On the Consistency of Bootstrap Testing for a Parameter on the Boundary of the Parameter Space

It is well-known that with a parameter on the boundary of the parameter space, such as in the classic cases of testing for a zero location parameter or no ARCH effects, the classic nonparametric bootstrap - based on unrestricted parameter estimates - leads to inconsistent testing. In contrast, we show here that for the two aforementioned cases a nonparametric bootstrap test based on parameter estimates obtained under the null - referred to as 'restricted bootstrap' - is indeed consistent. While the restricted bootstrap is simple to implement in practice, novel theoretical arguments are required in order to establish consistency. In particular, since the bootstrap is analyzed both under the null hypothesis and under the alternative, non-standard asymptotic expansions are required to deal with parameters on the boundary. Detailed proofs of the asymptotic validity of the restricted bootstrap are given and, for the leading case of testing for no ARCH, a Monte Carlo study demonstrates that the bootstrap quasi-likelihood ratio statistic performs extremely well in terms of empirical size and power for even remarkably small samples, outperforming the standard and bootstrap Lagrange multiplier tests as well as the asymptotic quasi-likelihood ratio test.