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

Paper: arXiv 2312.14765

Authors: Patrick Kurth, Max Nendel, Jan Streicher

Abstract

We present a statistical test that can be used to verify supervisory requirements concerning overlapping time windows for the long-term calibration in rating systems. In a first step, we show that the long-run default rate is approximately normally distributed with respect to random effects in default realization. We then perform a detailed analysis of the correlation effects caused by the overlapping time windows and solve the problem of an unknown distribution of default probabilities for the long-run default rate. In this context, we present several methods for a conservative calibration test that can deal with the unknown variance in the test statistic. We present a test for individual rating grades, and then pass to the portfolio level by suitably adapting the test statistic. We conclude with comparative statics analysing the effect of persisting customers and the number of customers per reference date.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 4.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical derivation for hypothesis testing in credit rating calibration under overlapping time horizons and unknown variances. However, it lacks extensive empirical backtesting on real-world portfolio data, leaning heavily on analytical formulation and comparative statics.

Research Flowchart

  flowchart TD
    A[Research Goal: Test long-term calibration in rating systems with overlapping time windows] --> B[Inputs: Default rates, rating grades & portfolio data across reference dates]
    B --> C[Step 1: Prove long-run default rate is approximately normally distributed]
    C --> D[Step 2: Analyze correlation effects from overlapping windows & solve unknown PD distribution]
    D --> E[Step 3: Develop conservative calibration test for individual rating grades]
    E --> F[Step 4: Adapt test statistic for portfolio-level testing]
    F --> G[Outcomes: Robust hypothesis test & comparative statics on customer persistence/volume]