Testing when a parameter is on the boundary of the maintained hypothesis

This paper considers testing problems where several of the standard regularity conditions fail to hold. We consider the case where (i) parameter vectors in the null hypothesis may lie on the boundary of the: maintained hypothesis and (ii) there mas be a nuisance parameter that appears under the alternative hypothesis, but not under the null. The paper establishes the asymptotic null and local alternative distributions of quasi-likelihood ratio, rescaled quasi-likelihood ratio, Wald, and score tests in this case. The results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models. Examples treated in the paper are: (i) tests of the null hypothesis of no conditional heteroskedasticity in a GARCH(1,1) regression model and (ii) tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with (possibly) correlated random coefficients.