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Higher-order residual analysis for AR-ARCH models with the TR test

Ramsey and Rothman (1996) design a statistical device to test for time reversibility (TR test). They claim that their procedure is not powerful against ARCH-type alternatives, which also allows Rothman (1998) to propose a strategy based on the TR test to detect bilinear and threshold autoregressive (TAR) models. However, this work shows through Monte Carlo analysis that the size of the TR test may be seriously distorted by this class of uncorrelated non-i.i.d. processes, and not necessarily time irreversible.