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LM-Tests for Linearity Against Smooth Transition Alternatives: A Bootstrap Simulation Study

By Jonathan B. Hill

Abstract

The universal method for testing linearity against smooth transition autoregressive (STAR) alternatives is the linearization of the STAR model around the null nuisance parameter value, and performing F-tests on polynomial regressions in the spirit of the RESET test. Polynomial regressors, however, are poor proxies for the nonlinearity associated with STAR processes, and are not consistent (asymptotic power of one) against STAR alternatives, let alone general deviations from the null. Moreover, the most popularly used STAR forms of nonlinearity, exponential and logistic, are known to be exploitable for consistent conditional moment tests of functional form, cf. Bierens and Ploberger (1997). In this paper, pushing asymptotic theory aside, we compare the small sample performance of the standard polynomial test with an essentially ignored consistent conditional moment test of linear autoregression against smooth transition alternatives. In particular, we compute an LM sup-statistic and characterize the asymptotic p-value by Hansen's (1996) bootstrap method. In our simulations, we randomly select all STAR parameters in order not to bias experimental results based on the use of "safe", "interior" parameter values that exaggerate the smooth transition nonlinearity. Contrary to past studies, we find that the traditional polynomial regression method performs only moderately well, and that the LM sup-test out-performs the traditional test method, in particular for small samples and for LSTAR processes.Smooth transition AR, consistent conditional moment test, Lagrange Multiplier, bootstrap

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