Testing under non-standard conditions in frequency domain: with applications to Markov regime-switching models of exchange rates and federal funds rate

We propose two test statistics in the frequency domain and derive their exact asymptotic null distributions under the condition of unidentified nuisance parameters. The proposed methods are particularly applicable in unobserved components models. Also, it is shown that the tests have considerable power when applied to a class of Markov regime switching models. We show that, after transforming the Markov regime switching model into the frequency domain representation we only have to face the issue of unidentified nuisance parameters in a nonlinear context. The singularity problem disappears. Compared to Hansen's (1992, 1996) LR-bound test of the same Markov regime switching model, our LM test performs better in terms of finite sample power, except in the special case of the Markov switching model in which the model becomes a Normal mixture model. Our test needs only a one-dimensional grid search while Hansen's (1992, 1996) test requires a three-dimensional grid search. The LM test is applied to Markov regime switching models of exchange rates and the Federal Funds rate. We used the same exchange rates data in Engel and Hamilton (1990). The null of random walk is not rejected in the exchange rates model. The null is rejected for the Federal Funds rate in subsample periods 1955:1-1979:9 and 1982:10-1995:11.