Efficient Robust Parameter Identification in Generalized Kalman Smoothing Models

Dynamic inference problems in autoregressive (AR/ARMA/ARIMA), exponential smoothing, and navigation are often formulated and solved using state-space models (SSMs), which allow a range of statistical distributions to inform innovations and errors. In many applications, the main goal is to identify not only the hidden state, but also additional unknown model parameters (e.g., AR coefficients or unknown dynamics). We show how to efficiently optimize over model parameters in SSM that use smooth process and measurement losses. Our approach is to project out state variables, obtaining a value function that only depends on the parameters of interest, and derive analytical formulas for first and second derivatives that can be used by many types of optimization methods. We illustrate this by implementing Newton, Gauss–Newton, and quasi-Newton algorithms on a numerical example. The approach can be used with smooth robust penalties such as Hybrid and the Student's T, in addition to classic least squares. We use the approach to estimate robust AR models and long-run unemployment rates with sudden changes.