Structural VAR Models and Applications Laurent Ferrara1 1 University

of Paris West

M2 EIPMC 2015

Overview of the presentation

1. Structural Vector Auto-Regressions I I

Definition Identification

2. Impulse response function (IRF) I I

Concept General IRF

3. Applications 3.1 Smets-Tsataronis (1995) 3.2 Kilian (AER, 2009)

Usefulness of Structural-VAR

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See notes

IRF from SVAR with Inflation, Unemployment and FFR

SVAR: Causal mechanisms in a stylized economy I

Assume that the GDP growth gt is affected by some real shocks ur ,t following: gt = −0.3(it−1 − πt ) + ur ,t

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where it is the nominal interest rate, πt is the inflation rate. Besides, assume that we have a simple Taylor rule governing short-term interest rates : it = 0.9it−1 + 1.5πt + ump,t

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where ump,t is a monetary-policy shock. The inflation rate is supposed to follow a backward-looking Phillips curve: πt = 0.9πt−1 + 0.2gt−1 + un,t where un,t is a cost-push shock.

SVAR: Causal mechanisms in a stylized economy

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The structural shocks ut = (ur ,t , ump,t , un,t ) are uncorrelated (i.e., the covariance matrix of ut , denoted with Ωu is diagonal)

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The structural shocks ut are serially uncorrelated (i.e. Cov (ut−k , ut ) = 0 for any t and k > 0).

SVAR: Causal mechanisms I

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 .

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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 .

SVAR: Causal mechanisms I

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 with  εi,t  =  0 1 1.5   ump,t  = B  επ,t 0 0 1 un,t With the procedure described above, one only gets estimate of Ωε where   εg ,t Ωε = Var  εi,t  . επ,t 

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 u r ,t ump,t . un,t an

SVAR: Causal mechanisms

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Note however that we must have Ωε = BΩu B 0 where Ωu is diagonal positive.

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In addition, given the “structural” framework, one knows that B is an upper-triangular matrix.

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The Cholesky decomposition can therefore be used to get the B matrix.

SVAR: Causal mechanisms

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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.

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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

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Formally, let assume that the residuals εt are some linear combinations of the structural shocks ut , that is: εt = But .

SVAR: Orthogonalization

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It has been shown previously that a SVAR is a structural model that draws from a theoretical framework.

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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.

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Consequently, to get the B matrix, one have to impose n(n − 1)/2 additional restrictions.

SVAR: Restrictions

There exist several of restrictions that can be implemented in a SVAR: restrictions: I

a short-run restriction prevents a structural shock from affecting an endogenous variable contemporaneously by setting to zero some entries of B.

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a long-run restriction prevents a structural shock from affecting an endogenous variable in a cumulative way.

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a sign restriction imposes negative or positive parameters in matrix B.

SVAR: Short-Run restrictions

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Short-run restrictions are simpler to implement than the other ones.

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There are particular cases in which some well-known matrix decompostion can be used to easily estimate some specific SVAR.

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Imagine a context in which you can argue that there exists an “ordering” of the shocks: I

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A first shock, u1,t ), can affect instantaneously (i.e., in t) only one of the endogenous variable (say, y1,t ); A second shock, u2,t , can affect instantaneously (i.e., in t) the first two endogenous variables (say, y1,t and y2,t ); ...

SVAR: Short-Run restrictions

Within this framework, the structural mechanism of variable ordering allows the creation of the B matrix   y1,t = c1 + u1,t      y2,t = c2 + β21 y1,t + u2,t y3,t = c3 + β31 y1,t + β32 y2,t + u3,t    ... = ...     yn,t = cn + βn1 y1,t + . . . + βn,n−1 yn−1,t + un,t where the ut are the structural shocks and are non-correlated

SVAR: Short-Run restrictions

The triangular structural model imposes the recursive causal ordering: y1 → y2 → . . . → yn I

y1 causes y2 , y3 , . . . , yn

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but y2 , y3 , . . . , yn do not cause y1

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y2 causes y3 , . . . , yn

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but y3 , . . . , yn do not cause y2

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...

Obviously, a different ordering of variables leads to a different result

Cholesky decomposition: Illustration I

Dedola and Lippi (“The Monetary Transmission Mechanism: Evidence from the Industries of Five OECD Countries”, 2005) estimate 5 SVAR for the US, the UK, Germany, France and Italy to analyse the monetary-policy transmission mechanisms

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SVAR(p = 5) estimation over the period 1975-1997

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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 and a monetary aggregate (except for the US).

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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.

Responses of Macro-variables to a monetary policy shock Figure 1

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

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

Smets and Tsatsaronis (1997) I

“Why does the yield curve predict economic activity? ” BIS Working Paper No.49.

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Objective: Investigating why the slope of the yield curve predicts future economic activity in Germany and the United States.

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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.

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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.

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

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