Estimation Bias and Inference in Overlapping Autoregressions: Implications for the Target‐Zone Literature*
| DOI | http://doi.org/10.1111/j.1468-0084.2007.00488.x |
| Date | 01 February 2008 |
| Author | Zsolt Darvas |
| Published date | 01 February 2008 |
1
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford, 2007. Published by Blackwell Publishing Ltd,
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OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 70, 1 (2008) 0305-9049
doi: 10.1111/j.1468-0084.2007.00488.x
Estimation Bias and Inference in Overlapping
Autoregressions: Implications for the Target-Zone
Literature*
Zsolt Darvas
Department of Mathematical Economics and Economic Analysis, Corvinus University of
Budapest, Budapest, Hungary (e-mail: zsolt.darvas@uni-corvinus.hu)
Abstract
Samples with overlapping observations are used for the study of uncovered interest
rate parity, the predictability of long-run stock returns and the credibility of exchange
rate target zones. This paper quantifies the biases in parameter estimation and size
distortions of hypothesis tests of overlapping linear and polynomial autoregressions,
which have been used in target-zone applications. We show that both estimation bias
and size distortions of hypothesis tests are generally larger, if the amount of overlap
is larger, the sample size is smaller, and autoregressive root of the data-generating
process is closer to unity. In particular, the estimates are biased in a way that makes
it more likely that the predictions of the Bertola–Svensson model will be supported.
Size distortions of various tests also turn out to be substantial even when using a
heteroskedasticity and autocorrelation-consistent covariance matrix.
I. Introduction
A time-series sample of a regression is said to be overlapping, when the dating of the
regressand leads the dating of the regressor by more than one period. There are at
least three main areas in empirical macroeconomics and finance where overlapping
*I thank Christopher Bowdler (editor), Casper G. de Vries,L´aszl´oHalpern, L´aszl´oHunyadi, G ´abor K˝orösi,
Helmut Lütkepohl, Judit Nem´enyi, Peter C. B. Phillips, Andr´as Simon, Andr´as Simonovits, J´anos Vincze,
two anonymous referees, and seminar participants at the Winter Symposium of the Econometric Society, the
Spring Meeting of Young Economists, and the Econometric Institute of the Erasmus University, Rotterdam
for comments and suggestions. The paper was finalized during my visit to the Tinbergen Institute, Rotterdam.
I thank the Tinbergen Institute for its hospitality. I am alone responsible for any errors and for the views
expressed in the paper.
JEL Classification numbers: C22, F31.
2Bulletin
samples are used: the analysis of uncovered interest rate parity (UIP), the predict-
ability of long-run equity returns and the credibility of exchange rate target zones.
In the first two of these three applications, multiperiod-ahead changes in asset prices
are regressed on another variable, that is, the future change in the exchange rate is
regressed on the current interest rate differential, and the future change in equity
prices is regressed on the dividend yield respectively. Target-zone applications, on
the other hand, are based on overlapping autoregressions.
The consequences of overlapping observations for hypothesis tests for the first
two applications have already been studied (see, e.g. Hansen and Hodrick, 1980;
Richardson and Stock, 1989; Richardson and Smith, 1991; Hodrick, 1992; Nelson
and Kim, 1993; Phillips, McFarland and McMahon, 1996; Smith and Yadav, 1996,
Kirby, 1997; Britten-Jones and Neuberger, 2004; Ang and Bekaert, 2007). These
studies either suggested methods to estimate the covariance matrix when the sample
is overlapping or studied the small-sample properties of the estimation and generally
reported size distortions of hypothesis tests.1Regardless of the covariance matrix
used, the consensus of the literature is the empirical failure of UIP.The predictability
of long-run equity excess returns (over the risk-free interest rate) had long been treated
as a stylized fact (see, e.g. Fama and French, 1988), but more recent studies showed
that these results were due to the small-sample properties of standard hypothesis tests,
and that the excess return predictability by the dividend yield is not significant (see
Ang and Bekaert, 2007, and references therein).
To our knowledge, however, the consequence of overlapping observations for
estimation and inference in target-zone applications, and more generally, in over-
lapping autoregressions has not yet been studied. Although the issue of overlapping
observations is referred to in most of the empirical target-zone applications with cau-
tionary warnings, the sole remedy for the problem has been the calculation of the
significance of the estimates by the heteroskedasticity and autocorrelation-consistent
(HAC) covariance matrix estimator of Newey and West (1987).2This paper aims to
fill this gap by studying the properties of overlapping autoregressions along three
main dimensions: (i) the amount of overlap; (ii) the root of the data-generating auto-
regressive process; and (iii) the sample size by Monte Carlo simulation.3In addition
to size properties of hypothesis tests, we also lay special emphasis on the biases of
parameter estimations.
1Estimation is usually done by least squares, with the exception of Phillips et al. (1996), who uses the fully
modified least absolute deviations (FM-LAD) regression procedure.
2This procedure was adopted in, for example, Frankel and Phillips (1992), Bertola and Svensson (1993),
Caramazza (1993), Lindberg, Söderlind and Svensson (1993), Svensson (1993), Rose and Svensson (1994,
1995), Thomas (1994), Werner (1996), Chen and Giovannini (1997), Knot, Sturm and de Haan (1998), and
Knot and Sturm (1999).
3The distributions can be derived analytically by extending the functional central limit theorem, which was
first applied to the problem of deriving the asymptotic distribution of parameter estimates when there is a unit
root by Phillips (1987). However, as our main goal is numerical characterization, we do not carry out this
exercise in this paper.
©Blackwell Publishing Ltd and the Department of Economics, University of Oxford 2007
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