Markov-Switching Models with Unknown Error Distributions: Identification and Inference Within the Bayesian Framework

3.1 Identification Issue #1: The Label Switching Problem

For our basic model, a typical way of labeling the states for S _t and D _t is given below:

where

The above labeling is not unique and the unconstrained parameter spaces for β′s and μ*′s (or h*²′s) contain K! and M! subspaces, respectively, each corresponding to different way to label states. As discussed in Stephens (2000) and Frühwirth-Schnatter (2001), when sampling from the unconstrained posterior via MCMC methods, it is impossible to know which component of the sampled parameter corresponds to which state due to potential label switching. Thus, as noted by Stephens (2000), summarizing joint posterior distributions by marginal distribution may lead to nonsensical answers due to the lack of identification.

The label switching problem is not an issue at all for the serially independent mixture indicator variable D _t , as we are not interested in the marginal distribution of μ′s or h ²′s, or in the inferences on D _t . Furthermore, the complete data likelihood f(y ₁, …, y _T |D ₁, …, D _T ; .) and the prior for D _t is invariant to the relabeling of the states in D _t . However, it is critical that we take care of the label switching problem for S _t during the MCMC procedure, given that we want to obtain inferences on S _t and the regime-specific parameters based on their marginal posterior distributions.

The label switching problem for S _t can be solved by imposing the following ordering constraint on the regime-specific parameters:

A conventional way to incorporate the above constraint in the MCMC sampler is to employ a rejection method after drawing {β ₁ β ₂ …β _K } jointly from the unconstrained joint posterior. However, note that the ordering constraint in equation (7) results in correlations among β _k , k = 1, 2, …, K. For example, the smaller the distances among the β parameters, the higher will be the correlations among them. Every time the β parameters are discarded and redrawn when the ordering constraint fails, we lose sample information about these correlations. This is why the rejection method may fail, especially when the distances among the β parameters are not large enough relative to the standard deviation of the error term.

For the permutation sampler proposed by Frühwirth-Schnatter (2001), the β parameters are first drawn from the unconstrained joint posterior. Then a suitable permutation of the labeling of the states is applied if the ordering constraint is violated. As the permutation is applied without discarding the β parameters drawn in this case, there is no loss of sample information unlike in the case of the rejection method. This is why the permutation sampler improves upon the rejection method, as illustrated by Frühwirth-Schnatter (2001). However, one potential drawback of the permutation sampler is that we need to set the marginal priors for β _k , k = 1, 2, …, K, to be identical, but independent. This is because the prior densities for the individual β coefficients should be permutation-invariant. For this reason, it would be impossible to specify a joint prior that appropriately reflects the potential correlations among the β parameters.

As an alternative method for dealing with the label switching problem, we consider the following transformation of the β parameters in equation (7):

An advantage of the above specification is that we can indirectly specify the potential prior dependence among β _k , k = 1, 2, …, K, by employing independent marginal priors for β ₁ and a _k , k = 2, 3, …, K. We first draw β ₁ conditional on a _k , k = 2, 3, …, K, and then, draw a _k for k = 2, 3, …, K, from appropriate truncated marginal posteriors conditional on β ₁ and a ̃ ≠ k = { a 2 , … , a k − 1 ; a k + 1 , … a K } . We can then recover the β ₂, …,β _K parameters based on equation (8). Here, an important issue to consider is that the likelihood function for a _k depends only on the observations for which S _t  = j, j = k, k + 1, …, K, while the likelihood function for β ₁ depends on all the observations in the sample. As in the case of the permutation sampler, there is no loss of sample information in the course of the proposed MCMC procedure. Unlike the case of the permutation sampler, however, the proposed prior and sampling procedure allow us to accommodate the non-sample information on the potential dependence among the β parameters that results from the inequality constraint in equation (7).^[3] Besides, the implementation of the proposed sampler is much easier than the permutation sampler, especially for a model like the one in our empirical application section, in which the regime-specific means are subject to structural breaks within the Markov-switching framework.

