Quantile regression for long memory testing: A case of realized volatility

A new class of quantile regression-based tests for fractional integration at individual and joint quantiles of a time series, thereby generalizing unit-root testing in this context, are introduced. The asymptotic null distributions of these tests are standard and free of nuisance parameters. The finite-sample validity of the approach is established through Monte Carlo simulations, which also provides evidence of power gains over least-squares-based procedures under non-Gaussian errors. An empirical application on daily realized volatility computed for a cross-section of individual stocks, on realized volatility of the S&P 500 index, and on option-based implied volatility is presented. The main finding is that the suitability of a fractionally integrated model with a constant order of integration around 0.4 cannot be rejected along the different percentiles of the distribution, providing in this way strong support for the existence of long memory in realized volatility from a completely new perspective.