A note on Phillips (1991): "A constrained maximum likelihood approach to estimating switching regressions"

Phillips [Phillips R.F., 1991. A constrained maximum likelihood approach to estimating switching regressions. Journal of Econometrics 48, 241-262] proposed a constrained maximum-likelihood approach to estimating the parameters in a switching regression model. In this note, we propose a new approach which leads to a proof of a more general result than Phillips's. Specifically, we prove that the Constrained MLE (CMLE) is still strongly consistent when the constant c decreases to 0 at the rate of as n increases to [infinity], with [alpha]>1. We also suggest a suitable [alpha], hence cn, for practice based on simulation results.