## The Annals of Statistics

### Sieve-based inference for infinite-variance linear processes

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 44, Number 4 (2016), 1467-1494.

Dates
Revised: November 2015
First available in Project Euclid: 7 July 2016

https://projecteuclid.org/euclid.aos/1467894705

Digital Object Identifier
doi:10.1214/15-AOS1419

Mathematical Reviews number (MathSciNet)
MR3519930

Zentralblatt MATH identifier
06624584

#### Citation

Cavaliere, Giuseppe; Georgiev, Iliyan; Taylor, A. M. Robert. Sieve-based inference for infinite-variance linear processes. Ann. Statist. 44 (2016), no. 4, 1467--1494. doi:10.1214/15-AOS1419. https://projecteuclid.org/euclid.aos/1467894705

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#### Supplemental materials

• Supplement to “Sieve-based inference for infinite-variance linear processes”. In this supplement, which contains additional theoretical results and proofs, we provide: a lemma with two tail inequalities regarding the series of the coefficients from the AR($\infty$) representations; a proof of Lemma 2 and corollaries from Section 6; proofs of the results given in Section 7.3.1; a discussion of multiple restrictions.