Maximum Likelihood Estimation of Regression Effects in State Space Models

Abstract Unknown parameters, including regression coefficients, in state space models can be estimated by maximum likelihood. An alternative approach is to augment the state vector to include regression coefficients. However, the state estimator obtained by the Kalman filter is numerically different from the maximum likelihood estimator. We address the discrepancy by a novel method based on proper distributions returned by the ordinary Kalman filter without dependency on diffuse initialization. We prove that maximizing a low-dimensional objective function that combines the likelihood, the filtering mean and variance can reproduce the high-dimensional maximum likelihood results.