VAR Models and Applications Laurent Ferrara1 (with the help of Jean-Paul Renne) 1 University

of Paris West

M2 EIPMC 2011

L. Ferrara (University of Paris West)

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Overview of the presentation

1

Vector Auto-Regressions Definition Estimation Tests

2

Impulse responses General concept Application to Structural VAR

3

Applications 1 2 3

Blanchard and Quah (1989) Smets and Tsatsaronis (1997) Dedola and Lippi (2005)

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Structural VAR Vector Auto-Regressions: Short introduction

The VAR are widely used in economic analysis. While simple and easy to estimate, they make it possible to conveniently capture the dynamics of multivariate systems. VAR popularity is mainly due to Sims (1980) influential work.

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Structural VAR Vector Auto-Regressions: Notations

let yt denote an (n × 1) vector of random variables. yt follows a pth order Gaussian VAR if, for all t, we have yt = c + Φ1 yt−1 + . . . Φp yt−p + εt where εt ∼ N(0, Ω). Consequently yt | yt−1 , yt−2 , . . . , y−p+1 ∼ N(c + Φ1 yt−1 + . . . Φp yt−p , Ω).  0 Denoting with Π the matrix c Φ1 Φ2 . . . Φp and with xt  0 0 0 0 yt−2 . . . yt−p the vector 1 yt−1 , 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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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (MLE)

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

T X t=1

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yt xt0

#" T X

#−1 xt xt0

(1)

.

t=1

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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (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 x t − Π0 x t yt − Π

0

Ω

−1



ˆ 0 xt + Π ˆ 0 xt − Π0 xt yt − Π



=

t=1 T  X

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

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 . L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (MLE)

T h X


i
0 = 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

ˆ − Π)Ω−1 (Π ˆ − Π)0 xt xt0 (Π

t=1

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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (MLE)

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

t=1

= trace

T X

! ˆ − Π)0 xt εˆ0 t Ω−1 (Π

t=1

ˆ − Π)0 = trace Ω−1 (Π

T X

! xt εˆ0 t

t=1

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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (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 xt∗ = 0,ie ˆ when Π = Π. L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Maximum Likelihood Estimation (MLE)

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

−

1 X h 0 −1 i εˆt Ω εˆt . 2 t=1

ˆ is a symmetric positive definite matrix. Fortunately, it turns out Ω that that the unrestricted matrix that maximizes the latter expression is a symmetric postive definite matrix. Indeed, T T X ∂`(Ω) T 1X 0 ˆ0 = 1 = Ω0 − εˆt εˆt =⇒ Ω εˆt εˆ0t . ∂Ω 2 2 T t=1

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t=1

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Structural VAR Vector Auto-Regressions: Likelihood-Ratio test

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

−

1 X h 0 ˆ −1 i εˆt Ω εˆt . 2 t=1

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Structural VAR Vector Auto-Regressions: Likelihood-Ratio test

The last term is T X

ˆ −1 εˆt εˆ0t Ω

i hP T 0Ω ˆ −1 εˆt ε ˆ = trace t=1 t

t=1

" = trace

T X

# " # T X −1 0 −1 0 ˆ ˆ Ω εˆt εˆt = trace Ω εˆt εˆt

t=1

t=1

h

 i ˆ = Tn. ˆ −1 T Ω = trace Ω Therefore ˆ Ω) ˆ = −(Tn/2) log(2π) + (T /2) log Ω ˆ −1 − Tn/2. L(YT ; Π, which is easy to calculate. L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Likelihood-Ratio test

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 ). ˆ0 and L ˆ1 the maximum Let us respectively denote with L log-likelihoods obtained withp0 and p1 lags. Under the null hypothesis, we have     ˆ1 − L ˆ0 = T log Ω ˆ −1 − log Ω ˆ −1 2 L 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 ).

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Structural VAR Vector Auto-Regressions: Unconditional variance

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

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 

yt∗ yt∗



  Φ1 Φ2 · · · Φp    1  0 ··· 0      ∗  =  + y +   . t−1 .    0  . 0 0  0 0 0 1 0 ∗ ∗ ∗ = c + Φyt−1 + εt

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c 0 .. .



