Test for serial independence based on quadratic forms

In time series analysis, tests for serial independence, symmetry, and goodness-of-fit based on divergence measures, such as the Kullback-Leibler divergence or Hellinger distance are currently receiving much interest (see Granger, Maasoumi, Racine (2004) as a recent example). We consider replacing the divergence measures in these tests by kernel-based positive definite bilinear forms. In this way we avoid the common practice of using plug-in estimators. Our approach separates the problem of consistent estimation of the divergence measure from that of estimating the underlying joint densities consistently. We construct a test for serial independence on the basis of the introduced bilinear forms. Optimal bandwidth selection is a common problem in the nonparametric econometrics. To confront this problem we use an adaptive bandwidth procedure over a range of different bandwidth values. In order to produce an exact test, a permutation procedure is favoured over the use of asymptotic theory. Our results are illustrated with simulations for various data generating processes relevant to financial econometrics. We compare the performance of our test with existing nonparametric tests for serial independence. For certain class of processes our approach produces higher power in comparison with BDS test and the test of Granger, Maasoumi and Racine