Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*
| Published date | 01 October 2011 |
| DOI | http://doi.org/10.1111/j.1468-0084.2010.00630.x |
| Date | 01 October 2011 |
| Author | Leslie G. Godfrey |
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 73, 5 (2011) 0305-9049
doi: 10.1111/j.1468-0084.2010.00630.x
Robust Non-nested Testing for Ordinary Least
Squares Regression when Some of the Regressors are
Lagged Dependent VariablesÅ
Leslie G. Godfrey
Department of Economics, University of York, Heslington, York. YO10 5DD, UK
(e-mail: leslie.godfrey@york.ac.uk)
Abstract
The problem of testing non-nested regression models that include lagged values of the
dependent variable as regressors is discussed. It is argued that it is essential to test for error
autocorrelation if ordinary least squares and the associated Jand Ftests are to be used.
A heteroskedasticity–robust joint test against a combination of the artificial alternatives
used for autocorrelation and non-nested hypothesis tests is proposed. Monte Carlo results
indicate that implementing this joint test using a wild bootstrap method leads to a well-
behaved procedure and gives better control of finite sample significance levels than
asymptotic critical values.
I. Introduction
The problem of testing a model in the presence of a non-nested alternative has proved
to be of importance in both applied and theoretical econometric analyses; see McAleer
(1995) and Pesaran and Weeks (2001) for surveys and comments. McAleer reports that,
of the various methods that have been proposed for testing non-nested regression models
after ordinary least squares (OLS) estimation, the J-test of Davidson and MacKinnon
(1981) is the one most often used by applied workers. In order to establish the asymptotic
validity of the J-test, Davidson and MacKinnon make the classical assumptions that all
regressors are exogenous and the errors are normally and independently distributed (NID)
with common variance and zero means. However, as shown in MacKinnon et al. (1983),
the J-test remains asymptotically valid when the errors are independently and identically
distributed (IID), but not necessarily normal, and some of the regressors are lagged values
of the dependent variable, provided there is dynamic stability.
Although the assumptions used in MacKinnon et al. (1983) are much weaker than those
in Davidson and MacKinnon (1981), the requirement that the errors of the model under test
be IID is clearly inconsistent with modern views about best practice techniques for applied
work; see, for example, Hansen (1999) and Stock and Watson (2006) in which empirical
ÅI am grateful to Peter Burridge and two referees for their helpful comments.
JEL Classification numbers: C12, C15, C52.
652 Bulletin
workers are urged to adopt heteroskedasticity-robust methods. Choi and Kiefer propose
‘robust tests that generalize the J-test ... for non-nested dynamic models with unknown
serial correlation and conditional heteroscedasticity’; see Choi and Kiefer (2008). Choi
and Kiefer seek to obtain a robust OLS-based J-test by using heteroskedasticity and auto-
correlation consistent (HAC) methods, based upon the non-standard (fixed-b) asymptotic
theory for HAC tests discussed in Kiefer, Vogelsang and Bunzel (2000) and Kiefer and
Vogelsang (2002a,b, 2005).
It is important to note that the asymptotic theory for robust OLS-based tests in Choi
and Kiefer (2008) does not apply to all types of dynamic regression models. Regression
models can be referred to as being dynamic when the regressors include lagged exogenous
variables and/or lagged values of the dependent variable. The strategy for testing advo-
cated by Choi and Kiefer is appropriate when all regressors can be taken to be current or
lagged values of strictly exogenous variables. However, it cannot be employed to obtain
valid HAC tests of the significance of OLS estimates in the combined presence of lagged
dependent variables and serially correlated errors. The root of the problem for the HAC
method and the stimulus for proposing a different procedure is that standard and fixed-b
asymptotics both require that the OLS estimator be consistent; for discussions of the former
and latter theories, see Greene (2008) and Kiefer and Vogelsang (2005), respectively.
In general, OLS estimators will be inconsistent when the errors are autocorrelated and
there are lagged values of the dependent variable in the regressors. Consequently, if OLS-
based tests of non-nested regression models with lagged dependent variables are required,
it is not possible to allow the presence of unspecified forms of autocorrelation and so the
assumption of serial independence is vital for the asymptotic validity of such tests. Thus,
the advice given in Choi and Kiefer (2008, p. 11) that ‘an empirical researcher need not
test the existence of serial correlation’ is inappropriate in such situations and such a test is
instead essential when some of the regressors are lagged values of the dependent variable.
Since the consistency of OLS estimators, when the regressors include lagged dependent
variables, requires that the model under test has the correct regression function and that its
errors have no autocorrelation, the data consistency of both of these assumptions should
be checked. Given that two assumptions are under test, an applied worker can use either
a joint test or two separate tests. It is argued below that, in the context of the problem
examined in this paper, a joint test is more appropriate. Although robustness to error auto-
correlation cannot be achieved when OLS estimation is used and the regressors include
lagged dependent variables, asymptotic robustness of tests to heteroskedasticity is still
feasible and desirable. The joint test is, therefore, constructed using a covariance matrix
that is consistent under either unspecified forms of heteroskedasticity or homoskedasticity,
provided regularity conditions are satisfied. The heteroskedasticity-robust joint tests can
be implemented using either asymptotic critical values or a wild bootstrap approach and
Monte Carlo evidence is provided that supports the use of the latter.
II. Models and assumptions
Consider two competing non-nested regression models written as
H1:yt=x
t1+u1t,(1)
and
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