Adaptive Rate-Optimal Detection of Small Autocorrelation Coefficients

A new test is proposed for the null of absence of serial correlation. The test uses a data-driven smoothing parameter. The resulting test statistic has a standard limit distribution under the null. The smoothing parameter is calibrated to achieve rate-optimality against several classes of alternatives. The test can detect alternatives with many small correlation coefficients that can go to zero with an optimal adaptive rate which is faster than the parametric rate. The adaptive rate-optimality against smooth alternatives of the new test is established as well. The test can also detect ARMA and local Pitman alternatives converging to the null with a rate close or equal to the parametric one. A simulation experiment and an application to monthly financial square returns illustrate the usefulness of the proposed approach.