On the Quantile Regression Based Tests for Asymmetry in Stock Return Volatility

This paper attempts to examine whether the asymmetry of stock return volatility varies with the level of volatility. Thus, quantile regression based tests (ρ‐tests) are presupposed. These tests differ from the diagnostic tests introduced by Engle and Ng (1993) insofar as they can provide a complete picture of asymmetries in volatility across quantiles of variance distribution and, in case of non‐normal errors, they have improved power due to their robustness against non‐normality. A small Monte Carlo evidence suggests that the Wald and likelihood ratio (LR) tests out of ρ‐tests are reasonable, showing that they outperform the Lagrange multiplier (LM) test based on least squares residuals when the innovations exhibit heavy tail. Using the normalized residuals obtained from AR(1)‐GARCH(1, 1) estimation, the test results demonstrated that only the TOPIX out of six stock‐return series had asymmetry in volatility at moderate level, while all stock return series except the FAZ and FA100 had more significant asymmetry in volatility at higher levels. Interestingly, it is clear from the empirical findings that, like hypothesis of leverage effects, volatility of the TOPIX, CAC40, and, MIB tends to respond significantly to extremely negative shock at high level, but is not correlated with any positive shock. These might be valuable findings that have not been seriously considered in past research, which has focussed only on mean level of volatility.