Research Article A Dependent Hidden Markov Model of Credit Quality

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1 International Stochastic Analysis Volume 2012, Article ID , 13 pages doi: /2012/ Research Article A Dependent Hidden Marov Model of Credit Quality Małgorzata Witoria Koroliewicz Centre for Industrial and Applied Mathematics, School of Mathematics and Statistics, University of South Australia, City West Campus, GPO Box 2471, Adelaide, SA 5001, Australia Correspondence should be addressed to Małgorzata Witoria Koroliewicz, malgorzata.oroliewicz@unisa.edu.au Received 29 February 2012; Accepted 11 May 2012 Academic Editor: Yaozhong Hu Copyright q 2012 Małgorzata Witoria Koroliewicz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited. We propose a dependent hidden Marov model of credit quality. We suppose that the true credit quality is not observed directly but only through noisy observations given by posted credit ratings. The model is formulated in discrete time with a Marov chain observed in martingale noise, where noise terms of the state and observation processes are possibly dependent. The model provides estimates for the state of the Marov chain governing the evolution of the credit rating process and the parameters of the model, where the latter are estimated using the EM algorithm. The dependent dynamics allow for the so-called rating momentum discussed in the credit literature and also provide a convenient test of independence between the state and observation dynamics. 1. Introduction Credit ratings summarise a range of qualitative and quantitative information about the credit worthiness of debt issuers and are therefore a convenient signal for the credit quality of the debtor. The estimation of credit quality transition matrices is at the core of credit ris measures with applications to pricing and portfolio ris management. In view of pending regulations regarding the calculation of capital requirements for bans, there is renewed interest in efficiency of credit ratings as indicators of credit quality and models of their dynamics Basel Committee on Baning Supervision 1. In the study of credit quality dynamics, it is convenient to assume that the credit rating process is a time-homogeneous Marov chain, with past changes in credit quality characterised by a transition matrix. The assumptions of time homogeneity and Marovian behaviour of the rating process have been challenged by some empirical studies; see, for example, Bangia et al. 2 or Lando and Sødeberg 3. In particular, it has been proposed that ratings exhibit rating momentum or drift, where a rating change in response to a

2 2 International Stochastic Analysis change in credit quality does not fully reflect that change in credit quality. As pointed out by Löffler in 4, 5, these violations of information efficiency could be the result of some of the agencies rating policies, namely, rating through the cycle and avoiding rating reversals. In recent years, a number of modelling alternatives were suggested to address departures from the Marov assumption. In Frydman and Schuermann 6, a mixture of two independent continuous time homogeneous Marov chains is proposed for the ratings migration process, so that the future distribution of a firm s ratings depends not only its current rating but also on the past history of ratings. Wendin and McNeil 7 suppose that credit ratings are subject to both observed and unobserved systematic ris. Rating transition patters e.g., rating momentum are captured within the context of a generalised linear mixed model GLMM that is estimated using Bayesian techniques. Stefanescu et al. 8 propose a Bayesian hierarchical framewor, based on Marov Chain Monte Carlo MCMC techniques, to model non-marovian dynamics in ratings migrations. In Wozabal and Hochreiter 9, a coupled Marov chain model is introduced to model dependency among rating migrations of issuers. In this paper we follow the hidden Marov model HMM approach taen in Koroliewicz and Elliott 10 and assume that the true credit quality evolution can be described by a Marov chain but we do not observe this Marov chain directly. Rather, it is hidden in noisy observations represented by posted credit ratings. The model is formulated in discrete time, with a Marov chain of true credit quality observed in martingale noise. However, we suppose that noise terms of the signal and observation processes are not independent, which allows for the presence of rating momentum in posted credit ratings. Application of such dependent hidden Marov model dynamics to modelling credit quality appears to be new. We employ hidden Marov filtering and estimation techniques described in Elliott et al. 11 and use the filter-based EM Expectation Maximization algorithm to estimate the parameters of the model. By construction parameters are revised as new information is obtained and so the resulting filters are adaptive and self-tuning. The paper is organized as follows. In Section 2 we describe a hidden Marov model HMM of credit quality and in Section 3 the dependent dynamics. Recursive filters are given in Section 4 and the parameter estimation procedure is described in Section 5. Section 6 provides an implementation example. 2. Dynamics of the Marov Chain and Observations Here we briefly describe a hidden Marov model as given in Chapter 2 of Elliott et al. 11. Formally, a discrete-time, finite-state, time homogeneous Marov chain is a stochastic process {X } with the state space S {1, 2,...,N} and a transition matrix A a ji 1 i,j N. Without loss of generality, we can assume that the elements of S are identified with the standard unit vectors {e 1,e 2,...,e N },e i 0,...,0, 1, 0,...,0 R N. Write F σ{x 0,X 1,...,X } for a filtration {F } models all possible histories of X.The relationship between the state process at time and the state of the process at time 1is then given by E X 1 X AX. Define V 1 X 1 AX R N. Then, the semimartingale representation of the chain X is X 1 AX V 1, 0, 1,..., 2.1 where V 1 is a martingale increment with E V 1 F 0 R N.

