Multivariate backtests and copulas for risk evaluation

TL;DR

This paper introduces a method using bivariate copulas and probability integral transforms to validate multivariate risk forecasts, demonstrating the effectiveness of Student copulas and historical innovation-based forecasts in financial time series.

Contribution

It develops a framework for validating bivariate risk forecasts using copulas and transforms, with empirical evidence on financial data showing the effectiveness of Student copulas and historical models.

Findings

Student copula models dependencies well regardless of correlation

Historical innovation-based forecasts perform correctly out-of-sample

Removing heteroskedasticity is essential for stationarity in analysis

Abstract

Risk evaluation is a forecast, and its validity must be backtested. Probability distribution forecasts are used in this work and allow for more powerful validations compared to point forecasts. Our aim is to use bivariate copulas in order to characterize the in-sample copulas and to validate out-of-sample a bivariate forecast. For both set-ups, probability integral transforms (PIT) and Rosenblatt transforms are used to map the problem into an independent copula. For this simple copula, statistical tests can be applied to validate the choice of the in-sample copula or the validity of the bivariate forecast. The salient results are that a Student copula describes well the dependencies between financial time series (regardless of the correlation), and that the bivariate forecasts provided by a risk methodology based on historical innovations performs correctly out-of-sample. A prerequisite is to remove the heteroskedasticity in order to have stationary time series, in this work a long-memory ARCH volatility model is used.