3.2 Identification Issue #2: Disentangling the Markov-Switching Variable (S _t ) from the Mixture Indicator Variable (D _t )

In this section, we consider the identification of the latent Markov-switching variable S _t from the latent and serially independent mixture indicator variable D _t in equation (3). For this purpose, we substitute equation (8) into equation (3) to obtain

where β ̄ = β ̄ 2   β ̄ 3 … β ̄ K ′ , with β ̄ k = ∑ j = 2 k a j , k = 2, 3, …, K; S ̄ t = S 2 , t … S K , t ′ , with S _k,t , k = 2, 3, …, K, being defined in equation (6); and ε t = β 1 + σ ε t * . We can then approximate ɛ _t by the following mixture of normals:

Note that the dynamics of S _t , given the transition probabilities in equation (4), can be represented by the following VAR process for S ̄ t :^[4]

where the elements of the (K − 1) × 1 vector Q ₀ and the (K − 1) × (K − 1) matrix Q ₁ are functions of the transition probabilities. We define C as the collection of the eigenvectors of the Q ₁ matrix. By pre-multiplying both sides of equation (11) by C ⁻¹, and then, by rearranging the terms in the resulting equation and equation (9), we obtain:

where β ̄ * = C ′ β ̄ ; S ̄ t * = C − 1 S t * ; Λ₀ = C ⁻¹ Q ₀; Λ₁ = C ⁻¹ Q ₁; C = diag{λ ₂, λ ₃, …, λ _K }, with λ _k , k = 2, 3, …, K, referring to the eigenvalues of Q ₁; and ν ̄ t * = C − 1 ν ̄ t .

Then, by noting that ɛ _t in equation (12) can be approximated by the mixture of normals as in equation (10) (i.e. ε t = ∑ m = 1 M μ m D m , t + ∑ m = 1 M h m 2 D m , t ε t * , ε t * ∼ i . i . d . N ( 0,1 ) ), where D _m,t as defined in equation (6) is serially independent and by noting that Λ₁ in equation (12) is diagonal, we can rewrite equations (12) and (13) as follows:

where β ̄ k * is the (k − 1)th row of β ̄ * ; S ̄ k , t * refers to the (k − 1)th row of S ̄ t * ; g _m  = Pr[D _t  = m]; and ν ̄ k , t * , k = 2, 3, …, K, and η _m,t , m = 1, 2, …, M, are discrete and serially independent.

Equation (14) tells us that we have a K-state first-order Markov-switching process for S _t and a mixture of M normals for ɛ _t only when the following conditions hold:

suggesting that all the eigenvalues of the Q ₁ matrix in equation (11) should be non-zero.

It is easy to show that the model is not identified when equation (15) does not hold. For example, suppose that λ _K  = 0 and λ _k  ≠ 0, for k = 2, 3, …, K − 1. Then, equation (14) can be written as

where μ M + 1 * = β ̄ K * ; g _M+1 = Λ _K,0; D M + 1 , t = S ̄ K , t * ; h M + 1 * = 0 ; and the transition probabilities for S _t and the mixture probabilities for D _t are redefined accordingly. Note that equation (14’) describes a model with a (K − 1)-state Markov-switching process for S _t and a mixture of (M + 1) normals for ɛ _t . The likelihood value for the model in Equation (14’) is exactly the same as that for the model in equations (14), and this is a typical example of non-identification.

For economic data, a negative serial correlation in S _t does not seem to make a lot of sense. We thus impose the constraints that λ _k  > 0, k = 2, …, K, in order to achieve the identification. For this purpose, We set the prior mean of p _S,jj to be larger than 0.5 for j = 1, 2, …, K, as this is the sufficient condition for λ _k  > 0, k = 2, 3, …, K. Then, once the transition probabilities are drawn conditional on S _t , t = 1, 2, …, T, we can construct the Q ₁ matrix in equation (11) and calculate its eigenvalues λ _k , k = 2, 3, …, K. Then, if the identifying constraints are not satisfied, we redraw S _t , t = 1, 2, …, T, and the corresponding transition probabilities until the constraints are satisfied.