3 International Stochastic Analysis 3 Suppose we do not observe X directly. Rather, we observe a process Y such that Y 1 c X,ω 1, 0, 1,..., 2.2 where c is a function with values in a finite set and {ω } is a sequence of i.i.d. random variables independent of X. Random variables {ω } represent the noise present in the system. Suppose the range of c consists of M points which are identified with unit vectors {f 1,f 2,...,f M },f j 0,...,0, 1, 0,...,0 R M. Write Y σ{y 0,Y 1,...,Y }, G σ{x 0,...,X,Y 0,...,Y }. 2.3 These increasing families of σ-fields are filtrations representing possible histories of the state process X, the observation process Y, and both processes X, Y.Writec ji P Y 1 f j X e i, 1 i N, 1 j M, for the probability of observing a state f j when the signal process is in fact in state e i. Then, it can be shown that E Y 1 X CX, where C c ji 1 i,j M is a matrix with c ji 0and M j 1 c ji 1. Define W 1 Y 1 CX. The semimartingale representation of the process Y is Y 1 CX W 1, 0, 1,..., 2.4 where W is a martingale increment with E W 1 G 0 R M. In our context, the process Y represents posted credit ratings and X true credit quality. For reasons which will become apparent in the next section, we assume one-period delay between X and Y. In summary, the model for the Marov chain X hidden in martingale noise is as follows. Hidden Marov Model HMM) Under a probability measure P, X 1 AX V 1 ) signal equation, true credit quality, Y 1 CX W 1 ) observation equation, posted rating. 2.5 A and C are matrices of transition probabilities whose entries satisfy N a ji 1, a ji 0; j 1 M c ji 1, c ji j 1 V and W are martingale increments satisfying E V 1 F 0, E W 1 G Parameters of this model are a ji, 1 i, j N and c ji, 1 j M, 1 i 1.

4 4 International Stochastic Analysis 3. Dependent Dynamics The situation considered in this section is that of a hidden Marov model HMM for which the noise terms in the state and observation processes are possibly dependent. The dynamics of the state process X and the observation process Y are as given in Section 2. However, the noise terms V and W are not independent. Instead, we suppose that the joint distribution of Y and X is given by Y 1 X 1 SX Γ 1, 0, 1,..., 3.1 where S s rji denotes a M N N matrix mapping R N into R M R N and s rji P Y 1 f r,x 1 e j X e i ), 1 r M, 1 i, j N. 3.2 Γ 1 is a {G }-martingale increment with E Γ 1 G 0. Write 1 1, 1,...,1 for the vector in R N or R M depending on the context. Then, for 1 R M, 1,SX AX and for 1 R N, SX, 1 CX, where, denotes the scalar product in R M and R N, respectively. Write γ rji P Y 1 f r X 1 e j,x e i s rji /a ji,andlet C be the M N N matrix γ rji, 1 r M, 1 i, j N. Then it can be shown that Y 1 C X 1 X W 1, where E W 1 G 0. In summary, the model is now as follows. Dependent Hidden Marov Model Dependent HMM) Under a probability measure P, X 1 AX V 1, Y 1 C X 1 X ) W A and C are matrices of transition probabilities whose entries satisfy N a ji 1, j 1 M γ rji 1. r V and W are martingale increments satisfying E V 1 F 0, ] E [ W 1 G Parameters of this model are a ji, 1 i, j N, c ji, 1 j M, 1 i 1, and s rji, 1 i, j N, 1 r M. We are in a situation analogous to the dependent hidden Marov model case discussed in Chapter 2, Section 10 of Elliott et al. 11.Thedifference is that we are assuming dynamics where the observation Y depends on both X and X 1. In other words, we suppose

