On the Model-Based Interpretation of Filters and the Reliability of Trend-Cycle Estimates

The paper is concerned with a class of trend cycle filters, encompassing popular ones, such as the Hodrick-Prescott filter, that are derived using the Wiener-Kolmogorov signal extraction theory under maintained models that prove unrealistic in applied time series analysis. As the maintained model is misspecified, inference about the unobserved components, and in particular their first two conditional moments, given the observations, are not delivered by the Kalman filter and smoother or the Wiener-Kolmogorov filter for the maintained model. The paper proposes a model based framework according to which the same class of filters is adapted to the particular time series under investigation; via a suitable decomposition of the innovation process, it is shown that any linear time series with ARIMA representation can be broken down into orthogonal trend and cycle components, for which the class of filters is optimal. Finite sample inferences are provided by the Kalman filter and smoother for the relevant state space representation of the decomposition. In this framework it is possible to discuss two aspects of the reliability of the signals’ estimates: the mean square error of the final estimates and the extent of the revisions. The paper discusses and illustrates how the uncertainty is related to features of the series and the design parameters of the filter, the role of smoothness priors, and the fundamental trade-off between the uncertainty and the magnitude of the revisions as new observations become available.