INTEREST-RATE RULES: Framework for Analysis 7

The resulting historical time series for the two disturbances st could then be identified with the residuals of the model’s structural equations further.

If the model fit the properties of the U.S. time series perfectly, the vector st constructed in this way would be orthogonal to the identified monetary policy disturbance In practice, the right hand side of (1.26) does depend upon the first element of the vector of VAR residuals etl which we identify as the monetary policy disturbance. Perhaps more troubling is the observation that if the real disturbances st are generated by a law of motion of the kind implied by conjoining (1.26) with equation (1.25) for the evolution of Zu then we should not expect all three of the independent structural disturbances et that matter for the evolution of st to be revealed by data on the three variables in Zt alone. (This is because one of the VAR innovations corresponds to the monetary policy shock, so that only the other two orthogonal innovations can reveal information about the real disturbances.)

But this would mean that forecasts of the future values of the variables in Zt using the VAR should not correspond, in principle, to the expectations of these variables conditional upon the public’s information set (assuming that the public has complete information about the structural disturbances); and thus our method for identifying the historical series for our structural equation residuals would not be internally consistent.

We prefer instead to work with a theoretical model not subject to this last problem, i.e., one in which the evolution of the real disturbances st depends only upon two orthogonal disturbances each period, which then should in principle correspond to the two VAR residuals e2t and ёзt. The structural disturbances st of our theoretical model then have moments that do not correspond precisely to those of the residuals of our model equations; but this discrepancy will exist only insofar as our model (quite apart from the law of motion chosen for the structural disturbances) is in fact inconsistent with the estimated VAR (and in particular, with the estimated impulse responses to a monetary policy shock). We accordingly consider a law of motion
where В is the same as in (1.25), and W corresponds to U with the first column deleted.