5 International Stochastic Analysis 5 that the current credit rating contains information about both current and previous credit quality, thus allowing for the situation where a rating does not immediately reflect all available information about credit quality, as indicated by a number of empirical studies see, e.g., Lando and Sødeberg 3. Putdifferently, in this model X and observation Y jointly depend on X 1, which means that, in addition to previous period s credit quality, nowledge of current credit rating carries information about current credit quality. Moreover, probabilities γ rji provide the distribution of the next period s credit rating given both current and next period s credit quality, thus allowing us to capture rating momentum or rating drift. In the following sections we will presents estimates for the state of the Marov chain X, the number of jumps from one state to another, the occupation time of X in any state, the number of transitions of the observation process Y into a particular state of X, andthe number of joint transitions of X and Y. We will then use the filter-based EM expectation maximization algorithm as described in Elliott et al. 11, to obtain optimal estimates of the model, maing it adaptive or self-tuning. Note that if the noise terms in the state X and observation Y are independent, we have P Y 1 f r,x 1 e j X e i ) P Y 1 f r X e i ) P X 1 e j X e i ). 3.6 Hence if the noise terms are independent, s rji c rj a ji 3.7 for 1 r M, 1 i, j N. Consequently, a test of independence is to chec whether parameter estimates satisfy ŝ rji ĉ rj â ji Recursive Filter Following Elliott et al. 11, suppose that under some probability measure P on Ω, F, {Y } is a sequence of i.i.d. uniform variables, that is, P Y 1 f j G P Y 1 f j 1/M. Further, under P, X is Marov chain independent of Y, with state space S {e 1,...,e N } and transition matrix A a ji.thatis,x 1 AX V 1, where E V 1 G E V 1 F 0 R N. Suppose C γ rji, 1 r M, 1 i, j N, is a matrix with γ ji 0, and M j 1 γ rji 1. Define λ l M M j 1 C X l X l 1,f j Y l,f j and Λ l 1 λ l. Define a new probability measure P by putting dp/dp G Λ. Then, under P, X remains a Marov chain with transition matrix A and P Y 1 f r X 1 e j,x e i γ rji. That is, under P, X 1 AX V 1 and Y 1 C X 1 X W 1. Suppose we observe Y 0,...,Y, and we wish to estimate X 0,...,X.Thebest mean square estimate of X given Y σ{y 0,...,Y } is E X Y R N. However, P is a much

6 6 International Stochastic Analysis easier measure under which to wor. Using Bayes Theorem as described in Elliott et al. 11, we have E X Y ] E [Λ X Y ]. 4.1 E [Λ Y Write q : E Λ X Y R N. q is then an unnormalized conditional expectation of X given the observations Y.NotethatE Λ Y q, 1, where 1 1, 1,...,1 R N.It then follows that E X Y q q, Hence, to estimate E X Y we need to now the dynamics of q. Using the methods of Elliott et al. 11, the following recursive formula for q 1 is obtained: q 1 MY 1 S q Parameter Estimates To estimate parameters of the model, matrices A, C, and S, we need estimates of the following processes: L ijr T ir J ij X n 1,e i X n,e j, 1 i, j N, n 1 O i X n 1,e i, 1 i N, n 1 X n 1,e i Y n,f r, 1 i N, 1 r M, n 1 X n 1,e i X n,e j Yn,f r, 1 r M, 1 i, j N. n The above processes are interpreted as follows: J ij is the number of jumps of X from state e i to state e j up to time. O i is the amount of time, up to time 1, X has spent in state e i. T ir is the number of transitions, up to time, from state e i to observation f r. L ijr is the number of jumps of X from state e i to state e j while Y was in state f r up to time. Note that N j 1 Jij M j 1 Tij M N r 1 j 1 Lijr O i.

7 International Stochastic Analysis 7 Consider first the jump process {J ij }. We wish to estimate Jij given the observations Y 0,...,Y. As in the case of a filter for the state X described in Section 4, thebest meansquare estimate is [ ] [ ] E Λ E J ij J ij Y Y [ ] E Λ Y : σ ) J ij q, We wish to now how σ J ij is updated as time passes and new information arrives. However, as noted in Elliott et al. 11, weworwithσ J ij X E Λ J ij X Y rather than σ J ij E Λ J ij Y, in order to obtain closed-form recursions. The quantity of interest, namely, σ J ij, is then readily obtained as σ J ij σ J ij X, 1. We have σ J ij X ) 1 MY 1 Sσ J ij X ) q,e i M M ) s rji Y 1,f r e j. r Similarly, we consider the best mean square estimates of O i, Tjr,andLrji given Y : [ ] [ ] E Λ O i E O i Y Y [ ] E Λ Y [ ] [ ] E Λ E T jir T jir Y Y [ ] E Λ Y [ ] [ ] E Λ E L ijr L ijr Y Y [ ] E Λ Y : σ ) O i q, 1, : σ ) T ir q, 1, : σ ) L ijr q, Recursive formulae for the processes ] σ O i X [Λ ) : E O i X Y, [ σ T ir X ) : E Λ T ir X Y ], [ ] σ L ijr X ) : E Λ L ijr X Y 5.5

