Parameters on the boundary in predictive regression

TL;DR

This paper studies bootstrap inference in predictive regressions with parameters on the boundary, addressing challenges from boundary constraints and non-stationarity, and proposes a modified bootstrap scheme.

Contribution

It introduces a data-dependent shift in the bootstrap parameter space to improve inference when parameters lie on the boundary and addresses non-stationarity issues.

Findings

Modified bootstrap scheme is valid under non-stationarity.

Boundary parameters cause the bootstrap distribution to be random.

The approach extends to broader boundary-inference problems.

Abstract

We consider bootstrap inference in predictive (or Granger-causality) regressions when the parameter of interest may lie on the boundary of the parameter space, here defined by means of a smooth inequality constraint. For instance, this situation occurs when the definition of the parameter space allows for the cases of either no predictability or sign-restricted predictability. We show that in this context constrained estimation gives rise to bootstrap statistics whose limit distribution is, in general, random, and thus distinct from the limit null distribution of the original statistics of interest. This is due to both (i) the possible location of the true parameter vector on the boundary of the parameter space, and (ii) the possible non-stationarity of the posited predicting (resp. Granger-causing) variable. We discuss a modification of the standard fixed-regressor wild bootstrap scheme where the bootstrap parameter space is shifted by a data-dependent function in order to eliminate the portion of limiting bootstrap randomness attributable to the boundary, and prove validity of the associated bootstrap inference under non-stationarity of the predicting variable as the only remaining source of limiting bootstrap randomness. Our approach, which is initially presented in a simple location model, has bearing on inference in parameter-on-the-boundary situations beyond the predictive regression problem.