Single and multiple error state-space models for signal extraction

We compare the results obtained by applying the same signal-extraction procedures to two observationally equivalent state-space forms. The first model has different errors affecting the states and the observations, while the second has a single perturbation term which coincides with the one-step-ahead forecast error. The signals extracted from both forms are very similar but their variances are drastically different, because the states for the single-source error representation collapse to exact values while those coming from the multiple-error model remain uncertain. The implications of this result are discussed both with theoretical arguments and practical examples. We find that single error representations have advantages to compute the likelihood or to adjust for seasonality, while multiple error models are better suited to extract a trend indicator. Building on this analysis, it is natural to adopt a ‘best of both worlds’ approach, which applies each representation to the task in which it has comparative advantage.     

Decomposition of a state-space model with inputs

This article shows how to compute the in-sample effect of exogenous inputs on the endogenous variables in any linear model written in a state–space form. Estimating this component may be either interesting by itself, or a previous step before decomposing a time series into trend, cycle, seasonal and error components. The practical application and usefulness of this method is illustrated by estimating the effect of advertising on the monthly sales of Lydia Pinkham's vegetable compound.     

Decomposition of Seasonality and Long‐term Trend in Seismological Data: A Bayesian Modelling of Earthquake Detection Capability

This study demonstrates the decomposition of seasonality and long‐term trend in seismological data observed at irregular time intervals. The decomposition was applied to the estimation of earthquake detection capability using cubic B‐splines and a Bayesian approach, which is similar to the seasonal adjustment model frequently used to analyse economic time‐series data. We employed numerical simulation to verify the method and then applied it to real earthquake datasets obtained in and around the northern Honshu island, Japan. With this approach, we obtained the seasonality of the detection capability related to the annual variation of wind speed and the long‐term trend corresponding to the recent improvement of the seismic network in the studied region.