8 8 International Stochastic Analysis are as follows: σ O i X ) 1 MY 1 Sσ O i X ) q M,e i M j 1 M ) s rji Y 1,f r e j, σ T ir X ) 1 MY 1 Sσ T ir X ) M q,e i N s rji e j Y 1,f r, ) σ L ijr X ) 1 MY 1 Sσ L ijr X q,e i Msrji Y 1,f r ej. r 1 j As in the case of the number of jumps of the state process X, quantities of interest σ O i, σ T ir,andσ L ijr are obtained by taing inner products with 1 1, 1,...,1 : σ O i) σ O i X ), 1, σ T ir) σ T ir X ), 1, 5.7 σ L ijr) σ L ijr X ), 1. The model is determined by parameters θ {a ji, 1 i, j N; c ji, 1 i N, 1 j M; s rji, 1 r M, 1 i, j N}. These satisfy a ji 0, N a ji 1, c ji 0, j 1 M c ji 1, s rji 0, j 1 M N s rji 1. r 1 j We want to determine a new set of parameters θ {â ji, 1 i, j N; ĉ ji, 1 i N, 1 j M}; ŝ rji, 1 r M, 1 i, j N} given the arrival of new information embedded in the values of the observation process Y. This requires maximum lielihood estimation. As in 11, we proceed by using the filter-based EM Expectation Maximization algorithm, which retains the well-established statistical properties of the EM algorithm while reducing memory costs and thus allowing for faster computation see, e.g., Krishnamurthy and Chung 12. Consider first the parameter a ji. Suppose that, under measure P θ,xis a Marov chain with transition matrix A a ji. We define a new probability measure P θ such that, under P θ, Xis a Marov chain with transition matrix  â ji,thatis, P θ X 1 e j X e i ) âji, 5.9 â ji 0, N j 1 âji 1. Define Λ N l 1 r,s 1 Λ 0 1, ) ) X l,e s X l 1,e r. âsr a sr 5.10 In case a ji 0 tae â ji 0andâ sr /a sr 1.

9 International Stochastic Analysis 9 Define P θ by setting dp θ/dp θ F Λ. It can then be shown that, under P θ, Xis a Marov chain with transition matrix  â ji. Moreover, given the observations up to time, {Y 0,Y 1,...,Y }, and given the parameter set θ {a ji, 1 i, j N; c ji, 1 i N, 1 j M}, the EM estimates â ji are given by â ji σ J ij) σ O i Consider now the parameter c ji. Suppose that, under measure P θ, Y 1 CX W 1, 5.12 where C c ji. We define a new probability measure P θ as follows. Put Λ N l 1 r,s 1 Λ 0 1, ĉsr c sr ) X l 1,e r Y l,f s ) In case c ji 0 tae ĉ ji 0andĉ sr /c sr 1. Define P θ by setting dp θ/dp θ G Λ. Again it can be shown that, under P θ, Y 1 ĈX Ŵ 1, 5.14 that is, P θ Y 1 f s X e r ĉ sr. Moreover, given the observations up to time, {Y 0,Y 1,...,Y }, and given the parameter set θ {a ji, 1 i, j N; c ji, 1 i N, 1 j M}, the EM estimates ĉ ji are given by ĉ ji σ T ij) σ O i Finally, consider the parameter s rji. A new probability measure P θ is defined by putting Λ M N l 1 r 1i,j 1 ŝrji s rji Λ 0 1, ) Yl,f r Xl,e j Xl 1,e i In case s rji 0 tae ŝ rji 0andŝ rji /s rji 1. Define P θ by setting dp θ/dp θ G Λ. Then, under P θ, Y 1 X 1 ŜX Γ 1,thatis, P θ Y 1 f r,x 1 e j X e i ) ŝrji. 5.17

