When testing for Markov switching in mean or intercept of an autoregressive
process, it is important to allow for serial correlation under the null
hypothesis of linearity. Otherwise, a rejection of linearity could merely reflect
misspecification of the persistence properties of the data, rather than any inherent nonlinearity. However, Monte Carlo analysis reveals that the Carrasco, Hu, and Ploberger (Optimal test for Markov Switching parameters, conditionally accepted at Econometrica, 2012) test for Markov switching has low power for empirically relevant data-generating processes when allowing for serial correlation under the null. By contrast, a parametric bootstrap likelihood ratio test of Markov switching has higher power in the same setting. Correspondingly, the bootstrap likelihood ratio test provides stronger support for a Markov-switching mean in an application to an autoregressive model of quarterly US real GDP growth.