DYNAMIC REGRESSION AND FILTERED DATA SERIES - A LAPLACE APPROXIMATIONTO THE EFFECTS OF FILTERING IN SMALL SAMPLES

It is common for an applied researcher to use filtered data, like seas onally adjusted series, for instance, to estimate the parameters of a dynamic regression model. In this paper, we study the effect of (linea r) filters on the distribution of parameters of a dynamic regression m odel with a lagged dependent variable and a set of exogenous regressor s. So far, only asymptotic results are available. Our main interest is to investigate the effect of filtering on the small sample bias and m ean squared error, In general, these results entail a numerical integr ation of derivatives of the joint moment generating function of two qu adratic forms in normal variables. The computation of these integrals is quite involved. However, we take advantage of the Laplace approxima tions to the bias and mean squared error, which substantially reduce t he computational burden, as they yield relatively simple analytic expr essions. We obtain analytic formulae for approximating the effect of f iltering on the finite sample bias and mean squared error. We evaluate the adequacy of the approximations by comparison with Monte Carlo sim ulations, using the Census X-ll filter as a specific example.