The log of the determinant of the autocorrelation matrix for testing goodness of fit in time series

A finite sample modification of a test by Peña and Rodríguez is proposed. The new modified test is asymptotically equivalent but it has a more intuitive explanation and it can be 25% more powerful for small sample size than the previous one. The test statistic is the log of the determinant of the mmth autocorrelation matrix. We propose two approximations by using the Gamma and the Normal distributions to the asymptotic distribution of the test statistic. It is shown that, depending on the model and sample size, the proposed test can be up to 50% more powerful than the Ljung and Box, Monti and Hong tests, and for finite sample size is always better than the previous Peña–Rodríguez test. This modified test is applied to the detection of several types of nonlinearity by using either the autocorrelation matrix of the squared or the absolute values of the residuals. It is shown that, in general, the new test is more powerful than the one by McLeod and Li.