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PARAMETERS ON THE BOUNDARY IN PREDICTIVE REGRESSION

Published online by Cambridge University Press:  21 February 2025

Giuseppe Cavaliere Open the ORCID record for Giuseppe Cavaliere [Opens in a new window] ,
Iliyan Georgiev Open the ORCID record for Iliyan Georgiev [Opens in a new window]  and
Edoardo Zanelli Open the ORCID record for Edoardo Zanelli [Opens in a new window]
Giuseppe Cavaliere*
Affiliation:
University of Bologna, University of Exeter
Iliyan Georgiev
Affiliation:
University of Bologna
Edoardo Zanelli
Affiliation:
University of Bologna
*
Address correspondence to: Giuseppe Cavaliere, Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna, Italy; email: giuseppe.cavaliere@unibo.it.
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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.

Type
MISCELLANEA
Information
Econometric Theory , First View , pp. 1 - 31
Copyright
© The Author(s), 2025. Published by Cambridge University Press

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Footnotes

We thank Yixiao Sun (co-editor), two anonymous referees, Rasmus Søndergaard Pedersen and Mervyn Silvapulle for comments. Financial support from the Italian Ministry of University and Research (PRIN 2020 Grant 2020B2AKFW) is gratefully acknowledged.

References

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