On the Behaviour of Phillips–Perron Tests in the Presence of Persistent Cycles
| DOI | http://doi.org/10.1111/obes.12091 |
| Author | A. M. Robert Taylor,Paulo M. M. Rodrigues,Tomás Del Barrio Castro |
| Published date | 01 August 2015 |
| Date | 01 August 2015 |
495
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 77, 4 (2015) 0305–9049
doi: 10.1111/obes.12091
On the Behaviour of Phillips–Perron Tests in the
Presence of Persistent Cycles*
Tom ´
as Del Barrio Castro†, Paulo M. M. Rodrigues‡ and
A. M. Robert Taylor§
†Department of Applied Economics, University of the Balearic Islands, Spain
(e-mail: tomas.barrio@uib.es)
‡Banco de Portugal, NOVA School of Business and Economics, Universidade Nova de
Lisboa, CEFAGE, Lisboa, Portugal (e-mail: pmrodrigues@bportugal.pt)
§Essex Business School, University of Essex, Colchester, CO4 3SQ, UK
(e-mail: rtaylor@essex.ac.uk)
Abstract
In this paper, we analyse the impact of persistent cycleson the well-known semi-parametric
unit root tests of Phillips and Perron (1988, Biometrika, Vol. 75, pp. 335–346). It is shown,
both analytically and through Monte Carlo simulations, that the presence of complex(near)
unit roots can severely bias the size properties of these tests. Given the popularity of these
tests with applied researchers and their routine presence in most econometric software
packages, the results presented in this paper suggest that practitioners should treat the
outcomes of these tests with some caution when applied to data which display a strong
cyclical component.
I. Introduction
Cyclical behaviour is an inherent feature of many macroeconomic, financial and other time
series. For example, Canova (1996) discusses the literature on Bayesian learning (Nyarko,
1992) and on noise traders in financial markets (Campbell and Kyle, 1993) where models
which generate irregularly spaced but significant cycles in economic activity and asset
prices are proposed. The presence of cycles is also documented in the political economy
literature (e.g. electoral cycles in government variables, Alesina and Roubini, 1992) and
naturally arises in the business cycle literature (Pagan, 1997).
Interestingly, the existence of both complex and real unit (or near-unit) roots can in-
duce growth cycles similar to those observed in economic data (Allen, 1997). For exam-
ple, Bierens (2001) finds that business cycles may indeed be due to complex unit roots.
Shibayama (2008) analyses inventories and monetary policy and detects complex roots
JEL Classification numbers: C12, C22
*We thank the Co-Editor, JonathanTemple, and two anonymous referees for their helpful and constructive com-
ments and suggestions on earlier versions of this paper.
496 Bulletin
generating cycles of around 55–70 months, which he argues are close to business cycle
lengths. Azariadis, Bullard and Ohanian (2004) provide an overview of evidence on the
eigenvalues from theAR and VAR representations of macroeconomic variables; they argue
that economic time series typically possess an autocorrelation structure that has complex or
negative eigenvalues in empirical reduced form. A further example is provided by Estrella
(2004), who uses autoregressive processes with complex roots to examine the cyclical be-
haviour of optimal bank capital, where a cycle is defined as a particular predictable pattern
which unfolds over time.
As a result, given the empirical relevance of persistent cycles, studying their impact on
the performance of pretesting procedures, in particular on the limiting null distributions and
finite-sample properties of zero frequency unit root test statistics, is of particular practical
relevance (see also Castro, Rodrigues and Taylor, 2013).
This paper contributes to the literature in two ways. First it provides asymptotic and
finite-sample results for the ordinary least squares (OLS) estimator of the parameter of
a nearly integrated first-order autoregressive (AR(1)) model driven by shocks which are
generated according to a near-integrated cyclical process; that is, a process characterized by
a second-order autoregressive structure with complex roots in the neighbourhood of unity.
A second contribution relates to the discussion of the behaviour of the well known and
widely used Phillips–Perron (PP) unit root tests (Phillips, 1987, Phillips and Perron, 1988)
in this important context. Although a lot of research has focused on the developmentof unit
root tests with improved size and power properties, the early tests introduced by Dickey
and Fuller (1979) and Phillips and Perron (1988) are still among the most widely used. One
of the main reasons is that these procedures are typically readily available to researchers
in most econometric software packages, and another important reason results from their
simplicity of application. Hence, empirical work that investigates the (non-)stationarity
properties of data frequently reports applications of the PP test; as an interesting recent
example see Erten and Ocampo (2013). However, in this paper we will show that care
needs to be taken when the PP tests are applied since the presence of persistent cycles can
seriously affect the finite-sample and asymptotic properties of these tests (see Castro et al.,
2013, for analysis of the implications for other unit root test procedures).
The remainder of the paper is organized as follows.In section II, we present our reference
time series model which allowsfor persistent cycles (cyclical near-unit roots) and we briefly
outline the PP unit root tests. Section III details the asymptotic behaviour of the PP unit root
statistics and associated tests when persistent cycles are present in the data. In section IV,
we report finite-sample simulation results relating to the performance of the least squares
estimator from a first order autoregression and of the PP tests when persistent cycles
are present in the data. Section V concludes. All proofs are collected in an mathematical
appendix.
II. The model and unit root tests
The time series model
Autoregressive (AR) processes with roots on the complex unit circle are non-stationary
and display persistent cyclical behaviour similar to that of persistent business cycles (see
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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