  We present supremum Lagrange Multiplier tests to compare a linear ARMA
specification against its threshold ARMA extension. We derive the asymptotic
distribution of the test statistics both under the null hypothesis and
contiguous local alternatives. Moreover, we prove the consistency of the tests.
The Monte Carlo study shows that the tests enjoy good finite-sample properties,
are robust against model mis-specification and their performance is not
affected if the order of the model is unknown. The tests present a low
computational burden and do not suffer from some of the drawbacks that affect
the quasi-likelihood ratio setting. Lastly, we apply our tests to a time series
of standardized tree-ring growth indexes and this can lead to new research in
climate studies.
