the last term can take is obtained when ˆ xt∗ = 0,ie when Π = Π.

Vector Auto-Regressions: MLE I

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ˆ the MLE of is the matrix Assume that we have computed Π, ` ˆ that maximizes Ω → L(YT ; Π, ˆ Ω). Ω ˆ t , we have Denoting with εˆt the estimated residual yt − Πx ˆ L(YT ; Π,Ω) = −(Tn/2) log(2π) + (T /2) log Ω−1 T

−

1 X  0 −1  εˆt Ω εˆt . 2 t=1

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ˆ is a symmetric positive definite matrix. Fortunately, it turns Ω out that that the unrestricted matrix that maximizes the latter expression is a symmetric postive definite matrix. Indeed, T T T 0 1X 0 1 X 0 ∂`(Ω) 0 ˆ = Ω − εˆt εˆt =⇒ Ω = εˆt εˆt . ∂Ω 2 2 T t=1

t=1

Vector Auto-Regressions: Likelihood-Ratio test

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The simplicity of the VAR framework and the tractability of its MLE contribute to convenience of various econometric tests. We illustrate this here with the likelihhod ratio test.

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The maximum value achieved by the MLE is ˆ −1 ˆ ˆ L(YT ; Π,Ω) = −(Tn/2) log(2π) + (T /2) log Ω T

−

1 X h 0 ˆ −1 i εˆt Ω εˆt . 2 t=1

Vector Auto-Regressions: Likelihood-Ratio test I

The last term is T X

ˆ −1 εˆt εˆ0t Ω

= trace

hP

T ˆ −1 εˆt ˆ0t Ω t=1 ε

i

t=1

" = trace

T X

# ˆ −1

Ω

εˆt εˆ0t

" ˆ −1

= trace Ω

t=1

T X

# εˆt εˆ0t

t=1

h  i ˆ = Tn. ˆ −1 T Ω = trace Ω I

Therefore ˆ Ω) ˆ = −(Tn/2) log(2π) + (T /2) log Ω ˆ −1 − Tn/2. L(YT ; Π, which is easy to calculate.

Vector Auto-Regressions: Likelihood-Ratio test

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For instance, assume that we want to test the null hypothesis that a set of variable follows a VAR(p0 ) against the alternative specification of p1 lags (with p1 > p0 ). Let us respectively denote with Lˆ0 and Lˆ1 the maximum log-likelihoods obtained withp0 and p1 lags. Under the null hypothesis, we have     ˆ −1 ˆ −1 2 Lˆ1 − Lˆ0 = T log Ω − log Ω 1 0 which asymptotically has a χ2 distribution with degrees of freedom equal to the number of restrictions imposed under H0 (compared with H1 ), ie n2 (p1 − p0 ).

Vector Auto-Regressions: Criteria I

In a VAR, adding lags quickly consume degrees of freedom. If lag length is p, each of the n equations contains n × p coefficients plus the intercept term.

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Adding lengths improve in-sample fit, but is likely to result in over-parameterization and affect the out-of-sample prediction performance.

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To select appropriate lag length, some criteria can be used (they have to be minimized) AIC SBIC where N = n × p 2 + p.

2 ˆ = log Ω + N T log T ˆ = log Ω + N T

Vector Auto-Regressions: Granger Causality I

Granger (1969) developed a method to analyze the causal relationship among variables systematically.

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The approach consists in determining whether the past values of y1,t can help to explain the current y2,t .

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Let us denote three information sets I1,t

= {y1,t , y1,t−1 , . . .}

I2,t

= {y2,t , y2,t−1 , . . .}

It I

= {y1,t , y1,t−1 , . . . y2,t , y2,t−1 , . . .} .

We say that y1,t Granger-causes y2,t if E [y2,t | I2,t−1 ] 6= E [y2,t | It−1 ] .

Vector Auto-Regressions: Granger Causality I

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To get the intuition behind the testing procedure, consider the following bivariate VAR(p) process: y1,t

= Φ10 + Σpi=1 Φ11 (i)y1,t−i + Σpi=1 Φ12 (i)y2,t−i + u1,t

y2,t

= Φ20 + Σpi=1 Φ21 (i)y1,t−i + Σpi=1 Φ22 (i)y2,t−i + u2,t .

Then,y1,t does not Granger-cause y2,t if Φ21 (1) = Φ21 (2) = . . . = Φ21 (p) = 0.

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Therefore the hypothesis testing is ( H0 : Φ21 (1) = Φ21 (2) = . . . = Φ21 (p) = 0 HA : Φ21 (1) 6= 0 or Φ21 (1) 6= 0 or . . . Φ21 (p) 6= 0.

