Unit Root Inference for Non-Stationary Linear Processes Driven by Infinite Variance Innovations

Abstract

The contribution of this paper is two-fold. First, we derive the asymptotic null distribution of the familiar augmented Dickey-Fuller [ADF] statistics in the case where the shocks follow a linear process driven by in…nite variance innovations. We show that these distributions are free of serial correlation nuisance parameters but depend on the tail index of the in…nite variance process. These distributions are shown to coincide with the corresponding results for the case where the shocks follow a …nite autoregression, provided the lag length in the ADF regression satis…es the same o(T 1=3 ) rate condition as is required in the …nite variance case. In addition, we establish the rates of consistency and (where they exist) the asymptotic distributions of the ordinary least squares sieve estimates from the ADF regression. Given the dependence of their null distributions on the unknown tail index, our second contribution is to explore sieve wild bootstrap implementations of the ADF tests. Under the assumption of symmetry, we demonstrate the asymptotic validity (bootstrap consistency) of the wild bootstrap ADF tests. This is done by establishing that (conditional on the data) the wild bootstrap ADF statistics attain the same limiting distribution as that of the original ADF statistics taken conditional on the magnitude of the innovations.

Unit Root Tests and Heavy‐Tailed Innovations

“…In the pure finite variance case ( ), Chang and Park () show that provided the lag truncation parameter, p T in , satisfies the rate condition as , and that standard summability and moment conditions hold, that and have the usual pivotal Dickey–Fuller limiting null distributions (which are functionals of a standard Brownian motion process) regardless of any weak dependence present in u t . Cavaliere et al () demonstrate that the same rate condition is sufficient in the pure infinite variance case, . In particular, they show that under the summability conditions of Assumption .5, this rate condition on p T ensures that and have the same limiting null distributions when weak dependence is present as in the case where u t is i.i.d.…”

“…The same convergence holds for the ASD estimator, , based on . Specifically, in the pure infinite variance context, the consistency of the estimators , j =1,…, p T , and the convergence are established in Cavaliere et al (). Although and need to be re‐normalised to achieve non‐trivial convergence, no re‐normalisations are needed in and because the contributions of a T cancel out, as does in the limits of the statistics.…”

“…More recently, Samarakoon and Knight () consider M ‐based testing for unit roots in finite‐order autoregressive processes driven by infinite variance innovations. Bootstrap‐based unit root tests are proposed by Horváth and Kokoszka (), Moreno and Romo () and Cavaliere et al (); the latter focusing on the popular augmented Dickey–Fuller (ADF) tests of Dickey and Fuller (). Chan et al () derive results for inference in a near‐integrated first‐order autoregressive process with i.i.d.…”

Tests of Unit Root Hypothesis With Heavy-Tailed Heteroscedastic Noises

“…heavy-tailed innovations; see Horváth and Kokoszka (2003) and Moreno and Romo (2012) for early work. Zhang and Chan (2020) extended the results in Cavaliere, Georgiev and Taylor (2018) to the case where the noise is from standard GARCH model, but the simulation results in their work indicate that the aforementioned wild bootstrap method can not deal with the heavy-tailed GARCH noises. Recently, Huang et al (2020) proposed a novel empirical-likelihood-based method to construct a unified test for model (1.1) with the noise following from the standard GARCH(p, q) model, namely,…”

“…To bypass the problem in heavy-tailed time series, one popular approach is to use the bootstrap or subsampling method to approximate the critical values. For example, Cavaliere, Georgiev and Taylor (2018) proposed a sieve wild bootstrap method to obtain the null distribution of augmented DF (ADF) test when the noise is a linear process driven by the i.i.d. heavy-tailed innovations; see Horváth and Kokoszka (2003) and Moreno and Romo (2012) for early work.…”