Greetings. Suppose we have a model with an explicitly specified stochastic trend, as detailed in Section 5 of “A Guide to Specifying the Observation Equations for the Estimation of DSGE Models”. Further suppose that we demean the data, so that the observation equation (for, say, output) takes the form

where g_t is the rate of technological progress and g_bar is the mean. Here, y_t represents the detrended output y_t=Y_t/X_t, so that g_t=\log(X_t/X_{t-1}).

Now, suppose that we want to calculate the forecast error variance decomposition of output. Generally speaking, under the detrending procedure y_t moves in lockstep with X_t and therefore cannot be used.

Though y_t^{obs} adjusts for the detrending procedures, it is expressed as a growth rate, whereas we want a unit in levels. So, it seems that we really want the FEVD of the cumulative response of y_t^{obs}.

In the presence of a deterministic trend, like Smets and Wouters, the observation equation (demeaned) would simply be

and we could simply calculate the FEVD of log y_t, as they do in the paper.

So, am I correct that for FEVD–and for impulse responses, for that matter–we should compute the FEVD of the cumulative response of y_t^{obs} and, if so, what is the easiest way to calculate that in a Dynare mod file?

Warm regards,

Mario