Cannot have more observed variables than shocks?


This document claims that, on page 15, that in a Dynare estimiation:
• Cannot have more observed variables than shocks in your model
source: … mation.pdf

Is the above claim true?

Yes, that is true. The problem is known as “stochastic singularity”. The problem is that in a linear model if you have fewer shocks than observables, there will be an exact linear combination implied between observables. That in turn means that the density of observing those variables is either 1 (the exact linear combination in the model holds in the data; rarely the case) or 0 (the exact linear combination does not hold; the typical case). As you can imagine with a log density of minus infinity, estimation will not work.

I see. Is there a way to achieve overidentification then, i.e.,having more observables than shocks?
For example, with SMM in dynare.

I would not call that

Overidentification would mean that you have more restrictions than estimated parameters, which is the case with full information Bayesian estimation. You are targeting the full density of the data, i.e. the means and all variances and covariances at all leads and lags. Your data does not feature more restrictions than that. The problem with stochastic singularity is that your model may not be able to capture the actual density of the data, because likelihood implied by the model is singular/degenerate. Doing SMM with more shocks than observables would not really be a way out, because stochastic singularity will imply that the moments for the additional variable you want to plug in are an exact linear combination of the moments of the other variables. Thus, you could e.g. not use the optimal weighting matrix, because it will be singular as well.

A better approach is to ask yourself why you want more observables than shocks, i.e. why does your model not have more sources of stochastic variation explaining the data. Related to this: stochastic singularity is only a problem if the data does not satisfy the linear restriction implied by the model. Why does your data not satisfy this restriction? Often people then argue that the reason is measurement error in the data. Adding such a measurement error would alleviate stochastic singularity and may be a way to process for you. See also Pfeifer (2013): “A Guide to Specifying Observation Equations for the Estimation of DSGE Models” on this