VAR Models and Applications

εt 0 .. .

    

0

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Structural VAR Vector Auto-Regressions: Unconditional variance

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

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Structural VAR Vector Auto-Regressions: Criteria

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. Adding lengths improve in-sample fit, but is likely to result in over-parameterization and affect the out-of-sample prediction performance. To select appropriate lag length, some criteria can be used (they have to be minimized) 2 ˆ AIC = log Ω + N T log T ˆ SBIC = log Ω N + T where N = n × p2 + p. L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Granger Causality

Granger (1969) developed a method to analyze the causal relationship among variables systematically. The approach consists in determining whether the past values of y1,t can help to explain the current y2,t . Let us denote three information sets  I1,t = y1,t , y1,t−1 , . . .  I2,t = y2,t , y2,t−1 , . . .  It = 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 .

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Structural VAR Vector Auto-Regressions: Unconditional variance

To get the intuition behind the testing procedure, consider the following bivariate VAR(p) process: y1,t y2,t

= Φ10 + Σpi=1 Φ11 (i)y1,t−i + Σpi=1 Φ12 (i)y2,t−i + u1,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. 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. L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Unconditional variance

Rejection of H0 implies that some of the coefficients on the lagged y1,t ’s are statistically significant. This can be tested using the F -test or asymptotic chi-square test. (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).

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Structural VAR Vector Auto-Regressions: Impulse responses

Objective: analyzing the effect of a given shock on the endogenous variables. Let us consider a random variable yt that presents the following Wold’s decomposition: yt =

∞ X

Φk εt−k .

k=0

The impulse response function of the shock εt on yt , yt+1 , . . . is given by the matrices Φk . Formally, the impulse response of the shock εt on the variable y is defined as ∂yt+k = Φk . ∂εt L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Impulse responses

Dynamics of yt , yt+1 , yt+2 , . . . when εt = 1, εt+1 = 0, εt+2 = 0, . . .

!! !" !#

"# !

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"#

%$"#$&

"#$'

"#$(

"#$)

"#$*

!

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Structural VAR Vector Auto-Regressions: Impulse responses

Exercise 2 Consider the process yt = 1 + 0.9yt−1 + εt . Compute the unconditional mean and variance of yt . Exercise 2 Consider the process yt = 1 + 0.5yt−1 + 0.4yt−2 + εt . Draw the impulse response of yt to εt (up to yt+3 ). What is the cumulated impact of a shock (εt = 1, εt+1 = 0, εt+2 = 0, . . .) on yt ? L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Causal mechanisms

Assume that the GDP growth gt is affected by some real shocks ur ,t following gt = −0.3(it−1 − πt ) + ur ,t where it denotes the nominal interest rate and πt denotes the inflation rate. Besides, assume that we have it πt

= 0.9it−1 + 1.5πt + ump,t = 0.9πt−1 + 0.2gt−1 + un,t

where ump,t and un,t are respectively some monetary-policy and cost-push shocks. The strucural shocks ut are uncorrelated (i.e., the covariance matrix of ut , denoted with Ωu is diagonal) and ut is serially uncorrelated (i.e. Cov (ut−k , ut ) = 0 for any t and k > 0). L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions: Causal mechanisms

The “structural” model reads   gt = −0.3(it−1 − πt ) + ur ,t it = 0.9it−1 + 1.5πt + ump,t   πt = 0.9πt−1 + 0.2gt−1 + un,t . To get it in a “reduced-form”, let us substitute πt in the right-hand sides of the first two equations:   gt = 0.06gt−1 − 0.3it−1 + 0.27πt−1 + 0.3un,t + ur ,t it = 0.9it−1 + 1.35πt−1 + 0.3gt−1 + ump,t + 1.5un,t   πt = 0.9πt−1 + 0.2gt−1 + un,t .

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Structural VAR Vector Auto-Regressions: Causal mechanisms

Exercise 3 Write the model in matrix form. Is this economy stationary? Propose a way of estimating the model. How to recover the structural shocks ut ?

