Bootstrap Inference on the Boundary of the Parameter Space with Application to Conditional Volatility Models

36 Pages Posted: 5 Dec 2018

See all articles by Giuseppe Cavaliere

Giuseppe Cavaliere

University of Bologna - Department of Economics

Heino Bohn Nielsen

University of Copenhagen - Department of Economics

Rasmus Søndergaard Pedersen

University of Copenhagen

Anders Rahbek

University of Copenhagen - Department of Statistics and Operations Research; University of Copenhagen - Department of Economics

Date Written: November 12, 2018

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 finite-order ARCH models with coefficients subject to inequality constraints. Extensive Monte Carlo simulations illustrate that the proposed bootstrap has attractive finite sample properties both under the null and under the alternative hypothesis.

Keywords: Inference on the boundary, Nuisance parameters on the boundary, ARCH models, Bootstrap

JEL Classification: C12, C22

Suggested Citation

Cavaliere, Giuseppe and Nielsen, Heino Bohn and Pedersen, Rasmus Søndergaard and Rahbek, Anders, Bootstrap Inference on the Boundary of the Parameter Space with Application to Conditional Volatility Models (November 12, 2018). Available at SSRN: https://ssrn.com/abstract=3282935 or http://dx.doi.org/10.2139/ssrn.3282935

Giuseppe Cavaliere (Contact Author)

University of Bologna - Department of Economics ( email )

Bologna
Italy
+390512098489 (Phone)

Heino Bohn Nielsen

University of Copenhagen - Department of Economics ( email )

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.
Denmark

Rasmus Søndergaard Pedersen

University of Copenhagen

Nørregade 10
Copenhagen, DK-1165
Denmark

Anders Rahbek

University of Copenhagen - Department of Statistics and Operations Research

Universitetsparken 5
DK-2100
Denmark
+45 3532 0682 (Phone)

University of Copenhagen - Department of Economics

Øster Farimagsgade 5
Bygning 26
1353 Copenhagen K.
Denmark

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