Beg, A. B. M. Rabiul A., Silvapulle, Mervyn J., and Silvapulle, Paramsothy (2001) Tests against inequality constraints when some nuisance parameters are present only under the alternative: Test of ARCH in ARCH-M models. Journal of business and economic statistics, 19 (2). pp. 245-253.

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Abstract

In this article we develop improved statistical tests for situations satisfying the following two nonstandard conditions simultaneously: (a) Some nuisance parameters become unidentified under the null hypothesis, and (b) the alternative hypothesis is restricted in the sense that it has inequality constraints/multiparameter one-sided hypotheses. In the statistical and econometric literature, inference problems under these two nonstandard conditions have been studied separately but not simultaneously. For example, procedures to deal with the nonstandard condition (a) only have been studied by Bera and Ra and by Andrews and Ploberger; surveys of test procedures to deal with (b) only may be found in the work of Robertson, Wright, and Dykstra. A main contribution of this article is that, by pooling the ideas and insights from both these areas of literature, we develop new tests to deal with (a) and (b) simultaneously. Based on the approach that we take, we would conjecture that our tests should perform better than other tests that are available for tests under the nonstandard conditions (a) and (b). As an example, we consider the problem of testing whether or not the error variance is constant over time, in an ARCH-in-Mean (ARCH-M) model. We use this example to motivate and explain our ideas. A data example illustrates the application of the test in a simple situation. In a simulation study, we observed that the new test procedures proposed here performed better than the other available ones for this problem.

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