Signal extraction and filtering by linear semiparametric methods

Signal extraction deals with weighting the available observations in order to estimate a latent feature of interest. A signal extraction method is linear if the feature is measured by a possibly time-varying linear combination of the available observations. Linear methods play an important role since they are well understood, easy to apply, and are a key ingredient in more elaborate nonlinear and non-Gaussian models. The focus is on the main methods for inference about parametric and semiparametric unobserved components models formulated as linear mixed models and state space models and establish the connections between best linear unbiased prediction, penalised least squares and recursive methods of signal extraction. The methods are illustrated with reference to the traditional problem of extracting the cycle and the trend from economic time series.