Recoverability

When can structural shocks be recovered from observable data? We present a necessary and sufficient condition that gives the answer for any linear model. Invertibility, which requires that shocks be recoverable from current and past data only, is sufficient but not necessary. This means that semi-structural empirical methods like structural vector autoregression analysis can be applied even to models with non-invertible shocks. We illustrate these results in the context of a simple model of consumption determination with productivity shocks and non-productivity noise shocks. In an application to postwar U.S. data, we find that non-productivity shocks account for a large majority of fluctuations in aggregate consumption over business cycle frequencies.