# Identification in DSGE-VAR models when the number of variables in the DSGE model exceeds the number of variables in the VAR model

Given a log-linearised DSGE model: Y^{dsge}_t = M(\theta)Y^{dsge}_{t-1} + G(\theta)e_t and a corresponding VAR model Y^{var}_t = A_0 + Y^{var}_{t-1} A_1 + u_t, identification of the VAR model as explained by Del Negro and Schorfheide’s paper is kindof a map between e_t and u_t (details ignored), like we do in BVARs.

But in their paper, Y^{dsge}_t = [x_t, \pi_t, R_t] has the same number of variables as Y^{var}_t = [\Delta lnx_t,\Delta lnp_t, lnR_t] . So the link between e_t and u_t can easily be seen.

In the example on dynare wiki, however, Y^{dsge}_t has 10 variables ( var a g mc mrs n winf pie r rw y;). And given, varobs pie r rw y;, I guess Y^{var}_t has 4 variables. Maybe I am wrong here.

But if Y^{var}_t = [pie r rw y]’, then some of the shocks in the DSGE model are kinda not directly related to the innovations in the VAR model. Here, for example, e_t = {e_a, e_g, e_lam, e_ms} in the DSGE model and u_t = {u_pie, u_r, u_rw, u_y} in the VAR model. Thus, if my Y^{var}_t here is correct. I can link some of shocks, but not all, particularly e_g.

Or maybe one does not need to link all the shocks in the DSGE model to the innovations in the VAR model. Thus, after you link some of them that you are interested in, you can pick the rest arbitrarily so that they match the number of variables in the VAR model?

Typically, your DSGE model implies a (potentially infinite order) VAR in the observables. You are considering that one, not the full DSGE model. You need as many observables as shocks.

Many thanks for the reply prof. Pfeifer.

But the shocks need not be necessarily related to the observables, right?. For example, assuming we have monetary policy shock, productivity shock, and government spending shock. The observables can be, say, output, export, and exchange rate? Assuming these variables are in the model.

I also wanted to know if variables in the DSGE model and the VAR model are always the same. It seems to me not always.

In the command varobs pie r rw y;, for example, does the VAR contain only these 4 variables or all variables in the DSGE model?

There is almost never a one to one mapping that one shocks is associated with a variable. The big question is always whether the observables are useful for recovering the shocks you are interested in.

All the variables of the VAR must be contained in the DSGE model. Apart from that, I don’t think there are big restrictions on what is observed (except from each shock affecting at least one variable in VAR so as to not have stochastic singularity)

Hi prof. Pfeifer, may I kindly ask this question again.

You said before that ‘All the variables of the VAR must be contained in the DSGE model’. I read the papers by Stéphane Adjemian et.al and Del Negro and Schorfheide [2004], but it appears in both papers that the number of variables in the DSGE is the same as the number of variables in the VAR.

In dynare, is there a way to have fewer variables in the VAR model less than variables in the DSGE model? Thanks

Oh Yeah, I see that now. Many thanks. So the variables declared in the varobs command determines the structure of the VAR. For example if we have two observed variables like varobs yobs xobs, then the VAR here is bivariate, right?