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

Abstract

It is a well-established fact that testing a null hypothesis on the boundary of the parameter space, with an unknown number of nuisance parameters at the boundary, is infeasible in practice in the sense that 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 mulitiple) 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 …nite-order ARCH models with coe¢ cients subject to inequality constraints. Extensive Monte Carlo simulations illustrate that the proposed bootstrap has attractive …nite sample properties both under the null and under the alternative hypothesis.

Specification Tests for GARCH Processes with Nuisance Parameters on the Boundary

“…n follows from that of φn , and hence (S.18) holds. Since c n converges to 0 at a rate slower than n −1/2 , (S.19) follows by arguing as in the proof of Lemma 1 in Cavaliere et al (2022). Hence, by a triangular array extension of the proof of Theorem 2 of Francq and Zakoian (2007), under Assumption (A5), we obtain that (S.17) holds; e.g., by arguing as in the proof of Proposition 3.2 in Hidalgo and Zaffaroni (2007); see also the discussion under Assumption E2 in Andrews (1997).…”

A Primer on Bootstrap Testing of Hypotheses in Time Series Models: With an Application to Double Autoregressive Models

“…In most cases, however, the null hypothesis restricts only a subset of the parameters. An example is testing if a parameter is on the boundary of the parameter space when the remaining parameters might be on the boundary, as in Cavaliere, Nielsen, Pedersen and Rahbek (2019). In this case, the limiting distribution of the bootstrap statistic depends on the asymptotic properties of the estimators used to generate the bootstrap data.…”

A residual bootstrap for conditional Value-at-Risk

“…Parameters on the boundary yield non-standard problems, which require special treatment. Cavaliere et al (2022) provide bootstrap inference on the boundary of the parameter space with application to conditional volatility models. We return to this issue in our empirical application in Remark 5.…”

“…Whereas in the former the bootstrap observations are generated recursively using the estimated volatility dynamics, the latter design keeps the dynamics of the bootstrap samples fixed at the value of the original series. Further recent applications of fixed or recursive bootstrap designs or variants thereof to conditional volatility models can be found in Hetland et al (2021), Francq and Zakoïan (2022) and Cavaliere et al (2022).…”