On the Importance of the First Observation in GLS Detrending in Unit Root Testing
| DOI | http://doi.org/10.1111/obes.12050 |
| Published date | 01 February 2015 |
| Author | Joakim Westerlund |
| Date | 01 February 2015 |
152
©2013 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 77, 1 (2015) 0305–9049
doi: 10.1111/obes.12050
On the Importance of the First Observation in GLS
Detrending in Unit Root Testing*
Joakim Westerlund†
†Financial Econometrics Group, School of Accounting, Economics, and Finance, 70 Elgar
Road, Burwood Highway, VIC 3125, Deakin University, Melbourne, Australia,
(e-mail: j.westerlund@deakin.edu.au)
Abstract
First-differencing is generally taken to imply the loss of one observation, the first, or at
least that the effect of ignoring this observation is asymptotically negligible. However, this
is not always true, as in the case of generalized least squares (GLS) detrending. In order to
illustrate this, the current article considers as an example the use of GLS detrended data
when testing for a unit root. The results show that the treatment of the first observation is
absolutely crucial for test performance, and that ignorance causes test break-down.
I. Introduction
Consider the time series variable {Xs}t
s=1. The present article originates with the following
very basic question: how should wedefine {Xs}t
s=1, where Xs=Xs−Xs−1? In particular,
while the definitions of X2,…, Xtare clear, it is less obvious how to treat X1. One way
is to say that, as accumulation should undo first-differencing, we must have X1=X1,
because only then will it be true that t
s=1Xs=Xt. Hence, according to this, we have
Xs=Xs−Xs−1for s=2,…,tand X1=X1.(1)
However, this definition is probably more the exception rather than the rule. Indeed, the by
far most common approach, is to simply ignore X1, as when defining
Xs=Xs−Xs−1for s=2,…,tand X1=0, (2)
in which case t
s=1Xs=t
s=2Xs=Xt−X1. The obvious rationale for this in the
unit root case is that as the relevant quantity here is not t
s=1Xsbut rather T−1/2times
this sum, the effect of ignoring X1is negligible; T−1/2t
s=2Xs=T−1/2(Xt−X1)=
T−1/2Xt+op(1). This is logical; if the data are Op(√T), as in the unit root case, the
treatment of the first observation, which is only Op(1), should not matter.
One situation when this issue becomes relevant is when performing ‘generalized least
squares (GLS) detrending’ (Elliott, Rothenberg and Stock, 1996). In this case, the
relevant (quasi) differenced variable is given by ¯
Xt=Xt−¯
Xt−1, where ¯
is such that
*The author would like to thank Anindya Banerjee and twoanonymous referees for many useful comments and
suggestions.
JEL Classification numbers: C12, C13, C33.
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