Bootstrap Co‐integration Rank Testing: The Effect of Bias‐Correcting Parameter Estimates

Published date01 October 2015
Date01 October 2015
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
doi: 10.1111/obes.12090
Bootstrap Co-integration Rank Testing: The Effect of
Bias-Correcting Parameter Estimates*
Giuseppe Cavaliere†, A. M. Robert Taylor‡ and Carsten
Department of Statistical Sciences, University of Bologna, Bologna, Italy
Essex Business School, University of Essex, Colchester, CO4 3SQ, UK
§Department of Economics, University of Mannheim, Mannheim, Germany
IAB Nuremberg, Nuremberg, Germany
Bootstrap-based methods for bias-correcting the first-stage parameter estimates used in
some recently developed bootstrap implementations of co-integration rank tests are inves-
tigated. The procedure constructs estimates of the bias in the original parameter estimates
by using the average bias in the corresponding parameter estimates taken across a large
number of auxiliary bootstrap replications. A number of possible implementations of this
procedure are discussed and concrete recommendations made on the basis of finite sam-
ple performance evaluated by Monte Carlo simulation methods. The results show that
bootstrap-based bias-correction methods can significantly improve the small sample per-
formance of the bootstrap co-integration rank tests.
I. Introduction
The use of tests based on the likelihood ratio (LR) principle for determining the co-
integration rank of a VAR system of I(1) variables, see Johansen (1996), is now common-
place in empirical research in macroeconomics and finance. However, the finite sample
properties of these tests, when based on asymptotic critical values, can be quite poor;
see, in particular, Johansen (2002) and the references therein. This has prompted a num-
ber of recent studies to propose bootstrap implementations of the LR co-integration rank
tests with the aim of delivering tests with empirical rejection frequencies (ERFs) closer to
*We thank the editor,Anindya Banerjee, and an anonymous referee for very helpful suggestions. Cavaliere and
Taylor thank the Danish Council for Independent Research, Sapere Aude |DFF Advanced Grant (Grant number
12-124980) for financial support. Cavaliere also thanks the Italian Ministry of Education, University and Research
(MIUR), PRIN project ‘Multivariate statistical models for risk assessment’ for financial support. Trenkler thanks
the Deutsche Forschungsgemeinschaft (DFG) who supported the research through the SFB 884 ‘Political Economy
of Reforms’. Parts of this paper were written while Cavaliere was affiliated with the University of Copenhagen,
Department of Economics, as visiting Professor.
Bootstrap co-integration rank testing 741
the nominal level; see, in particular, van Giersbergen (1996), Swensen (2006), Cavaliere,
Rahbek and Taylor (2010a) and Cavaliere, Rahbek and Taylor (2012).
Monte Carlo results reported in Swensen (2006) and Cavaliereet al. (2012) suggest that
the aforementioned bootstrap LR tests do indeed appear to yield significant improvements
on the small sample performance of the asymptotic LR tests, particularly so when smaller
the sample size the larger the dimension of the VAR system under study. However, their
results also show that significant size distortions remain in the bootstrap tests when theVAR
process contains stationary dynamics, most notably where these display strong positive
autocorrelation, such that we have a near-I(2) system.These bootstrap procedures are both
based around the use of bootstrap sample data formed using estimates of the stationary
dynamics obtained from the original data. As a consequence, their efficacy will clearly
be related to the degree of finite sample bias present in these estimates. Swensen (2006)
and Cavaliere et al. (2010a) use unrestricted estimates of the stationary dynamics while
Cavaliere et al. (2012) estimate the dynamics under the co-integration rank restriction
of the null hypothesis being tested. Cavaliere et al. (2012) demonstrate the theoretical
validity (meaning that, for the class of data-generating process (DGPs) considered, the
bootstrap attains the same first-order limiting null distribution as the LR statistic) of the
latter approach and that it delivers superior finite sample properties to the approach outlined
in Swensen (2006) and Cavaliere et al. (2010a), the theoretical validity of which also
remains to be established. We therefore focus attention on the approach of Cavaliere et al.
(2012) in this paper.
In the context of providing estimated confidence intervals for impulse response func-
tions from VAR models, be they estimated in levels, first differences or co-integrated VAR
form, Kilian (1998) shows that these can be quite inaccurate in small samples owing to
the finite sample bias seen in the estimates of the lag coefficient matrices characterizing
the VAR model; see also the simulation results in Engsted and Pedersen (2011).As Kilian
(1998) further notes, this bias is systematic and, as a consequence, bootstrap data gener-
ated conditional on biased point estimates will tend to result in an even greater bias in the
resulting bootstrap estimates of the lag coefficient matrices, relative to the true parameters.
This leads to the potential for a standard bootstrap confidence interval to be less accurate
than that based on the original estimates. In order to improve upon the accuracy of the
bootstrap confidence levels, Kilian (1998) proposes the so-called bootstrap-after-bootstrap
(BaB) method. Here the bootstrap data are generated not using the original point estimates
from the VAR model but on bias-corrected estimates which are themselves obtained by
bootstrap methods.
The basic idea underlying the bias correction used in the BaB approach of Kilian (1998)
is as follows. Suppose one estimates the matrix of parameters on the lagged dependent
variable in a stationary VAR model with one lag [VAR(1)] and is interested in obtaining
an estimate of the finite sample bias inherent in this estimate. The BaB approach then
uses some form of resampling, for example i.i.d. resampling, from the residuals from this
estimated VAR(1) model to obtain a set of bootstrap innovations.A bootstrap analogue of
the original sample data is then constructed using a recursion derived from these bootstrap
innovations and the estimated lag parameter matrix. A VAR(1) is then estimated on the
resulting bootstrap data using the same estimation method as was applied to the original
data. This bootstrap procedure is replicated a large number, say B1times. Since we have
©2015 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd

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