Recently, at work, I developed a project to estimate probabilities of default (PDs) using Markov Chains. We have a portafolio of loans classified in an internal rating (ratings from "Bucket" 1 to 10, where bucket10 is the state of default). The state of default is absorbent and the higher the Bucket, the higher the PD for a loan.
First, I estimated a monthly transition matrix, $M_{monthly}$, with the observed monthly transitions between buckets in a determined period. Then, to get annual PDs, I elevated to the 12 power this matrix.
However, my boss asked to make a yearly transtition matrix, $M_{yearly}$, (using observed YEARLY transitions between buckets) to compare it to $M_{monthly}^{12}$.
My question is, how can I compare them and tell if they are statistically similar? Do I compare all the entries of the matrices or only the last column of the matrices (the columns of the PDs)?
I have looked to different norms and the goodness-of-fit $X^2$ but Im not sure which is the most aproppiate test I can use.
I'll appreciate your help.
Thanks in Advance.