Detecting Serial Dependence in Tail Events

A test for serial independence is proposed which is related to the BDS test but focuses on tail event probabilities rather than probabilities near the center of the distribution. The motivation behind this approach is to obtain a test more suitable for detecting structure in the tails, such as remaining ARCH or GARCH type structure in standardized residuals of financial time series. The new test can be implemented easily by slight modification of the standard BDS test, and is also suitable for model identification. The BDS test and the modified version are compared numerically. To enable fair power comparisons, both tests are implemented as exact level Monte Carlo tests, enabling power calculations of the tests at identical actual sizes. The Monte Carlo implementation allows the use of test statistics which are considerably simpler than for the standard BDS test. For all nonlinear stochastic time series models examined the power of the new test is found to be uniformly larger over all practically reasonable values of the bandwidth parameter. The test is illustrated with an empirical application.