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We present a statistical test that can be used to verify supervisoryrequirements concerning overlapping time windows for the long-term calibrationin rating systems. In a first step, we show that the long-run default rate isapproximately normally distributed with respect to random effects in defaultrealization. We then perform a detailed analysis of the correlation effectscaused by the overlapping time windows and solve the problem of an unknowndistribution of default probabilities for the long-run default rate. In thiscontext, we present several methods for a conservative calibration test thatcan deal with the unknown variance in the test statistic. We present a test forindividual rating grades, and then pass to the portfolio level by suitablyadapting the test statistic. We conclude with comparative statics analysing theeffect of persisting customers and the number of customers per reference date.

A hypothesis test for the long-term calibration in rating systems with overlapping time windows