Persistence-Robust Break Detection in Predictive CoVaR Regressions

Abstract

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.

Figures (14)

• Figure 1: Level of the VIX from 2005--2014.
• Figure 2: Top panel: Plot of the function $s\mapsto s^2(1-s)^2[\widehat{\bm \gamma}_n(0,s) - \widehat{\bm \gamma}_n(s,1)]^\prime \bm{\mathcal{N}}_{n,\bm \gamma}^{-1}(s)[\widehat{\bm \gamma}_n(0,s) - \widehat{\bm \gamma}_n(s,1)]$ for $\alpha=\beta=0.95$. The 5%-critical value is indicated by the dashed horizontal line. Middle panel: Plot of the function $s\mapsto s^2(1-s)^2[\widehat{\bm \alpha}_n(0,s) - \widehat{\bm \alpha}_n(s,1)]^\prime \bm{\mathcal{N}}_{n,\bm \alpha}^{-1}(s)[\widehat{\bm \alpha}_n(0,s) - \widehat{\bm \alpha}_n(s,1)]$ for $\alpha=0.95$. The 5%-critical value is indicated by the dashed horizontal line. Bottom panel: Rolling-window estimates of the slope coefficient $\beta_{1,t}$ of the VIX in the linear predictive CoVaR regression \ref{['eq:CoVaR appl']}. The rolling window estimates are based on 500 daily returns.
• Figure 3: Panel (a): Empirical size of test based on $\mathcal{U}_{n,\bm \alpha}$ for the quantile regression. Panel (b): Empirical size of test based on $\mathcal{U}_{n,\bm \gamma}$ for the CoVaR regression. In both panels, size is plotted as a function of the autoregressive parameter $r$ of the (I0) predictors. The dashed vertical lines correspond to the values of $r_n=1-n^{-0.5}$ in the (NS) setting. The dotted horizontal lines indicate the tests' nominal level of 5%.
• Figure 4: Panel (a): Empirical power of test based on $\mathcal{U}_{n,\bm \alpha}$ for the quantile regression. Panel (b): Empirical power of test based on $\mathcal{U}_{n,\bm \gamma}$ for the CoVaR regression. In both panels, power is plotted as a function of $\Delta$, i.e., the deviation from the null. The dotted horizontal lines indicate the tests' nominal level of 5%. Predictors are stationary with $r=0.5$ (such that $\kappa=0$).
• Figure 5: Panel (a): Empirical power of test based on $\mathcal{U}_{n,\bm \alpha}$ for the quantile regression. Panel (b): Empirical power of test based on $\mathcal{U}_{n,\bm \gamma}$ for the CoVaR regression. In both panels, power is plotted as a function of $\Delta$, i.e., the deviation from the null. The dotted horizontal lines indicate the tests' nominal level of 5%. Predictors are near-stationary with $r_n=1-n^{-0.5}$ (such that $\kappa=1/2$).

Theorems & Definitions (29)

• Theorem 1
• Corollary 1
• Remark 1: Computation of test statistic
• Remark 2: Mixed-persistence predictors
• Corollary 2
• Theorem 2
• Corollary 3
• Corollary 4
• Corollary 5
• Theorem 3