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Structural VAR Vector Auto-Regressions:Impulse responses

In matrix form        gt 0.06 −0.3 0.27 gt−1 εg,t  it  =  0.3 0.9 1.35   it−1  +  εi,t  πt 0.2 0 0.9 πt−1 επ,t 

      εg,t 1 0 0.3 u r ,t u r ,t with  εi,t  =  0 1 1.5   ump,t  = B  ump,t . επ,t 0 0 1 un,t un,t With the procedure described above, one only gets an estimate of Ωε where   εg,t Ωε = Var  εi,t  . επ,t L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions:Impulse responses

Note however that we must have Ωε = BΩu B 0 where Ωu is diagonal positive. In addition, given the “structural” framework, one knows that B is an upper-triangular matrix. The Choleshy decomposition can therefore be used to get the B matrix.

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Structural VAR Vector Auto-Regressions:Impulse responses

Whereas the VAR model is able to capture efficiently the interactions between the different variables, it does not allow to reveal the underlying causal mecanisms since two different causal schemes can correspond to the same reduced forms. By taking into account certain economic relationships, a Structural VAR model (SVAR) makes it possible to identify structural shocks while letting play the interactions between the different variables (see Gali, 1992 or Gerlach and Smets 1995). Formally, let assume that the residuals εt are some linear combinations of the structural shocks ut , that is: εt = But .

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Structural VAR Vector Auto-Regressions:Impulse responses

How it has been shown previously, a SVAR is based on a structural model that draws from a theoretical framework. As a starting point, we always have Ωε = BΩu B 0 that provides us with n(n + 1)/2 restrictions to recover the B matrix. Consequently, to get the B matrix, one have to impose n(n − 1)/2 additional restrictions. There exist two kinds of restrictions that can be easily implemented in a SVAR: short-run and long-run restrictions: a short-run restriction prevents a structural shock from affecting an endogenous variable contemporaneously; a long-run restriction prevents a structural shock from affecting an endogenous variable in a cumulative way.

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Structural VAR Vector Auto-Regressions:Impulse responses

Concretely, the short-run restrictions consists in setting to zero some entries of B. The long-run restrictions require additional computations to be applied. More precisely, one needs to implement the computation of the cumulative effect of one of the structural shocks ut on one of the endogenous variable. Assume that we have the (reduced-form) VAR yt = c + Φ1 yt−1 + . . . Φp yt−p + εt . As was shown previously, one can always write a VAR(p) as a VAR(1), by stacking yt , yt−1 , . . . yt−p+1 in a vector yt∗ . Consequently, let us consider only the VAR(1) case: yt = c + Φyt−1 + εt . L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions:Impulse responses

Once more, let us consider the Wold’s form of yt : yt

= c + εt + Φ (c + εt−1 ) + . . . + Φk (c + εt−k ) + . . . = c + But + Φ (c + But−1 ) + . . . + Φk (c + But−k ) + . . .

Consequently, the cumulated effect of the first structural shock u1,t on the endogenous variables is obtained by computing   γ  0    (B + ΦB + . . . + Φk B + . . .)  .   ..  0 if the initial shock of u1,t is of magnitude γ. L. Ferrara (University of Paris West)

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Structural VAR Vector Auto-Regressions:Impulse responses

In this context, consider the following long-run restriction: the j th structural shock does not affect,in a cumulative way, the i th endogenous variable. Then, denoting with Θ the matrix (I + Φ + . . . + Φk + . . .)B, it comes that the entry (i, j) of Θ must be equal to zero.

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Structural VAR A simple SVAR

It can be noted that short-run restrictions are simpler to implement than long-run one. There are particular cases in which some well-known matrix decompostion can be used to easily estimate some specific SVAR. Imagine a context in which you can argue that there exists an “ordering” of the shocks: A first shock (say, ε1,t ) can affect instantaneously (i.e., in t) only one of the endogenous variable (say, y1,t ); A second shock (say, ε2,t ) can affect instantaneously (i.e., in t) the first two endogenous variables (say, y1,t and y2,t ); ...

