## 1. INTRODUCTION

US unemployment is characterized by relatively brief periods of rapid economic contraction (rising unemployment) followed by relatively extended periods of slow economic expansion (falling unemployment). In the recent literature, several studies (Rothman, 1998; Montgomery *et al.*, 1998; Koop and Potter, 1999; Van Dijk *et al.*, 2002) have attempted to capture this salient feature by means of well-known nonlinear models such as threshold autoregression (TAR), the closely related logistic smooth transition autoregression (LSTAR), and Markov switching autoregression (MSAR). The first two papers attempt to base model choice on a comparison of forecasting performance.

There are two important conceptual differences between the MSAR and the TAR or LSTAR models. First, MSAR incorporates less prior information than TAR and LSTAR. Indeed, a filtered or smoothed regime probability in an MSAR model can be interpreted as a transition function which is estimated flexibly from the data. By contrast, specifying the transition function in a TAR or LSTAR model necessitates the choice of a transition variable (a difficult problem). Secondly, regime changes are predetermined in a TAR or LSTAR model, but are exogenous in the MSAR: in the latter, even if the model parameters were known, these changes could not be predicted with certainty from past data due to the presence of additional disturbances (in the Markov evolution equation). It is of interest to investigate whether the added flexibility and complexity of the MSAR model result in a superior predictive ability.

For the reasons given by West and McCracken (1998), it may be important to base such an investigation on small-sample predictive densities that take parameter uncertainty into account. In the context of maximum likelihood estimation, this requires using the bootstrap, and involves the repeated estimation of nonlinear models by local optimization algorithms. Convergence difficulties may make this impractical: see, for example, Chan and McAleer (2002, 2003). These difficulties do not arise if Bayesian methods are used: Markov chain Monte Carlo (MCMC) is used for simulating the joint posterior, and the resulting parameter replications are used for dynamic simulations of future observations.

In a Bayesian context, a prior identification constraint on the parameters of Markov switching models should be imposed; if this is not done, a multimodal posterior is obtained, except by spurious accident when mixing in the MCMC sampler is poor. This constraint can take the form θ_{1} < θ_{2} < · < θ_{K}, where θ_{i} is a particular population parameter in regime *i* and *K* is the number of regimes. The permutation sampler proposed by Frühwirth-Schnatter (2001) provides an effective procedure both for choosing an appropriate prior identification constraint, and for subsequently imposing this constraint. An MCMC posterior simulator for LSTAR models has been proposed by Lopes and Salazar (2006).

Among the authors mentioned in the first paragraph, only Montgomery *et al.* (1998) investigate the forecasting performance of an MSAR model; and only Koop and Potter (1999) fully rely on Bayesian methods. Even though the MSAR model in Montgomery *et al.* (1998) is estimated by the Gibbs sampler, the presentation is frequentist: the authors do not present their prior specification (including those aspects of the prior that are relevant for model identification) and only discuss point estimates and point forecasts. Koop and Potter (1999) provide a thorough Bayesian treatment of a TAR model of US unemployment; however, they do not compare its forecasting performance with that of an MSAR model.

An LSTAR formulation can approximate the TAR models used in Montgomery *et al.* (1998) and Koop and Potter (1999), but can be estimated with standard econometric software (contrary to the TAR and MSAR models), and is therefore particularly convenient. Van Dijk *et al.* (2002) have been the only authors to investigate the forecasting accuracy of an LSTAR model of the US unemployment rate. However, they do not update the parameter estimates as new observations become available, presumably for the reasons given in our third paragraph.

As pointed out by Koop and Potter (1999), there are important benefits in using a logistic transformation of the unemployment rate. This transformation not only guarantees that predictions are restricted to the unit interval (an important consideration if the emphasis is on predictive densities), but also removes the strong residual leptokurticity which plagues a model estimated from untransformed data. Among the four contributions mentioned in the first paragraph, only the paper by Koop and Potter (1999) uses such a transformation.

On these grounds, and since the permutation sampler has only recently become available, it may be argued that the potential of the LSTAR and MSAR models for predicting the US unemployment rate should be examined in more detail, and that true Bayesian predictive densities should be used in the investigation. This is the twofold objective of this paper.

An outline follows. Section 2 presents an MCMC posterior simulator for LSTAR models. It differs from the previous one in two respects. First, an independence Metropolis–Hastings chain is used, rather than the random walk chain used by Lopes and Salazar (2006). Secondly, the autoregressive order *p* and transition delay parameter *d* are assumed to be fixed (whereas one of the algorithms proposed by Lopes and Salazar is defined on a space that includes *p* and *d*). In our approach, we propose to choose *p* and the transition variable (or function) according to the criterion of highest marginal likelihood; some potential advantages are discussed. Section 2 therefore also describes our application of the bridge sampling method of Meng and Wong (1996) to the estimation of marginal likelihoods in a STAR model.

Section 3 briefly describes the MCMC estimation of the MSAR model and the bridge sampling estimation of marginal likelihoods for this model.

Section 4 presents estimated marginal likelihoods for 54 possible LSTAR, MSAR, and autoregressive (AR) models, where the dependent variable is a logistic transformation of the monthly US unemployment rate; a sensitivity analysis with respect to the prior parameters is done.

Section 5 discusses Bayesian misspecification diagnostics for the LSTAR and MSAR models that were found, in Section 4, to have the highest marginal likelihoods. The diagnostics are based on posterior predictive *p*-values for three relevant misspecification indicators.

Section 6 presents the MCMC estimates of the chosen LSTAR model and of its MSAR counterpart; some economic implications of the estimates are discussed.

Section 7 presents, for comparison purposes, maximum likelihood estimates of the models in Section 6, and diagnostics based on generalized residuals.

Finally, Section 8 attempts to discriminate between the MSAR, the LSTAR, and a benchmark AR model by means of simulated out-of-sample prediction exercises. For each model, Bayesian predictive densities are estimated from expanding windows of observations and for horizons of 1 to 6 months. Diagnostics based on probability integral transforms (Diebold *et al.*, 1998; Berkowitz, 2001), on one of the test statistics proposed by Diebold and Mariano (1995), and on efficiency tests based on regressions of observations on point predictions are reported; versions of the efficiency tests are analyzed from both classical and Bayesian standpoints. Section 9 concludes.