Risk minimization for time series binary choice with variable selection

This paper considers the problem of predicting binary choices by selecting from a possibly large set of candidate explanatory variables, which can include both exogenous variables and lagged dependent variables. We consider risk minimization with the risk function being the predictive classification error. We study the convergence rates of empirical risk minimization in both the frequentist and Bayesian approaches. The Bayesian treatment uses a Gibbs posterior constructed directly from the empirical risk instead of using the usual likelihood-based posterior. Therefore these approaches do not require a correctly specified probability model. We show that the proposed methods have near optimal performance relative to a class of linear classification rules with selected variables. Such results in classification are obtained in a framework of dependent data with strong mixing.