"Frontmatter". In: Analysis of Financial Time Series

136 NONLINEAR TIME SERIESMarkov chain to govern the transition from one conditional mean function to another.This is different from that of a SETAR model for which the transition is determinedby a particular lagged variable. Consequently, a SETAR model uses a deterministicscheme to govern the model transition, whereas an MSA model uses a stochasticscheme. In practice, the stochastic nature of the states implies that one is never certainabout which state x t belongs to in an MSA model. When the sample size islarge, one can use some filtering techniques to draw inference on the state of x t .Yetas long as x t−d is observed, the regime of x t is known in a SETAR model. This differencehas important practical implications in forecasting. For instance, forecasts ofan MSA model are always a linear combination of forecasts produced by submodelsof individual states. But those of a SETAR model only come from a single regimeprovided that x t−d is observed. Forecasts of a SETAR model also become a linearcombination of those produced by models of individual regimes when the forecasthorizon exceeds the delay d. It is much harder to estimate an MSA model than othermodels because the states are not directly observable. Hamilton (1990) uses the EMalgorithm, which is a statistical method iterating between taking expectation andmaximization. McCulloch and Tsay (1994) consider a Markov Chain Monte Carlo(MCMC) method to estimate a general MSA model. We discuss MCMC methods inChapter 10.McCulloch and Tsay (1993) generalize the MSA model in Eq. (4.16) by letting thetransition probabilities w 1 and w 2 be logistic, or probit, functions of some explanatoryvariables available at time t −1. Chen, McCulloch, and Tsay (1997) use the ideaof Markov switching as a tool to perform model comparison and selection betweennon-nested nonlinear time series models (e.g., comparing bilinear and SETAR models).Each competing model is represented by a state. This approach to select a modelis a generalization of the odds ratio commonly used in Bayesian analysis. Finally,the MSA model can easily be generalized to the case of more than two states. Thecomputational intensity involved increases rapidly, however. For more discussions ofMarkov switching models in econometrics, see Hamilton (1994, Chapter 22).Example 4.4. Consider the growth rate, in percentage, of U.S. quarterly realgross national product (GNP) from the second quarter of 1947 to the first quarterof 1991. The data are seasonally adjusted and shown in Figure 4.3, where a horizontalline of zero growth is also given. It is reassuring to see that a majority ofthe growth rates are positive. This series has been widely used in nonlinear analysisof economic time series. Tiao and Tsay (1994) and Potter (1995) use TAR models,whereas Hamilton (1989) and McCulloch and Tsay (1994) employ Markov switchingmodels.Employing the MSA model in Eq. (4.16) with p = 4 and using a Markov ChainMonte Carlo method, which is discussed in Chapter 10, McCulloch and Tsay (1994)obtain the estimates shown in Table 4.1. The results have several interesting findings.First, the mean growth rate of the marginal model for State 1 is 0.909/(1 −0.265 − 0.029 + 0.126 + 0.11) = 0.965 and that of State 2 is −0.42/(1 − 0.216 −0.628 + 0.073 + 0.097) =−1.288. Thus, State 1 corresponds to quarters with positivegrowth, or expansion periods, whereas State 2 consists of quarters with negative