10 10 International Stochastic Analysis Given the observations up to time, {Y 0,Y 1,...,Y }, and given the parameter set θ {a ji, 1 i, j N; c ji, 1 i N, 1 j M; s rji, 1 r M, 1 i, j N}, the EM estimates ŝ rji are then given by ŝ rji σ L ijr) σ O i Implementation Example The dependent hidden Marov model Dependent HMM described in previous sections was applied to a dataset of Standard & Poor s credit ratings. Description of the data and implementation results are given below Data Description Our analysis taes advantage of the Standard & Poor s COMPUSTAT database, which contains rating histories for 1,301 obligors over the period Standard & Poor s 13. The universe of obligors is mainly large US and Canadian corporate institutions. The obligors include industrials, utilities, insurance companies, bans and other financial institutions, and real-estate companies. The COMPUSTAT database provides annual ratings. Every year each of the rated obligors is assigned to one of the Standard and Poor s 7 rating categories, ranging from AAA highest rating to CCC lowest rating as well as D payment in default and the NR not rated state. We have a total of 19,515 firm-years in our sample. However, only 34% of those observations correspond to one of the eight Standard & Poor s rating labels in a given year. The remaining 66% of observations represent the so-called NR not rated status. As discussed in the literature, transitions to NR may be due to several reasons, such as expiration of the debt, calling of the debt, or the issuer deciding to bypass an agency rating see, e.g., Bangia et al. 2. Unfortunately, details of individual transitions to NR are not nown. Excluding NR, approximately 85% of the remaining ratings are in categories A down to B. The median rating is BB, the highest non-investment-grade rating. Approximately 1% of the observed ratings are AAA and 2% are defaults. The most common rating is B, two rating categories above default, which accounts for 25.5% of the observations Implementation Results Since individual firms generally experience few rating changes and changes that do occur are to neighbouring categories, we apply the Dependent HMM algorithm to an aggregate of firms in the dataset rather to allow for more observed transitions between rating categories and mae inferences possible. Specifically, we follow the filter-based cohort approach adopted in Koroliewicz and Elliott 10, and instead of estimating the distribution and parameters for the Marov chain X l for each firm l, we estimate the distribution and parameters for L l 1 Xl given the additivity of all stochastic processes discussed in Sections 4 and 5. Given the fairly large number of parameters to be estimated compared to the number of rating transitions in the dataset, we have reclassified all firms in the sample as IG investment grade,sg speculative grade, D, or NR and then applied the Dependent HMM

11 International Stochastic Analysis 11 Table 1: Estimates of matrices A, C,andS. a Estimated matrix A Estimated matrix C IG SG D NR IG SG D NR IG IG SG SG D D NR NR b Estimated matrix S IG category D category IG SG D NR IG SG D NR IG IG SG SG D D NR NR SG category NR category IG SG D NR IG SG D NR IG IG SG SG D D NR NR algorithm to the new dataset. This classification is motivated by the fact that a corporation which can issue higher rated debt usually receives better financing terms. Further, as a matter of policy or law, some institutional investors can only purchase investment-grade bonds. Hence it is often crucial for a borrower to maintain an investment-grade rating and so it is interesting to see if rating transition data reflects this. Each modified credit rating category IG, SG, as well as default D and NR, was identified with a unit vector in R 4. Given the relatively short time period, parameter estimates were updated with the arrival of every new observation for the 1,301 firms in the dataset. Repetition of the estimation procedures ensures that the model and estimates improve with each iteration. Estimated parameters of the model, namely, matrices Â, Ĉ, and S, are given in Table 1. Considering the estimated transition matrix Â, note that entries above the diagonal correspond to rating upgrades and those below the diagonal to rating downgrades. Nonzero transition probabilities are concentrated and highest on the diagonal and the second largest probability is in the last row, indicating that obligors generally either maintain their rating or enter the NR not rated category. Our results show that investment-grade firms generally hold on to their status. The probability of downgrade to speculative-grade status is estimated as 6.8%. However, for speculative-grade firms, the probability of upgrade to investmentgrade status is lower estimated probability of 1.8%. Speculative-grade firms tend to maintain their status or disappear from the dataset because of either default or withdrawn rating. The probability of transition to the NR status is higher for speculative-grade obligors 71.5% than for investment-grade obligors 52.4%.

