Koopman, S. and Wong, S. (2011). Kalman filtering and smoothing for model-based signal extraction that depend on time-varying spectra Journal of Forecasting, 30(1):147--167.

We develop a flexible semi-parametric method for the introduction of time-varying parameters in a model-based signal extraction procedure. Dynamic model specifications for the parameters in the model are not required. We show that signal extraction based on Kalman filtering and smoothing can be made dependent on time-varying sample spectra. Our new procedure starts with specifying the time-varying spectrum as a semi-parametric flexible spline function that can be formulated in state space form and can be treated by multivariate Kalman filter and smoothing methods. Next we show how a time series decomposition model can be made dependent on a time-varying sample spectrum in a frequency domain analysis. The key insight is that the spectral likelihood function depends on the sample spectrum. The estimates of the model parameters are obtained by maximizing the spectral likelihood function. A time-varying sample spectrum leads to a time-varying spectral likelihood and hence we obtain time-varying parameter estimates. The time series decomposition model with the resulting time-varying parameters reflect the time-varying spectrum accurately. This approach to model-based signal extraction includes a bootstrap procedure to compute confidence intervals for the time-varying parameter estimates. We illustrate the methodology by presenting a business cycle analysis for three quarterly US macroeconomic time series between 1947 and 2010. The empirical study provides strong evidence that the cyclical properties of macroeconomic time series have been changing over time. Copyright {\textcopyright} 2010 John Wiley \& Sons, Ltd.