Davies Test

Various types of switching models face a technical problem in testing for their very existence (vs a simpler linear model) because the more complicated model has one or more parameters which are unidentified under the null (that the second regime is unnecessary). This violates a key assumption in the theorem that gives rise to an asymptotic chi-squared distribution for the likelihood ratio test.

Davies(1987) proposes an adjustment to the critical value of the likelihood ratio test to correct for this, under certain conditions. Garcia and Perron(1996) employed this for the case of testing for a Markov Switching model vs a linear model and (citing G&P) it has been used by other authors in later papers on the subject. However, Garcia and Perron isn’t a proper use of Davies. First of all, Davies allows for only a single parameter that’s unidentified under the null, while a two-regime Markov switching has two. (Davies adjustment is based upon approximating an integral over that single parameter). It also only applies to a least squares model where the “regime” is determined (exactly) based upon that one parameter, while the Markov Switching model only provides a probability of each regime.

Hansen(1996) actually looks at the behavior of the Davies adjustment when applied to the simpler case of the threshold autoregression (which does meet the two requirements described above) and shows (analytically) that the likelihood ratio statistic in that case fails two other key assumptions required by Davies, and that (on simulation) it fails rather badly to provide a good approximation.

Formal testing procedures for the number of regimes (and, in particular, for one regime vs two) are generally quite complicated, involving simulations. Frühwirth-Schnatter(2006) cites a number of papers which have chosen the relatively simple Schwarz Bayesian Criterion (which requires only estimating both models). While the SBC has similar problems with the lack of identification under the null (in deriving the result that it gives an “asymptotic” estimate of the posterior odds), it seems to work adequately in practice.