Exercise 4 In such a context, what is the form of the matrix B? Suggest a methodology to estimate such a SVAR. L. Ferrara (University of Paris West)

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Structural VAR Cholesky decomposition: Illustration

Dedola and Lippi (“The Monetary Transmission Mechanism: Evidence from the Industries of Five OECD Countries”, 2005) estimate 5 structural VAR for the US, the UK, Germany, France and Italy to analyse the monetary-policy transmission mechanisms. They estimate an SVAR over the period 1975-1997, using 5 lags in VAR. The shock-identification scheme is based on Cholesky decompositions, the ordering of the endogenous variables being: the industrial production, the consumer price index, a commodity price index, the short-term rate, a monetary aggregate and (except for the US). This ordering implies that monetary policy reacts to the shocks affecting the first three variables but that the latter react to monetary policy with a one-period lag. L. Ferrara (University of Paris West)

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Structural VAR Responses of Macro-variables to a monetary policy shock

Figure 1

Responses of the main macro variables to a monetary policy shock !±"#$%&'(&)(#*))+)#,&'($-

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Structural VAR Responses of Macro-variables to a monetary policy shock

Note: The boxes in each column show the response of the VAR variables to a shock to the short term interest rate (equal to L. Ferrara of Paris West)by the SVAR VAR Modelsofand Applications EIPMC one (University standard deviation) yielded estimates Table 1. The error bands were computedM2 with Monte2011 Carlo

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Structural VAR Illustration: Blanchard and Quah (1989)

The article “The Dynamics Effects of Aggregate Demand and Supply Disturbances” (AER, 1989) implements long-run restrictions in a small-sized VAR. Two variables are considered GDP and unemployment. Consequently, the VAR is affected by two types of shocks. The authors want to identify supply shocks (that xan have a permanent effect on output) and demand shocks (that can not have a permanent effect on output).

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Structural VAR Illustration: Blanchard and Quah (1989)

The motivation of the authors regarding their long-run restrictions can be obtained from a traditional Keynesian view of fluctuations. The authors propose a variant of a model from Stanly and Fisher (1977) Yt

= Mt − Pt + a.θt

(2)

Yt

= Nt + θt

(3)

Pt

= Wt − θt n o = W | Et−1 Nt = N

(4)

Wt

(5)

To close the model, the aithors assume the following dynamics for the money supply and the productivity

L. Ferrara (University of Paris West)

Mt

= Mt−1 + εdt

θt

= θt−1 + εst

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Structural VAR Illustration: Blanchard and Quah (1989)

In this context, it can be shown that 4gnpt ut

= εdt − εdt−1 + a.(εst − εst−1 ) + εst = −εdt − aεst

Then, it appears that the demand shocks have no long-run impact on output. Besides, neither shocks have a long-run impact on unemployment. The endogenous variable is ( 4gnpt the logarithm of GNP.

ut ) where gdpt denotes

It is assumed to be stationary. Therefore, neither disturbances has a long-run effect on unemployment or the rate of change in output. The long-run restriction implies that the demand shocks also have no long-run effect on the output level gnp itself. L. Ferrara (University of Paris West)

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Structural VAR Illustration: Blanchard and Quah (1989)

Estimation data: quarterly, from 1950:2 to 1987:4. 8 lags.

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Structural VAR Dynamic effects of demand disturbances

THE AMERICAN ECONOMIC RE VIEW

mpositions only

amic effects of nces.

nd and Supply

SEPTEMBER 1989

1.40

1.20 1.000.80 -0.600.40

mand and supd in Figures 1 igures 1 and 2 of output and the horizontal s. Figures 3-6 , but now with ds around the

0.20 0.00

,,,

-0.20 0

10

20

30

40

-0.40

-0.60 FIGURE

1. RESPONSE TO DEMAND,-= -

=

OUTPUT,

UNEMPLOYMENT

a hump-shaped loyment. Their L. Ferrara (University of Paris West)

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, but now with ds Structural around the

-0.60 -

VAR FIGURE

1. RESPONSE TO DEMAND,-=

OUTPUT,

= UNEMPLOYMENT Dynamic effects of supply disturbances

a hump-shaped loyment. Their r quarters. The line to vanish The responses are mirror imn to this aspect cussing the ef-

allest when the allowing for a unemployment ecays the most change in the is allowed, the changes in untively unimporemand distur-

consistent with amic effects of

1.00 0.80 0.60 -/ 0.400.20 0.00 0 -0.20 -

10

20

l+q 30

ii

i 40

-0.40

-0.60FIGURE

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2.

