Abstract: We present supremum Lagrange multiplier tests for comparing a linear ARMA specification against its threshold ARMA extension. We derive the asymptotic distribution of the test statistics under both the null hypothesis and contiguous local alternatives, and prove the consistency of the tests. A Monte Carlo study shows that the tests enjoy good finite-sample properties and are robust against various forms of model mis-specification. Furthermore, the performance of the tests is not affected by the unknown order of the model. The tests have a low computational burden and do not suffer from some of the drawbacks that affect the quasi-likelihood ratio setting. Lastly, we present an application to a time series of standardized tree-ring growth indices. Our results can lead to new research in climate studies.

Keywords: Lagrange multiplier test, linearity testing, marked empirical processes, threshold ARMA models, tree-rings.