4.1 Specification for a General Model

Consider the following generalized model:^[5]

where ε t * is independently distributed; ϕ(L) = 1 − ϕ ₁ L − ϕ ₂ L ² − ⋯ − ϕ _p L ^p is a polynomial equation in the lag operator; all roots of ϕ(L) = 0 lie outside the complex unit circle; the transitional dynamics of S _t is specified in equation (4). We assume that W _t is independent of S _t and follows an N-state, first-order Markov-switching process with the following transition probabilities:^[6]

In order to avoid the non-identification resulting from the problem of label switching, we follow employ the following specifications for the β S t and σ W t 2 parameters:

which allow us to employ independent priors for {β ₁, a _k , k = 2, 3, …, K} and for σ 1 2 , ( 1 + b n ) , n = 2,3 , … , N . This way, we can also indirectly account for the dependence among β _k , k = 1, 2, …, K and the dependence among σ n 2 , n = 1, 2, …, N which result from the ordering constraints (β ₁ < β ₂ < … < β _K and σ 1 2 < σ 2 2 < … < σ N 2 ).

By substituting equation (18) into equation (16) and rearranging terms, we obtain:

where

Then, by defining e _t  = β ₁ + u _t , equation (19) can be rewritten as:^[7]

Model with Transformed Parameters

where ε t = ϕ ( 1 ) β 1 / g W t + u t * . The unknown distribution of ɛ _t |W _t can be approximated by the following Dirichlet process mixture of normals:^[8]

Dirichlet Process Mixture of Normals

where DP(., .) refers to the Dirichlet process;^[9] G ₀ and α are referred to as the base distribution and the concentration parameter, respectively.

The base distribution G ₀ is like the mean of the Dirichlet Process. In other words, the Dirichlet Process draws distributions around the base distribution the way a normal distribution draws real numbers around its mean. The concentration parameter α is like an inverse-variance of the Dirichlet Process. It describes the concentration of mass around the base distribution. In a Dirichlet Process mixture model, we can show that the probability of assigning an observation to a newly drawn distribution around the base distribution is α T − 1 + α . Therefore, the larger the α is, the higher the probability of assigning an observation to a new distribution, and thus prior mean of the number of mixture is higher.

We employ a Normal-Inverse Gamma distribution as the base distribution. This means that we use N λ 0 , ψ 0 h m 2 as the prior distribution for mixture mean μ _m , and we use I G ( ν h 2 , δ h 2 ) as the prior distribution for mixture variance h m 2 . These are the conjugate priors for the Dirichlet Process Mixture model. In the case of finite mixture, the joint distribution of μ m , h m 2 is given by G ₀, and thus, G = G ₀. The α parameter can be either fixed or random. In case the α parameter is random, it is common to employ a Gamma prior. Note that, conditional on g W t , μ D t , and h D t 2 , the first line in equation (22) implies

To complete the model, we employ the following priors for the parameters except those associated with the mixture of normals:

Other Priors

where Σ a ̃ is diagonal; 1[.] is the indicator function; S ϕ ̃ refers to the stationary region of ϕ ̃ ; IG(.) refers to the inverted Gamma distribution; and Dir(.) refers to the Dirichlet distribution. Following Section 3.2, we impose the constraints p _S,kk  > 0.5, k = 1, 2, …, K, and p _W,nn  > 0.5, n = 1, 2, …, N, in order to identify the Markov-switching processes S _t and W _t against the mixture indicator variable D _t .

4.2 MCMC Procedure

In this section, we present an MCMC procedure for estimating the model that consists of equations (21)–(24).