Frequency Response Analysis of State Space Models for Time Series Analysis

Abstract State space models are well‐suited for time series in which the evolution of variables cannot be well represented by deterministic basis functions. A key challenge in using state space models is the selection of the noise variance parameters. To better understand their impact on the model's filtering behavior, we derive the frequency response of several commonly used state space models by utilizing the connection between the Kalman smoother and regularized least squares problems. We show that the frequency response reveals distinguishing spectral features between competing state space models and explains the flattening of the log‐likelihood function when the ratio of observation to disturbance variance becomes large. Using Dutch tide gauge data, we illustrate how the frequency response can be used to make informed decisions about the variance parameters, which could in turn support more reliable interpretations of time series data.