Vector Auto-Regressions: Granger Causality

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Rejection of H0 implies that some of the coefficients on the lagged y1,t ’s are statistically significant.

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This can be tested using the F -test or asymptotic chi-square test. I

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(RSS−USS)/p The F -statistic is F = USS/(T −2p−1) (where RSS: restricted residual sum of squares, USS: unrestriced residual sum of squares) Under H0 , the F -statistic is distributed as F (p, T − 2p − 1) In addition, pF → χ2 (p).

Vector Auto-Regressions: Granger Causality

See RATS example on the US-EA GDP growth relationships

Vector Auto-Regressions: Impulse responses I

Objective: analyzing the effect of a given shock on the endogenous variables.

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Let the stationary VAR(p) system: yt = c + Φ1 yt−1 + . . . Φp yt−p + εt

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Assume the system receives a shock at t: εt = δ

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bf Definition A standard IRF (h, δ) describes the effects of the shock at date t + h compared to a zero-shock εt = 0, assuming that εt+h = 0 for all h > 0.

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The Generalized IRF by Koop, Pesaran and Potter (1996): GIRF (h, δ, Ft−1 ) = E {yt+h |εt = δ; εt+h = 0, h > 0; Ft−1 } − E {yt+h |εt+h = 0, h ≥; Ft−1 }

Vector Auto-Regressions: Impulse responses

Exemple of a centered univariate AR(1) : xt = φxt−1 + εt . Assume xt−1 = 0, thus xt = εt = δ. IRF (1, δ) = E (xt+1 |εt = δ, εt+1 = 0, Ft−1 )−E (xt+1 |εt = εt+1 = 0, Ft−1 ) IRF (1, δ) = φδ IRF (2, δ) = φ2 δ.... IRF (h, δ) = φh δ Remark : IRF is proportional to the size of the shock and independent of past history

Vector Auto-Regressions: Impulse responses

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Let us consider a stationary vector random variable yt that presents the following Wold’s decomposition: y t = εt +

∞ X

Ψj εt−j .

j=1 I

The hth impulse response function of the shock εt on yt , yt+1 , . . . is given by Ψh δ and vanishes as h → ∞

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Formally, the impulse response of the shock εt on the variable y is defined as ∂yt+h = Ψh . ∂εt

Vector Auto-Regressions: Impulse responses Dynamics of yt , yt+1 , yt+2 , . . . when εt = 1, εt+1 = 0, εt+2 = 0, . . .

!! !" !#

"# !

"#

%$"#$&

"#$'

"#$( !

"#$)

"#$*

Vector Auto-Regressions: Unconditional variance I

The unconditional matrix of variance-covariance of yt is Var (y ) = lim E0 ((yt − y¯t )(yt − y¯t )0 ) t→∞

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where y¯t denotes the unconditional mean of y .  0 0 0 . . . yt−p Let denote with yt∗ the vector yt0 yt−1 , we have       c Φ1 Φ2 · · · Φp εt  0   1  0  0 ··· 0      ∗   ∗ yt =  .  +   yt−1 +  ..   ..   0 . . . 0   0 .  yt∗

0 0 0 ∗ ∗ = c + Φyt−1 + εt ∗

1

0

0

Vector Auto-Regressions: Unconditional variance

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It is then easy to get the Wold’s decomposition of yt∗ :


∗ yt∗ = c ∗ + Φ c ∗ + Φyt−2 + ε∗t−1 + ε∗t = c ∗ + ε∗t + Φ(c ∗ + ε∗t−1 ) + . . . + Φk (c ∗ + ε∗t−k ) + . . . I

The ε∗t ’s being iid, we have Var (y ) = Ω + ΦΩΦ0 + . . . + Φk ΩΦ0k + . . .

Vector Auto-Regressions: Extensions

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Let yt denote an (n × 1) vector of random variables. yt is Gaussian VAR(p) with exogeneous variables xt = (xt1 , . . . , xtm ) of dimension m, for all t yt = Φ1 yt−1 + . . . Φp yt−p + Cxt + εt where εt ∼ N(0, Ω) and 

c11 . . .  .. C = . cij cn1 . . .

 c1m ..  .  cnm

Vector Auto-Regressions: Exemple n = 2

VAR(1)for yt = (y1,t , y2,t ):

y1,t

1 2 = φ11 y1,t−1 + φ12 y2,t−1 + c11 xt−1 + c12 xt−1 + ε1,t

y2,t

2 1 = φ21 y1,t−1 + φ22 y2,t−1 + c21 xt−1 + c22 xt−1 + ε2,t .

Vector Auto-Regressions: Extensions

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Bayesian VAR

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Non-linear VAR (Smooth-Transition VAR, Markov-Switching VAR)

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Factor-Augmented VAR