Bootstrap HAC Tests for Ordinary Least Squares Regression*
| Date | 01 December 2012 |
| DOI | http://doi.org/10.1111/j.1468-0084.2011.00671.x |
| Author | Francesco Bravo,Leslie G. Godfrey |
| Published date | 01 December 2012 |
903
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 74, 6 (2012) 0305-9049
doi: 10.1111/j.1468-0084.2011.00671.x
Bootstrap HAC Tests for Ordinary Least Squares
RegressionÅ
Francesco Bravo and Leslie G. Godfrey
Department of Economics and Related Studies, University of York, York, YO10 5DD, UK
(e-mails: francesco.bravo@york.ac.uk; leslie.godfrey@york.ac.uk)
Abstract
There is a need for tests that are derived from the ordinary least squares (OLS) estimators
of regression coefficients and are useful in the presence of unspecified forms of heter-
oskedasticity and autocorrelation. A method that uses the moving block bootstrap and
quasi-estimators in order to derive a consistent estimator of the asymptotic covariance
matrix for the OLS estimators and robust significance tests is proposed. The method is
shown to be asymptotically valid and Monte Carlo evidence indicates that it is capable of
providing good control of significance levels in finite samples and good power compared
with two other bootstrap tests.
I. Introduction
The primary aim of regression analysis is to specify and make inferences about the con-
ditional mean function. It is assumed in this study that the specified mean function is a
linear combination of exogenous regressors with observation-invariant coefficients and
that the error term is additive. The behaviour of the error term is, of course, important for
the properties of tests of hypotheses about the coefficients of the mean function. In the
past, textbook discussions of asymptotic tests derived from ordinary least squares (OLS)
results often relied upon the assumption that errors were independently and identically dis-
tributed (iid). However, the correct specification of the conditional mean function does not
automatically imply that the errors are either conditionally homoskedastic or uncorrelated.
Several authors have argued that, in the conditional model, the variances, like the mean
values, should be allowed to vary with regressors; see, for example, Stock and Watson
(2007). Also, errors may be autocorrelated when time series data are used. There is usually
little prior information about the precise ways in which the traditional assumption of iid
errors is violated. Consequently there is a need for OLS-based tests that are asymptotically
valid in the presence of unspecified forms of heteroskedasticity and autocorrelation, that
is to say, for heteroskedasticity and autocorrelation consistent (HAC) tests.
ÅWe are grateful to two anonymous referees, Peter Burridge, Russell Davidson, S´ılvia Gon¸calves, Joao Santos
Silva and Peter N. Smith for their helpful comments and suggestions.
JEL Classification numbers: C12, C22.
904 Bulletin
When deriving HAC tests, it is usual to employ regularity conditions that imply that the
OLS estimators of regression coefficients are consistent and asymptotically Normally dis-
tributed. Given these regularity conditions, a key step in a well-established approach to the
construction of a test statistic is to obtain a consistent estimator of the asymptotic covari-
ance matrix of the OLS estimators. If a HAC covariance matrix estimator (HACCME) is
available, a Wald-type statistic that is asymptotically distributed as chi-squared, when the
null hypothesis is true, can be calculated. Several influential articles have been published
on ways in which to obtain a HACCME to make use of asymptotic critical values from a
chi-squared distribution when conducting tests of hypotheses about regression coefficients;
see, for example, Andrews (1991), Andrews and Monahan (1992) and Newey and West
(1987). Advice about computing a HACCME is provided in Den Haan and Levin (1997).
In one important general approach to the construction of a HACCME that is described
in many texts, the user must specify a kernel function and a bandwidth smoothing parame-
ter. Examples of kernel functions are provided inAndrews (1991, p. 821). To establish the
consistency of the kernel-based HACCME in standard situations, it is sufficient to assume
that the bandwidth grows with the sample size, but at a slower rate, with the ratio of the
bandwidth to the sample size tending to zero as the sample size tends to infinity; see de
Jong and Davidson (2000, section 2) for a detailed analysis.
There is, however, considerable evidence from Monte Carlo studies that asymptotic
theory fails to provide an accurate and reliable guide to the finite-sample behaviour of
test statistics that use a kernel-based HACCME. The general finding is that, when the null
hypothesis is true, tests that use a kernel-based HACCME with asymptotic chi-squared
critical values have estimated rejection rates that are rather greater than the desired sig-
nificance levels, even when prewhitening is used to obtain improved performance; see,
for example, the estimates in Tables III–V in Kiefer et al. (2000). Clearly consistency of a
covariance matrix estimator is not sufficient to ensure adequate control of the finite sample
significance levels of HAC tests.
An approach that does not rely upon the consistency of a covariance matrix estimator is
proposed in Kiefer et al. (2000). It is shown in Kiefer and Vogelsang(2002a) that the HAC
robust tests described by Kiefer et al. are exactly equivalent to those that would be obtained
by using a Bartlett kernel function with bandwidth equal to sample size. Assuming that
bandwidth is equal to sample size violates the condition given in de Jong and Davidson
(2000) and produces large sample results that are quite different from those for asymptotic
chi-squared tests. However, while the test statistic derived in Kiefer et al. has an asymptotic
distribution under the null hypothesis that is non-standard, it is asymptotically pivotal; see
Kiefer et al. (2000, p. 695 and 700). It is, therefore, possible to obtain tables of estimates
of the non-standard asymptotic critical values that replace the standard chi-squared tables
associated with the use of a consistent covariance matrix estimator.
The test proposed in Kiefer et al. (2000) is generalized in Kiefer and Vogelsang(2002b).
Kiefer and Vogelsang provide an analysis that covers the use of various kernel functions,
each to be combined with bandwidth equal to sample size. The test statistics discussed
in Kiefer and Vogelsang (2002b) have non-standard asymptotic null distributions that
depend upon the kernel function. Thus the non-standard asymptotic critical values vary
with the choice of the kernel; see, for example, the estimates given in Table 1 of Kiefer
and Vogelsang (2002b, p. 1356).
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2011
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