Too Many Shocks Spoil the Interpretation

We show that when a model has more shocks than observed variables the estimated
filtered and smoothed shocks will be correlated. This is despite no correlation being
present in the data generating process. Additionally the estimated shock innovations
may be autocorrelated. These correlations limit the relevance of impulse responses,
which assume uncorrelated shocks, for interpreting the data. Excess shocks occur
frequently, e.g. in Unobserved-Component (UC) models, filters, including Hodrick-
Prescott (1997), and some Dynamic Stochastic General Equilibrium (DSGE) models.
Using several UC models and an estimated DSGE model, Ireland (2011), we
demonstrate that sizable correlations among the estimated shocks can result.