Revisiting Error‐Autocorrelation Correction: Common Factor Restrictions and Granger Non‐Causality*
| Date | 01 April 2009 |
| DOI | http://doi.org/10.1111/j.1468-0084.2008.00538.x |
| Published date | 01 April 2009 |
| Author | Aris Spanos,Anya McGuirk |
273
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2009. Published by Blackwell Publishing Ltd,
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 71, 2 (2009) 0305-9049
doi: 10.1111/j.1468-0084.2008.00538.x
Revisiting Error-Autocorrelation Correction:
Common Factor Restrictions and Granger
Non-CausalityÅ
Anya McGuirk† and Aris Spanos‡
†SAS Institute Inc., Cary, NC 27513, USA (e-mail: anya.mcguirk@sas.com)
‡Department of Economics, Virginia Tech, Blacksburg, VA 24061, USA
(e-mail: aris@vt.edu)
Abstract
The paper questions the appropriateness of the practice known as ‘error-auto-
correlation correcting’ in linear regression, by showing that adopting an AR(1) error
formulation is equivalent to assuming that the regressand does not Granger cause any
of the regressors. This result is used to construct a new test for the common factor
restrictions, as well as investigate – using Monte Carlo simulations – other potential
sources of unreliability of inference resulting from this practice. The main con-
clusion is that when the Granger cause restriction is false, the ordinary least square
and generalized least square estimators are biased and inconsistent, and using auto-
correlation-consistent standard errors does not improve the reliability of inference.
I. Introduction
The general theme of this paper concerns the potential unreliability of inference asso-
ciated with a modelling practice known as ‘error-fixing’: re-modelling the error term
to account for the results of a particular mis-specification (M-S) test. The unreliability
is primarily due to certain invalid restrictions which are implicitly (and inadvertently)
imposed on the observable vector process underlying the data. The focus of this paper
is on how the re-modelling of the white-noise error term in linear regression (LR),
using an AR(1) error formulation to account for residual autocorrelation, often gives
rise to unreliable inferences. Its primary contribution comes in the form of unveiling
ÅMany thanks are due to the Associate Editor for numerous suggestions that improved the clarity of the
discussion substantially.
JEL Classification numbers: C2, C4.
274 Bulletin
these implicit restrictions by deriving an equivalence result between the assumed
probabilistic structure of the error and the observable processes involved. In addi-
tion, the paper brings out several implications of this equivalence result which are
used to elucidate a number of modelling issues pertaining to this form of ‘error fixing’.
Since Sargan (1964), it has been known that re-modelling a white-noise error term
{ut,t∈N:=(1, 2, ...)}in the LR model:
yt=Txt+ut,t∈N,(1)
using an AR(1) error process:
ut=ut−1+t,||<1, t∈N,(2)
giving rise to the restricted dynamic linear regression (RDLR) model:
yt=Txt+yt−1−Txt−1+t,t∈N,(3)
is equivalent to adopting the dynamic linear regression (DLR) model:
yt=1yt−1+T
0xt+T
1xt−1+vt,t∈N,(4)
subject to the common factor (CF) restrictions:
1−01=0.(5)
Hendry and Mizon (1978) and Sargan (1980) investigated these restrictions further
and proposed several tests for assessing their validity (see also Hendry, 2003).
These restrictions turn out to be of crucial importance in empirical modelling
in practice because when invalid, the ordinary least square (OLS) estimator of is
not just inefficient (as traditionally presumed), but also inconsistent, giving rise to
highly unreliable inferences (see Spanos, 1986; Mizon, 1995). In view of such serious
implications, Spanos (1988) investigated the CF restrictions (5) as they relate to the
probabilistic structure of the observable vector process {Zt,t∈N},Zt:=(yt,XT
t)T,by
raising the question: ‘what kind of probabilistic structure does one implicitly impose
on {Zt,t∈N}, when one re-models a white-noise error into an AR(1) process (2)?’
He provided only sufficient conditions which amounted to ‘when all the variables
in Ztare mutually Granger non-causal, the CF restrictions (5) hold’. Mutual Granger
non-causality seems highly unrealistic for most economic data and this result raised
serious questions concerning the appropriateness of ‘error-autocorrelation correc-
tion’. This message was enhanced by Hoover (1988) and Mizon (1995) who provided
further evidence for the unreliability of inference by investigating the CF restrictions
and their implications using both real and simulated data. However, the above ques-
tion was not fully answered.
This paper provides an unequivocal answer to the above question by deriving
necessary and sufficient conditions. Specifically, we prove that by imposing the CF
restrictions (5), one implicitly assumes that: ytdoes not Granger cause any of the
regressors in Xt.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2009
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