Robust Adaptive Rate-Optimal Testing for the White Noise Hypothesis

A new test is proposed for the weak white noise null hypothesis. The test is based on an automatic choice of the order for a Box-Pierce or Hong test statistic. The simplest version of the test uses Lobato (2001) or Kuan and Lee (2006) HAC critical values but the procedure is flexible enough to improve the detection properties of any prescribed test. This can allow for instance to calibrate the test for optimal detection of specific alternatives as in Delgado and Velasco (2010a). The data-driven order choice is tailored to give a test which achieves adaptive rate-optimality against several classes of alternatives, namely (i) alternatives with a large enough number of autocorrelation coefficients converging to 0 faster than the parametric rate; (ii) alternatives with a "peak and valley" spectral density function. A simulation experiment leads to prefer the Box-Pierce version of the test, both under the null and the alternative. An application to daily exchange rate returns illustrates the usefulness of the proposed approach.