12 12 International Stochastic Analysis Table 2: Linear regression results. a Summary output Regression statistics Multiple R R square Adjusted R square Standard error Observations 64 b ANOVA df SS MS F Significance F Regression Residual Total c Coeff Std err t stat P value Lower Upper 95% 95% Intercept s rji Recall from Section 3 that, given estimates of matrices A and C, our Dependent HMM also provides the distribution of posted credit ratings at time 1 given true credit quality at times and 1, namely, estimates of conditional probabilities γ rji P Y 1 f r X 1 e j, X e i. To illustrate, consider a borrower with investment-grade true credit quality at times and 1. The probability that this borrower is assigned to a speculative-grade rating class is P Y 1 SG X 1 IG,X IG, which, given our model parameter estimates, is given by ŝ SG,IG,IG /â IG,IG 0.007/ Similarly, for a borrower whose true credit quality improves from SG to IG, the probability of being assigned to an IG rating class is given by P Y 1 IG X 1 IG, X SG, which we would estimate to be 0.007/ These estimates again suggest that rating agencies may be somewhat reluctant to downgrade upgrade borrowers from to investment grade, thus introducing a degree of rating momentum Test of Independence Recall that the Dependent HMM allows the noise terms in the state and observation processes to be possibly dependent. As indicated in Section 3, a convenient test of independence is to chec whether the estimated parameters of the model satisfy ŝ rji ĉ rj â ji. Given our estimates of matrices  and Ĉ, products ĉ rj â ji were calculated and then compared to corresponding entries of the estimated matrix Ŝ using linear regression. The regression results are given in Table 2. As indicated by the high F-statistic and high R 2 value 98.71%, the fitted regression model is significant. The slope estimate is very close to one with low standard error and P value of 0.000, while the intercept estimate is very close zero and not significant P value of These regression results suggest no major departures from independence, which seems to agree with findings in Kiefer and Larson 14

13 International Stochastic Analysis 13 that indicate the Marov assumption, implicit in most credit ris models, does not seem to be too wrong for typical forecast horizons. However, longer rating histories may be necessary to verify these results. 7. Conclusion We have proposed a Dependent Hidden Marov Model for the evolution of credit quality in discrete time with a Marov chain observed in martingale noise. We have applied the estimation techniques of hidden Marov models from Elliott et al. 11 to obtain the best estimate of the Marov chain representing true credit quality and estimates of the parameters. The estimation procedure was repeated to ensure that the model and estimates improved with each iteration. The model was applied to a dataset of Standard & Poor s issuer ratings and our preliminary results agree with some qualitative observations made in the literature regarding credit rating systems but also indicate no significant dependence in the dynamics of the state credit quality and observation credit rating processes. References 1 Basel Committee on Baning Supervision, Studies on validation of of internal rating systems, BCBS Publications 14, Ban for International Settlements, A. Bangia, F. X. Diebold, A. Kronimus, C. Schagen, and T. Schuermann, Ratings migration and the business cycle, with application to credit portfolio stress testing, Baning and Finance, vol. 26, no. 2-3, pp , D. Lando and T. M. Sødeberg, Analyzing rating transitions and rating drift with continuous observations, Baning and Finance, vol. 26, no. 2-3, pp , G. Löffler, An anatomy of rating through the cycle, Baning and Finance, vol. 28, no. 3, pp , G. Löffler, Avoiding the rating bounce: Why rating agencies are slow to react to new information, Economic Behavior and Organization, vol. 56, no. 3, pp , H. Frydman and T. Schuermann, Credit rating dynamics and Marov mixture models, Baning and Finance, vol. 32, no. 6, pp , J. Wendin and A. J. McNeil, Dependent credit migrations, Credit Ris, vol. 2, pp , C. Stefanescu, R. Tunaru, and S. Turnbull, The credit rating process and estimation of transition probabilities: A Bayesian approach, Empirical Finance, vol. 16, no. 2, pp , D. Wozabal and R. Hochreiter, A coupled Marov chain approach to credit ris modeling, Journal of Economic Dynamics and Control, vol. 36, no. 3, pp , M. W. Koroliewicz and R. J. Elliott, A hidden Marov model of credit quality, Economic Dynamics & Control, vol. 32, no. 12, pp , R. J. Elliott, L. Aggoun, and J. B. Moore, Hidden Marov Models: Estimation and Control, vol. 29, Springer, New Yor, NY, USA, V. Krishnamurthy and S. H. Chung, Signal processing based on hidden Marov models for extracting small channel currents, in Handboo of Ion Channels: Dynamics, Structure, and Applications, S. H. Chung, S. H. Andersen, and V. Krishnamurthy, Eds., Springer, Berlin, Germany, Standard & Poor s, Standard & Poor s COMPUSTAT North America User s Guide, Standard & Poor s, Englewood Colorado, N. M. Kiefer and C. E. Larson, A simulation estimator for testing the time homogeneity of credit rating transitions, Empirical Finance, vol. 14, no. 5, pp , 2007.

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