Dynamic Regression and Filtered Data Series: A Laplace Approximation to the Effects of Filtering in Small Samples
研究了线性滤波对动态回归模型参数在小样本下偏差和均方误差的影响，利用拉普拉斯近似简化计算，并通过蒙特卡洛模拟验证近似效果。
It is common for an applied researcher to use filtered data, like seasonally adjusted series, for instance, to estimate the parameters of a dynamic regression model. In this paper, we study the effect of (linear) filters on the distribution of parameters of a dynamic regression model with a lagged dependent variable and a set of exogenous regressors. So far, only asymptotic results are available. Our main interest is to investigate the effect of filtering on the small sample bias and mean squared error. In general, these results entail a numerical integration of derivatives of the joint moment generating function of two quadratic forms in normal variables. The computation of these integrals is quite involved. However, we take advantage of the Laplace approximations to the bias and mean squared error, which substantially reduce the computational burden, as they yield relatively simple analytic expressions. We obtain analytic formulae for approximating the effect of filtering on the finite sample bias and mean squared error. We evaluate the adequacy of the approximations by comparison with Monte Carlo simulations, using the Census X-11 filter as a specific example