Hamilton and Susmel(1994), and independently, Cai(1994), proposed a Markov switching ARCH model as an alternative to a (standard) GARCH model. In these models, instead of the lagged variance term providing the strong connection for volatility from one period to the next, a Markov model governs switches between several variance regimes. Cai’s variation on this is used in the SWARCH.RPF example while there’s a replication example for the Hamilton-Susmel paper.

 

Given the dominance of the GARCH model over the ARCH in analyzing volatility, one might ask why those authors chose to do an MS-ARCH rather than MS-GARCH. There’s a good practical reason for this—an exact analysis of an MS-GARCH model is nearly impossible because the lagged variance isn’t observable and depends upon the entire prior history of the regimes up until that time. By contrast, the MS-ARCH uses only a finite number of lagged (observable) residuals.

 

To make MS-GARCH feasible, the likelihood must be approximated by using some method to summarize the regime history in a finite number of lags. There are two principal “filter” methods that have been proposed: Gray(1996) collapses it right away, so there is just a single lagged variance, while Dueker(1997) collapses it after a one period lag, so each regime at \(t-1\) has its own variance. Of the two, Dueker’s is actually simpler to use in practice and more general and probably more accurate. Despite that, most other published work with MS-GARCH models has used Gray’s filter.

 

We should note at this point, that several published papers (and many proposed projects from RATS users) have used or tried to use MS-GARCH when some other form of switching model was more appropriate.

 

An interesting question is why the superior alternative was largely pushed aside. Probably the best explanation is that Dueker’s application was not particularly convincing. Dueker analyzed daily S&P 500 returns. Despite trying four different combinations of switching parameters, none of his MS-GARCH models proved to be clearly better than an analogous non-switching GARCH.

 

By contrast, Gray worked with weekly US T-bill yields over a sample which included some fairly dramatic (and readily identifiable) changes, and his switching models fit substantially better than their non-switching counterparts.

 

In the 2nd edition of the Structural Breaks and Switching Models e-course, we demonstrate how to use both filters and apply both to the same model and data set (Gray’s). Both filters give fairly similar results with two highly persistent regimes, but the Dueker filter produces an (approximated) log likelihood that is higher (by +10 on 1266 datapoints), most likely because the Dueker filter is more accurate (by carrying the history for one extra step).

 

However, it should also be noted that both filters with the same model and same data produce a second “mode” which has a much higher log likelihood (for Dueker’s filter, 206 vs 171, with an even larger gap for Gray’s). This other mode has regimes which aren’t persistent and show very odd behavior (one with a lagged variance coefficient of 2.5). We had noted this in the replication program for Gray’s paper that’s been distributed with RATS for many years. Upon more careful analysis (using a particle filter), it turns out that the two filters fail badly to produce a good approximation to the unobservable lagged variance when the regimes aren’t persistent, and that failure can produce a misleadingly high log likelihood.

 

If you’re interested in this type of model, our recommendation is first, to make sure that it’s really the type of model that will answer the questions you’re asking, and if it is, use the Dueker filter and check the results carefully.

 

Copyright © 2025 Thomas A. Doan