Gray JFE 1996

Gray(1996) was one of two papers (the other is Dueker(1997)) written in the mid 1990's proposing methods of implementing Markov Switching GARCH (MS-GARCH) models. Because the GARCH recursion depends upon the unobservable lagged variance (unlike an ARCH, which requires only observable lagged residuals), the exact analysis of MS-GARCH is infeasible. Gray and Dueker proposed different methods for collapsing the information in past regimes down to a manageable level. A detailed comparison of the Dueker and Gray filters is included in the Structural Breaks and Switching Models e-course.

Gray's paper applies a variety of models (both Markov switching and fixed regimes) to weekly data on one-month US Treasury Bill rates over the period from 1970 to 1994. This includes several episodes of anti-inflation monetary policy, which is reflected in the estimates in several of the models. In addition to a standard GARCH model, Gray includes a Generalized Regime Switching (or GRS) model where the variance evolves according to

(1) \({h_t}(i) = \sigma _i^2{r_{t - 1}} + {b_{1i}}\varepsilon _{t - 1}^2 + {b_{2i}}{h_{t - 1}}\)

that is, the "variance constant" in the GARCH recursion isn't constant, but changes with the lagged variance. Note that this is specific to this being a model of interest rates, and wouldn't be appropriate if the dependent variable were (say) stock market returns. Also note that while the theoretical description of this in the paper has the first term depending upon the square root of the lagged variance, the actual empirical work uses the straight lagged value. Gray's model has an AR(1) mean model, conditionally Normal residuals, with all parameters in both the mean and variance models switching with the regime; however, use of the Gray filter for handling the path-dependence in the MS-GARCH does not depend upon those specifics of the model. Gray's filter collapses the variance and residual immediately (so the lagged squared residual and lagged variance in (1) don't depend upon the lagged regime), while Dueker's waits a period.

The Gray paper (and example program) does quite a few models, starting with the very basic—the only thing common to all of these is the AR(1) mean model which fully switches with regime in any regime-switching model.

1.Single regime least squares

2.Markov switching "fixed variance" models.

3.Single regime GARCH

4.Markov switching GARCH

5.Single regime GRS

6.Markov switching GRS with constant transition probabilities

7.Markov switching GRS with time-varying transition probabilities

Numbers 4, 6 and 7 depend upon the use of the approximation for the likelihood. In all three cases, there is an alternative mode with a much higher log likelihood than the published one, but where the high log likelihood appears to be the result of a failure of the approximation. This makes estimating this model quite difficult, because simply doing what one can to maximize the (approximated) log likelihood doesn't necessarily give the "correct" results. The replication of the published results requires a two-step estimation process that enforces a high level of persistence in the regimes.