Persistence-Robust Break Detection in Predictive CoVaR Regressions

Forecasting risk (as measured by quantiles) and systemic risk (as measured by Adrian and Brunnermeiers’s (2016) CoVaR) is important in economics and finance. However, past research has shown that predictive relationships may be unstable over time. Therefore, this paper develops structural break tests in predictive quantile and CoVaR regressions. These tests can detect changes in the forecasting power of covariates, and are based on the principle of self-normalization. We show that our tests are valid irrespective of whether the predictors are stationary or near-stationary, rendering the tests suitable for a range of practical applications. Simulations illustrate the good finite-sample properties of our tests. Two empirical applications concerning equity premium and systemic risk forecasting models show the usefulness of the tests.

💡 Research Summary

The paper tackles a crucial gap in the literature on systemic risk measurement: while CoVaR (Conditional Value‑at‑Risk) has become a standard tool for quantifying the risk spillover from a distressed institution to the financial system, there has been no formal methodology to test whether the predictive relationship between a set of covariates and CoVaR remains stable over time. The authors develop a suite of structural break tests for both predictive quantile regressions and predictive CoVaR regressions that are robust to the persistence properties of the regressors.

Model framework
The authors consider a standard linear quantile regression for a “distress” variable (Y_t) and a linear CoVaR regression for a “systemic loss” variable (Z_t). The quantile regression is written as (Y_t = X_{t-1}’\alpha_0 + \varepsilon_t) with (Q_\alpha(\varepsilon_t|\mathcal{F}{t-1})=0). The CoVaR regression is (Z_t = X{t-1}’\beta_0 + \delta_t) together with the condition (\text{CoVaR}{\beta|\alpha}(\delta_t,\varepsilon_t|\mathcal{F}{t-1})=0). The vector of predictors (X_{t-1}) may be stationary, mildly integrated (near‑stationary), or a mixture of both.

Estimation
Quantile coefficients (\alpha_0) are estimated by minimizing the usual pinball loss. CoVaR coefficients (\beta_0) are estimated by a weighted pinball loss that only uses observations for which the distress variable exceeds its conditional (\alpha)-quantile, i.e., (Y_t > X_{t-1}’\hat\alpha_n). This reflects the definition of CoVaR as a conditional quantile given a distress event.

Testing methodology
The core of the paper is a self‑normalized (SN) structural break test. For any split point (s\in