Analysis of Financial Time Series, Third Edition

4.2 Nonlinearity Tests

In this section, we discuss some nonlinearity tests available in the literature that have decent power against the nonlinear models considered in Section 4.1. The tests discussed include both parametric and nonparametric statistics. The Ljung–Box statistics of squared residuals, the bispectral test, and the Brock, Dechert, and Scheinkman (BDS) test are nonparametric methods. The RESET test (Ramsey, 1969), the F tests of Tsay (1986, 1989), and other Lagrange multiplier and likelihood ratio tests depend on specific parametric functions. Because nonlinearity may occur in many ways, there exists no single test that dominates the others in detecting nonlinearity.

4.2.1 Nonparametric Tests

Under the null hypothesis of linearity, residuals of a properly specified linear model should be independent. Any violation of independence in the residuals indicates inadequacy of the entertained model, including the linearity assumption. This is the basic idea behind various nonlinearity tests. In particular, some of the nonlinearity tests are designed to check for possible violation in quadratic forms of the underlying time series.

Q-Statistic of Squared Residuals

McLeod and Li (1983) apply the Ljung–Box statistics to the squared residuals of an ARMA(p, q) model to check for model inadequacy. The test statistic is

where T is the sample size, m is a properly chosen number of ...