HEGY Tests in the Presence of Moving Averages*
| Date | 01 October 2011 |
| DOI | http://doi.org/10.1111/j.1468-0084.2011.00633.x |
| Author | Tomás Del Barrio Castro,Denise R. Osborn |
| Published date | 01 October 2011 |
691
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2011. Published by Blackwell Publishing Ltd,
9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 73, 5 (2011) 0305-9049
doi: 10.1111/j.1468-0084.2010.00633.x
HEGY Tests in the Presence of Moving Averages*
Tom ´
as Del Barrio Castro† and Denise R. Osborn‡
†Department of Applied Economics, University of the Balearic Islands, Palma de Mallorca, 07013,
Spain (e-mail: tomas.barrio@uib.es)
‡Economics, School of Social Sciences, University of Manchester, Manchester, M13 9PL, UK
(e-mail: denise.osborn@manchester.ac.uk)
Abstract
We analyze the asymptotic distributions associated with the seasonal unit root tests of
Hylleberg et al. (1990) for quarterly data when the innovations follow a moving average
process. Although both the t- and F-type tests suffer from scale and shift effects compared
with the presumed null distributions when a fixed order of autoregressive augmentation is
applied, these effects disappear when the order of augmentation is sufficiently large. How-
ever, as found by Burridge and Taylor(2001) for the autoregressive case, individual t-ratio
tests at the semi-annual frequency are not pivotal even with high orders of augmentation,
although the corresponding joint F-type statistic is pivotal. Monte Carlo simulations verify
the importance of the order of augmentation for finite samples generated by seasonally
integrated moving average processes.
I. Introduction
The predominant approach to testing for unit roots in the context of seasonal time series is
that of Hylleberg, Engle, Granger and Yoo [HEGY] (1990), who develop the approach of
Dickey and Fuller (1979) in this context. More specifically, HEGY provide tests for unit
roots at the zero and each seasonal frequency, within the overall null hypothesis that
seasonal (or annual) differencing is required to induce stationarity in a quarterly time
series. In common with Dickey and Fuller (1979), HEGY assume that the process has an
autoregressive (AR) form, and hence AR augmentation is used to account for serial cor-
relation. However, Burridge and Taylor (2001) show that even with appropriate augmen-
tation, AR innovations cause the HEGY t-ratio statistics associated with the complex
conjugate unit roots at the semi-annual frequency /2 to depend on nuisance parameters.
Nevertheless, the distributions of the t-ratio tests associated with frequencies zero and ,
together with the joint F-type test associated with frequency /2, remain pivotal in this
situation.
One implication of the analysis of Burridge and Taylor (2001) is that the consequences
of serial correlation need to be considered carefully when applying HEGYseasonal unit root
ÅThe authors thank Robert Taylorand a referee for useful comments on an earlier draft of this article. Also, Tom´as
del Barrio Castro gratefully acknowledges financial support from Ministerio de Educaci´on y Ciencia ECO2008
05215.
JEL Classification numbers: C12, C22.
To continue reading
Request your trial