A Powerful Portmanteau Test of Lack of Fit for Time Series

Abstract

A new portmanteau test for time series, more powerful than the tests of Ljung and Box and Monti, is proposed. The test is based on the mth root of the determinant of the mth autocorrelation matrix. It is shown that the proposed statistic is a function of all of the squared multiple correlation coefficients of the regressions of the residuals on their lags when the number of lags goes from 1 to m. It can also be written as a function of the first m partial autocorrelation coefficients. The asymptotic distribution of the test statistic is a linear combination of chi-squared distributions and can be approximated by a gamma distribution. It is shown, depending on the model and sample size, that this test can be up to 50% more powerful than the Ljung and Box and Monti tests. The test is applied to the detection of several types of nonlinearity by using the autocorrelation matrix of the squared residuals, and it is shown that, in general, the new test is more powerful than the test of McLeod and Li. An example is presented in which this test finds nonlinearity in the residuals of the sunspot series.

KEY WORDS:

• ARIMA model
• Autocorrelation coefficient
• Autocorrelation matrix
• Goodness of fit
• Heteroscedasticity model
• Nonlinearity test
• Partial autocorrelation