We introduce a new quantile regression approach to test for long memory in time series. The procedures proposed allow testing at individual and joint quantiles. The latter approach is useful to determine the order of integration over a set of percentiles, allowing in this way to more generally address the overall hypothesis of fractional integration. The 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 procedures under non-Gaussian errors. An empirical application of the new procedures on di§erent measures of daily realized volatility is presented. The main finding is that the suitability of a long-memory model with a constant order of integration around 0.4 cannot be rejected along the diferent percentiles of the distribution, providing in this way strong support to the existence of long memory
in realized volatility from a completely new perspective.