Exclusion restrictions as instrumental variables in VAR

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joe_p
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Joined: Fri Mar 25, 2022 2:01 pm

Exclusion restrictions as instrumental variables in VAR

Postby joe_p » Tue Apr 05, 2022 7:44 am

I'd like to know if my intuition behind exclusion restrictions that we place to identify the structural vector autoregressive (VAR) models is correct.
Let's say we have a very simple 2-dimensional system described by two equations (z,c). We identify the structural coefficients of the VAR by assuming that the contemporaneous correlation matrix A0 is lower triangular. We do this by applying the Cholesky decomposition to the estimated variance-covariance matrix of the VAR in reduced form. Thus, the just identified structural system will have a recursive structure, as follows:

A0 yt=∑p Ai yt−i + ϵt

[a11,0; a21, a22] [zt, ct]' = ∑pi Ai yt−i + ϵt
where A0 =[a11,0; a21, a22] is a 2X2 lower triangular matrix
and yt= [zt,ct]' is my vector of endogenous

This implies that the first variable does not respond contemporaneously to the second. The second, on the other hand, respond contemporaneously to the first. This means that the first variable is fully exogenous concerning the second. Let's say, I want to estimate the system equation by equation by OLS. I can do this for the first equation by regressing the first variable, i.e. z, on its lags and the lags of the second variable, i.e. c

However, I cannot estimate the second equation by OLS, since the regressor, z enters contemporaneously in the second equation, namely, it is endogenous. This is true because z is a linear combination of c and this implies that z is correlated with the error of the second equation (correct?). What I can do is run an IV regression by using the estimated residual of the first equation as an instrument for z. Indeed, this residual is correlated with z but uncorrelated with the residual of the second equation.

Is my intuition correct?

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