8.4 TESTING FOR NONLINEARITY

8.4 TESTING FOR NONLINEARITY

Nonlinear models typically nest linear models as special cases and therefore can be expected to provide better in-sample fits of the data simply because they adopt a more flexible functional form. Most applied studies using nonlinear forecasting models examine tests of the null of nonlinearity to motivate their use of such models, so it is helpful to briefly review such tests.

Tests of the null of a linear model against a nonlinear alternative that nests the linear specification often result in nonstandard problems since most nonlinear models have parameters that are not identified under the null hypothesis of a linear model. This problem can be illustrated using the nonlinear models introduced above. For the TAR model in (8.5), the threshold, s , is not identified since the model is the same for all s under the null that the parameters do not differ across regimes. The STAR models in (8.12) are linear if $\gamma =0;$ , in which case c is unidentified since the model is the same for all possible values for c. Similarly, for the Markov switching model, if the linear specifications are the same in each regime, the transition probabilities $\mathcal{P}ij$ are not identified.

This problem—illuminated in Davies (1977) and Davies (1987) and discussed extensively in the statistical and econometric literature since then—has three implications. First, Likelihood Ratio (LR), Wald and Lagrange Multiplier (LM) tests are typically not asymptotically equivalent under the null and alternative hypotheses; second, these tests are often no longer well approximated by the usual ${\chi }^2$ distributions; third, the LR statistic is no longer necessarily an optimal test.

The approach favored in the literature on STAR models is to take a first-order expansion around the model under the null $(\gamma =0)$ and test whether the additional terms that arise from the expansion, over and above the linear model, are significant. See Teräsvirta (2006) for further discussion.

One issue with this approach is that if the tests are rejected, it does not follow directly that a particular nonlinear model is appropriate. The tests that are usually employed have power against many forms of nonlinearity, and so a rejection does not imply a particular parametric form for the apparent nonlinearity and the forecaster does not know which of the above models, if any, would provide the best forecasts.