A NONPARAMETRIC BAYESIAN APPROACH TO DETECT THE NUMBER OF REGIMES IN MARKOV SWITCHING MODELS

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Acknowledgment

We thank D. Böhning for some useful comments on an earlier version of the paper and to ESEM 99 participants (Santiago de Compostela). The paper greatly benefitted from the suggestions of two anonymous referees and the Associate Editor. Thanks are also due to Jim Hamilton for providing the data used in Engel and Hamilton[2] and to Charles Engel for the data used in his 1994 article. Financial support from Italian MURST (grant “ex40%-1998—Econometric Modeling of Structural Change”) is gratefully acknowledged.

Notes

We thank an anonymous referee for pointing this out to us.

This issue should be kept separate from the estimation of the switching models in a Bayesian context; for example, Carter and Kohn22-23 and Shephard[24] propose Monte Carlo Markov Chain methods to estimate a general model which encompasses the switching model; Albert and Chib[25] and McCulloch and Tsay[26] use the Gibbs sampling to estimate the MS model; Hamilton himself[1] uses a quasi-Bayesian approach to bypass singularity problems in estimation. But in this context the number of regimes is considered fixed a priori.

Some results which we will be referring to are reported in Engel[27] and are not present in the 1994 version.

In the presence of an autoregressive component, the conditional independence of ( y _t |θ _t ) is lost. The rationale for working with a DGP with dependent observations is to check the robustness of our procedure relative to the violation of this assumption.

Even on a Pentium III with 1 Ghz processor, the time required to generate the results is extensive, given the high number of times the Gibbs sampler is run, so that experiments with a larger sample size would become very demanding.

We thank the Associate Editor for pointing out this possibility to us.

The log-likelihood function of a Markov switching model presents numerous local maxima, so that the final estimation depends on the starting values. To choose the starting values, we have selected various grids for unknown parameters, starting from the combination with highest log-likelihood. The grids are: μ₁ ∈ [0,5] and μ₂ ∈ [ − 5,0] both with step-length 1; σ² ∈ [8,28] with step-length 2; p ₁₁ ∈ [0.5,0.95] and p ₂₂ ∈ [0.5,0.95] both with step-length 0.05, for a total of 39,600 combinations.

We obtain the following values for the parameters (standard errors are in parentheses):

The asymptotic theory for ML estimation of MS models is by no means standard. The consistency of the ML estimator and the consistency and asymptotic normality of the pseudo-ML estimator are ensured under given conditions as shown by Francq and Roussignol.[10]

We have had numerical problems calculating the prior with 8 in the Appendix, using T = 214 as in Engel.[27] Following the result by Escobar and West[21] that the prior is not sensible to changes in T, when the sample size is sufficiently high, we have calculated the prior fixing T = 170.

The convergence is in distribution, so that every θ _i is considered sampled from the posterior distribution (𝚯 _T | Y _T ).