Open Access
August 2016 Sieve-based inference for infinite-variance linear processes
Giuseppe Cavaliere, Iliyan Georgiev, A. M. Robert Taylor
Ann. Statist. 44(4): 1467-1494 (August 2016). DOI: 10.1214/15-AOS1419


We extend the available asymptotic theory for autoregressive sieve estimators to cover the case of stationary and invertible linear processes driven by independent identically distributed (i.i.d.) infinite variance (IV) innovations. We show that the ordinary least squares sieve estimates, together with estimates of the impulse responses derived from these, obtained from an autoregression whose order is an increasing function of the sample size, are consistent and exhibit asymptotic properties analogous to those which obtain for a finite-order autoregressive process driven by i.i.d. IV errors. As these limit distributions cannot be directly employed for inference because they either may not exist or, where they do, depend on unknown parameters, a second contribution of the paper is to investigate the usefulness of bootstrap methods in this setting. Focusing on three sieve bootstraps: the wild and permutation bootstraps, and a hybrid of the two, we show that, in contrast to the case of finite variance innovations, the wild bootstrap requires an infeasible correction to be consistent, whereas the other two bootstrap schemes are shown to be consistent (the hybrid for symmetrically distributed innovations) under general conditions.


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Giuseppe Cavaliere. Iliyan Georgiev. A. M. Robert Taylor. "Sieve-based inference for infinite-variance linear processes." Ann. Statist. 44 (4) 1467 - 1494, August 2016.


Received: 1 January 2015; Revised: 1 November 2015; Published: August 2016
First available in Project Euclid: 7 July 2016

zbMATH: 06624584
MathSciNet: MR3519930
Digital Object Identifier: 10.1214/15-AOS1419

Primary: 62M10 , 62M15
Secondary: 62G09

Keywords: bootstrap , infinite variance , sieve autoregression , time series

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 4 • August 2016
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