Bootstrap inference on the boundary of the parameter space, with application to conditional volatility models

It is a well-established fact that ñwith an unknown number of nuisance parameters at the boundary ñtesting a null hypothesis on the boundary of the parameter space is infeasible in practice as the limiting distributions of standard test statistics are non-pivotal. In particular, likelihood ratio statistics have limiting distributions which can be characterized in terms of quadratic forms minimized over cones, where the shape of the cones depends on the unknown location of the (possibly multiple) model parameters not restricted by the null hypothesis. We propose to solve this inference problem by a novel bootstrap, which we show to be valid under general conditions, irrespective of the presence of (unknown) nuisance parameters on the boundary. That is, the new bootstrap replicates the unknown limiting distribution of the likelihood ratio statistic under the null hypothesis and is bounded (in probability) under the alternative. The new bootstrap approach, which is very simple to implement, is based on shrinkage of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at an appropriate rate. As an application of our general theory, we treat the problem of inference in Önite-order ARCH models with coe¢ cients subject to inequality constraints. Extensive Monte Carlo simulations illustrate that the proposed bootstrap has attractive Önite sample properties both under the null and under the alternative hypothesis.