Regularization and selection in Gaussian mixture of autoregressive models

Gaussian mixtures of autoregressive models can be adopted to explain heterogeneous behaviour in mean, volatility, and multi‐modality of the conditional or marginal distributions of time series. One important task is to infer the number of autoregressive regimes and the autoregressive orders. Information‐theoretic criteria such as aic or bic are commonly used for such inference, and typically evaluate each regime/autoregressive combination separately in order to choose an optimal model. However the number of combinations can be so large that such an approach is computationally infeasible. In this article we first develop a computationally efficient regularization method for simultaneous autoregressive‐order and parameter estimation when the number of autoregressive regimes is pre‐determined. We then propose a regularized Bayesian information criterion ( rbic ) to select the number of regimes. We study asymptotic properties of the proposed methods, and investigate their finite sample performance via simulations. We show that asymptotically the rbic does not underestimate the number of autoregressive regimes, and provide a discussion on the current challenges in investigating whether and under what conditions the rbic provides a consistent estimator of the number of regimes. We finally analyze U.S. gross domestic product growth and unemployment rate data to demonstrate the proposed methods. The Canadian Journal of Statistics 45: 356–374; 2017 © 2017 Statistical Society of Canada