Bootstrapping a Powerful Mixed Portmanteau Test for Time Series

A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. In this paper, we elaborate on the Toeplitz autocorrelation matrix to the autocorrelation and cross-correlation of residuals and squared residuals block matrix. We derive a new portmanteau test statistic using the log of the determinant of the mth autocorrelations and cross-correlations block matrix. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-squared distributions and can be approximated by a gamma distribution. This test is applied to identify the linearity and nonlinearity dependency of some stationary time series models. It is shown that the convergence of the new test to its asymptotic distribution is reasonable with higher power than other tests in many situations. We demonstrate the efficiency of the proposed test by investigating linear and nonlinear effects in Vodafone Qatar and Nikkei-300 daily returns.