This paper is concerned with Bayesian inference in hidden Markov models. Focusing on switching regression models, we propose a new methodology that delivers a joint estimation of the parameters and the number of regimes that have actually appeared in the studied sample. The only prior information that is required on the latter quantity is an upper bound. We implement a particle filter algorithm to compute the corresponding estimates. Applying this methodology to the information content of the yield curve regarding future inflation in four OECD countries, we show that the predictive content for given country and combination of maturities is subject to regime switching.
Many macroeconomic variables or structural relationships undergo episodes in which their behavior seems to be characterized by instability or important changes. In this respect, one may define instability as a switch from one period to another. The idea was first introduced by Quandt (1972), in the case of independent switches in a regression model. Goldfeld and Quandt (1973) and Lindgren (1978) have extended the analysis to Markov-chain regime-dependent-switching probabilities. Since the seminal contributions of Hamilton 1989 and Hamilton 1990, economists’ attention has been drawn to Markov-switching modeling of endogenous structural changes. Dynamic models with Markov switching have offered new perspectives in many economic areas such as business fluctuations and long-run trend in GNP (Hamilton, 1989), the behavior of foreign exchange rates and real interest rates (Garcia and Perron, 1996), the evolution of stock returns (Kim et al., 1998), etc. Kim and Nelson (1999) consider the advantages of a Bayesian approach when dealing with such models, and address the practical implementation through Gibbs sampling techniques. However, most of these papers invoke inference procedures which are valid for a given number of distinct regimes. In contrast, Chopin (2001) proposes a new general approach of discrete state-space models, which allows for a Bayesian joint estimation of the parameters and the number of distinct regimes featured by the studied data. We show in this paper how to adapt this approach to a switching regression model, and describe the corresponding implementation strategy, which relies on a particle filter algorithm.
To model abrupt changes in a given structural relationship, we introduce an unobserved discrete process, which gives the state of the system (regime) at date t. The hidden process may reflect changes in monetary policies, exchange rate regimes or any change in the economic environment. We focus here on the case where the hidden process is a Markov chain (hidden Markov models), and the structural relationship is linear (switching regression models). The divergence between our approach and those found in the literature reflects in part the use of the Bayesian framework. The latter is motivated by two shortcomings of the classical approach.
First, classical inference procedures often rely on asymptotic justifications, and therefore may show some fragility when applied to short series, notably in a regime-switching context, since it may be that only a few points of the studied sequence originate from a given regime. Second, in the classical approach, estimation of the state variables is conditional on the maximum likelihood estimates of the parameters. Here, parameters and regimes are jointly distributed random variables, i.e. estimate of each appropriately reflects the uncertainty of the others.
It must also be stressed that determining the number of regimes is quite an involved problem either in a classical context (Hansen 1992 and Hansen 1996; Hamilton, 1996) or in a Bayesian one (Chib, 1995; Carlin and Chib, 1995). This is because most inference procedures are only valid for a given number of regimes, while in practice if this number is misspecified, for instance if it is greater than necessary, extra regimes with no clear interpretation are often artificially created. These extra regimes may be discarded afterwards through some testing procedures, which unfortunately are often difficult to implement and may make the whole estimation process quite time-consuming. In contrast, our method provides in the same run (jointly with the other parameters) an estimate of the actual number of regimes, that is the number of distinct regimes that indeed have appeared in the studied sequence, given that it is supplied with a correct upper bound K of this number of regimes. As an illustration, we apply this methodology to the information content of the yield curve about future inflation in four OECD countries.
This paper is organized as follows. In Section 2, we discuss the Bayesian approach used in this paper. Thus we explain the state number determination for hidden Markov regression models, the prior modeling as well as the computational implementation of the particle filter. Section 3 presents a simple hidden Markov regression model for the information content of the yield curve regarding future inflation, describes the data and discusses the results. Section 4 briefly summarizes the main findings.
This paper gives new insights into the Bayesian analysis of hidden Markov models, with a particular focus on switching regression models. Mainly, the determination of the number of regimes that actually appear in a given sequence is more easily identified in our approach. This allows for a more precise characterization of the instability of a given structural relationship, as illustrated by our yield curve against future inflation data. Our method may show handful in a variety of other economic applications.
