Least Squares Regression: Graduation and Filters

This chapter provides an introduction to smoothing methods in time series analysis, namely local polynomial regression and polynomial splines, that developed as an extension of least squares regression and result in signal estimates that are linear combinations of the available information. We set off exposing the local polynomial approach and the class of Henderson filters. Very important issues are the treatment of the extremes of the series and real time estimation, as well as the choice of the order of the polynomial and of the bandwidth. The inferential aspects concerning the choice of the bandwidth and the order of the approximating polynomial are also discussed. We next move to semiparametric smoothing using polynomial splines. Our treatment stresses their relationship with popular stochastic trend models proposed in economics, which yield exponential smoothing filters and the Leser or Hodrick-Prescott filter. We deal with signal extraction filters that arise from applying best linear unbiased estimation principles to the the linear mixed model representation of spline models and establish the connection with penalised least squares. After considering several ways of assessing the properties of a linear filter both in time and frequency domain, the chapter concludes with a discussion of the main measurement issues raised by signal extraction in economics and the accuracy in the estimation of the latent signals.