Price discovery in the markets for credit risk: a Markov switching approach

A Construction of the likelihood function

Denote the vector of price series by y_t . Conditional on the unobserved state variable s_t , the density function of y_t is known. As the innovations are assumed to be conditionally normally distributed, the conditional density of y_t is given by

for j=1, 2 and Y_t–1={y₁, …, y_t–1}. Multiplying the conditional density in Equation (A.1) with the marginal density of s_t yields the joint density of y_t and s_t

for j=1, 2. Summing over all regimes allows to integrate out s_t , which results in the unconditional density of y_t :

This density corresponds to the average of the conditional densities weighted by the probability of the particular regime. To compute the weighting factors, we have to account for the Markov structure of s_t . In time t, the probability of regime j can be calculated as Pr(s_t =j|Y_t–1)=Pr(s_t =j|s_t–1=i)×Pr(s_t–1=j|Y_t–1), where the probabilities in the first term on the right hand side are summarized in the transition matrix P=[p111−p221−p11p22] with p_ij =Pr(s_t =j|s_t–1=i) ∀i, j∈1, 2.

Using Bayes’ theorem, the probability Pr(s_t =j|Y_t ) can be filtered out, which together with the transition matrix P delivers Pr(s_t+1=j|Y_t ). Iterating these steps until the last observation generates f(y_t |Y_t–1) for each point in time. Finally, the log-likelihood function is given by

It is maximised with respect to W, Ψ, p₁₁, and p₂₂.

The main difficulty of this optimization approach concerns the search of appropriate initial values. Due to the complexity of the log-likelihood function, finding the global peak proves to be challenging. In order to verify the maximization results and to find a more reliable method in terms of starting values, we recommend to apply the EM-algorithm.

B The EM-algorithm

In this study, the approach described by Herwartz and Lütkepohl (2014) is implemented. It represents the EM-algorithm enhanced by a Baum-Lindgren-Hamilton-Kim (BLHK) filter, which is explained by Krolzig (1997). Contrary to the approach of Herwartz and Lütkepohl (2014), we do not compute the residual series within the EM-algorithm, but estimate the VECM model in Equation (1) and subsequently use the VECM residuals in the optimization of the log-likelihood.

Let P denote the transition matrix, ι is a 2×1 vector of ones, and

ξt|s=[Pr(st=1|Ys)Pr(st=2|Ys)],

⊙ stands for element-wise multiplication,

⊘ stands for element-wise division and

⊗ for the Kronecker product.

1. Initialization of the starting values:

   P(0), ξ0|0, W0 and Ψ(0).

2. Expectation step:

   The expectation step of the EM-Algorithm begins with the filtering or a “forward recursion”

   ξt|t=ηt⊙Pξt−1|t−1ι′(ηt⊙Pξt−1|t−1) for t=1, …, T,

   where

   ηt=[f(yt|st=1, Yt−1)f(yt|st=2, Yt−1)],

   with

   f(yt|st=m, Yt−1)=(2π)− K2det(∑m)− 12exp{−12u′t∑m−1ut}  for m=1, 2.

   Then, the filtered probabilities are smoothed by means of “backward recursion”:

   ξt|T=(P′(ξt+1|T⊘Pξt|t))⊙ξt|t   for t=T−1, …, 0.

3. Maximization step:

   Along the maximization step, the transition matrix is estimated with the Hidden Markov Chain formula (Krolzig 1997)

   vec(P^′)=(∑t=1T−1ξt|T(2))⊘(ι⊗∑t=1Tξt|T)

   where

   ξt|T(2)=vec(P′)⊙[(ξt+1|T⊘Pξt|t)⊗ξt|t]   for t=0, …, T−1.

   Finally, the values of W and Ψ, which maximize the expected log-likelihood function, have to be found. The expected log-likelihood is given by

   logLEM=−12∑t=1T∑m=12ξmt|T[2log(2π)+log(|∑m|)+u′t∑m−1ut].

   Writing out the regimes yields

   =−12[2log(2π)∑t=1Tξ1t|T+∑ξ1t|Tlog(|WW′|)+tr((WW′)−1∑t=1Tξ1t|Tu^′tu^t)+2log(2π)∑t=1Tξ2t|T +∑ξ2t|Tlog(|WΨW′|)+tr((WΨW′)−1∑t=1Tξ2t|Tu^′tu^t)].

The likelihood function is then maximized subject to constraints on the W and Ψ matrices discussed in Section 2. In addition, a lower bound of 0.001 is imposed on the determinant of each covariance matrix. The estimates become the initial values for the next iteration:

P^=P(0),ξ0|T=ξ0|0,W^=W0, andΨ^=Ψ(0).

Steps 2 and 3 are iterated until convergence is achieved, e.g. until

∑i=12∑j=12|W^ij−Wij(0)|<0.00001.

Unfortunately, using numerical optimization does not guarantee monotonic convergence of the expected log-likelihood Schaefer (1997). This is the case in our study as shown by Figure B.1. It shows that using numerical optimization leads to non-monotonic convergence in our empirical application. Consequently we verify our results by applying both approaches, direct maximization of the loglikelihood function an the EM-algorithm outlined above.

Figure B.1:

Convergence of the log-likelihood function.

The graphic illustrates the convergence of the log-likelihood function for the estimation of Allianz.

Standard errors of the estimates are obtained by means of a parametric bootstrap. This bootstrap is a slightly adapted version of the procedure described in Grammig and Peter (2013). It proves to be convenient, bearing in mind the two-step estimation structure.

Estimation is done in Gauss and all program codes are available upon request.