RESPONSE TO SUPPLY, = UNEMPLOYMENT

VAR Models and Applications

OUTPUT,

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. In the intervening 7.60 output devia-VAR andStructural 7.40 peak reAt theAuto-Regressions:Impulse n. Vector responses ts an implied coef7.20 four, higher in ab7.00 Output-gap definition That 's coefficient. 1980 1970 1960 1950 is higher coefficient The output gap is the component of GNP that is explained FIGURE 7. OUTPUT FLUCTUATIONSABSENT than for demand DEMAND shocks. we expect. Suphat mand y to affect the relad employment, and 0.10 ttle or no change in 0.08

ns of Demand and rbances.

only by de-

0.06 0.04 0.02

0.007

v

amic effects of each -0.04 next step is to assess -0.06 n to fluctuations in -0.08 ent. We do this in -0.10 1980 1970 1960 1950 ormal, and entails a orical time-series of FIGURE 8. OUTPUT FLUCTUATIONSDUE TO of output to the DEMAND usiness cycles. The e decompositions of are stationof Models unemployment components VAR and demand entL.in Ferrara (University of Paris West) and Applications

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Structural VAR Variance decomposition

The k quarter-ahead forecast error in output is defined as the difference between the actual value of output and its VAR-based forecast. Variance decomposition Consider the VAR(1): yt = c + Φyt−1 + εt where the residuals are some linear combinations of structural shocks ut (εt = But ). Compute ξ k = yt+k − Et (yt+k ) with respect to ut+1 , ut+2 , . . . ut+k . How to compute the contribution of the i th structural shock on the j th endogenous variable?

L. Ferrara (University of Paris West)

VAR Models and Applications

M2 EIPMC 2011

44 / 55

Structural VAR Vector Auto-Regressions:Impulse responses

666

THE AMERICAN ECONOMIC RE VIEW

SEPTEMBER 1989

TABLE 2-VARIANCE DECOMPOSITION OF OUTPUT AND UNEMPLOYMENT (CHANGE IN OUTPUT GROWTH AT 1973/1974; UNEMPLOYMENT DETRENDED)

Percentage of Variance Due to Demand: Horizon (Quarters) 1 2 3 4 8 12 40

Output

Unemployment

99.0 (76.9,99.7) 99.6 (78.4,99.9) 99.0 (76.0,99.6) 97.9 (71.0,98.9) 81.7 (46.3,87.0) 67.6 (30.9,73.9) 39.3 (7.5,39.3)

51.9 (35.8,77.6) 63.9 (41.8,80.3) 73.8 (46.2,85.6) 80.2 (49.7,89.5) 87.3 (53.6,92.9) 86.2 (52.9,92.1) 85.6 (52.6,91.6)

TABLE 2A-VARIANCE DECOMPOSITION OF OUTPUT AND UNEMPLOYMENT (No DUMMY BREAK, TIME TREND IN UNEMPLOYMENT)

L. Ferrara (University of Paris West)

VAR Models and Applications

M2 EIPMC 2011

45 / 55

Structural VAR Smets and Tsatsaronis (1997)

“Why does the yield curve predict economic activity?” BIS Working Paper No.49. Objective: Investigating why the slope of the yield curve predicts future economic activity in Germany and the United States. Methodology: A structural VAR is used to identify aggregate supply, aggregate demand, monetary policy and inflation scare shocks and to analyse their effects on the real, nominal and term premium components of the term spread and on output. Findings: In both countries demand and monetary-policy shocks contribute to the covariance between output growth and the lagged term spread, while inflation scares do not.

L. Ferrara (University of Paris West)

VAR Models and Applications

M2 EIPMC 2011

46 / 55

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Structural VAR

Yield-curve slope (10yr-3mth) vs. Output gap, Euro area data, Source: OECD

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L. Ferrara (University of Paris West)

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