Estimating Credit Migration Generators Directly from Rating Data
Summary
The document discusses how to estimate continuous-time credit rating migration dynamics from observed rating histories. It contrasts discrete-time cohort transition matrices, formed from beginning-to-end rating changes over a period, with a continuous-time Markov model described by a generator matrix. The question is whether to first estimate a one-year transition matrix and then infer a generator, or to use all observed migration events.
The answer recommends fitting the generator directly to rating transition data. It notes that continuous-time maximum likelihood estimation is possible, while discrete observations may be handled with an expectation-maximization approach. Inferring a generator from an estimated transition matrix can encounter embeddability and identification problems. The response offers a concise methodological recommendation but does not detail the likelihood, data requirements, treatment of censored histories, or empirical comparison of estimators. Its practical value is in highlighting that a transition matrix may not uniquely or validly yield a continuous-time generator.
Key ideas
- A continuous-time Markov model represents rating migration with a generator matrix.
- The generator can be estimated directly from observed rating transition histories.
- Continuous-time maximum likelihood is one proposed estimation approach.
- An expectation-maximization method may be useful when observations are discrete.
- Inferring a generator from a transition matrix can face embeddability and identification issues.
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# Observed rating migration matrix to derive the generator matrix # Observed rating migration matrix to derive the generator matrix I am doing some reading on the derivation of credit rating migration/transition matrices and probability of default term structures. I understand that a homogeneous Markov chain can be either discrete-time or continuous-time. In the discrete-time case, the one year transition matrix can be derived using the cohort method which involves computing the proportion of observed migrations from the beginning of a year to the end of a year. In the continuous-time case, we need to derive a generator matrix G. I am not sure I understood it correctly, but the generator matrix needs to be derived from an observed transition matrix, correct? If yes, what is the best way of obtaining the observed transition matrix? Simply the cohort method (and afterwards some smoothing etcetera to ensure that the generator matrix exists) or should I take into account all rating migrations during the one year time interval? ## Answer by user60544 (score 1, accepted) https://quant.stackexchange.com/a/69324 In case an answer is still useful for you after 11 months -> No, a generator matrix can be directly derived from observed rating transition data from which a transition probability matrix can be derived using the matrix exponential. For continuous-time Markov chains direct MLE is possible (and best) for discrete, you can look into applying an EM algortihm (also considered the best). Trying to find a generator matrix from a transition probability matrix makes you run into several mathematical problems (embeddability, identification etc.), thus you should try to coordinate the generator matrix unto rating transition data directly.
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