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The probability of default (PD) estimation is an important process forfinancial institutions. The difficulty of the estimation depends on thecorrelations between borrowers. In this paper, we introduce a hierarchicalBayesian estimation method using the beta binomial distribution and consider amulti-year case with a temporal correlation. A phase transition occurs when thetemporal correlation decays by power decay. When the power index is less thanone, the PD estimator does not converge. It is difficult to estimate the PDwith limited historical data. Conversely, when the power index is greater thanone, the convergence is the same as that of the binomial distribution. Weprovide a condition for the estimation of the PD and discuss the universalityclass of the phase transition. We investigate the empirical default datahistory of rating agencies and their Fourier transformations to confirm theform of the correlation decay. The power spectrum of the decay history seems tobe 1/f, which corresponds to a long memory. But the estimated power index ismuch greater than one. If we collect adequate historical data,the parameterscan be estimated correctly.

Phase transition in the Bayesian estimation of the default portfolio