A Comparison of Sequential and Information‐based Methods for Determining the Co‐integration Rank in Heteroskedastic VAR Models
| Published date | 01 February 2015 |
| Author | Luca De Angelis,Anders Rahbek,A. M. Robert Taylor,Giuseppe Cavaliere |
| Date | 01 February 2015 |
| DOI | http://doi.org/10.1111/obes.12051 |
106
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 77, 1 (2015) 0305–9049
doi: 10.1111/obes.12051
A Comparison of Sequential and Information-based
Methods for Determining the Co-integration Rank in
Heteroskedastic VAR Models*
Giuseppe Cavaliere†, Luca De Angelis†, Anders Rahbek‡ and
A. M. Robert Taylor§
†Department of Statistical Sciences, University of Bologna, Via Belle Arti, 41, 40124,
Bologna, Italy (e-mail: giuseppe.cavaliere@unibo.it; l.deangelis@unibo.it)
‡Department of Economics, University of Copenhagen, Øster Farimagsgade 5, Building 26,
1353 Copenhagen K, Denmark (e-mail: anders.rahbek@econ.ku.dk),
§Essex Business School, University of Essex, Colchester, CO4 3SQ, UK
(e-mail: rtaylor@essex.ac.uk)
Abstract
In this article, we investigate the behaviour of a number of methods for estimating the
co-integration rank in VAR systems characterized by heteroskedastic innovationprocesses.
In particular, we compare the efficacy of the most widely used information criteria, such
as Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) , with
the commonly used sequential approach of Johansen [Likelihood-based Inference in Coin-
tegrated Vector Autoregressive Models (1996)] based around the use of either asymptotic
or wild bootstrap-based likelihood ratio type tests. Complementing recent work done for
the latter in Cavaliere, Rahbek and Taylor [Econometric Reviews (2014) forthcoming], we
establish the asymptotic properties of the procedures based on information criteria in the
presence of heteroskedasticity (conditional or unconditional) of a quite general and un-
known form. The relativefinite-sample properties of the different methods are investigated
by means of a Monte Carlo simulation study. For the simulation DGPs considered in the
analysis, we find that the BIC-based procedure and the bootstrap sequential test procedure
deliver the best overall performance in terms of their frequency of selecting the correct
co-integration rank across different values of the co-integration rank, sample size, station-
ary dynamics and models of heteroskedasticity. Of these, the wild bootstrap procedure is
perhaps the more reliable overall as it avoids a significant tendency seen in the BIC-based
method to over-estimate the co-integration rank in relatively small sample sizes.
*Wethank Anindya Banerjee and two anonymous referees for their helpful and constructive feedback on an earlier
version of this article. Cavaliere, Rahbek and Taylor thank the Danish Council for Independent Research, Sapere
Aude – DFF Advanced Grant (Grant no.: 12-124980), for financial support. Cavaliere and De Angelis thank the
Italian Ministry of Education, University and Research (MIUR), PRIN project ‘Multivariate statistical models for
risk assessment’, for financial support.
JEL Classification numbers: C30, C32.
Determining co-integration rank in heteroskedastic VAR models 107
I. Introduction
It is now well-known that sequential procedures based on the asymptotic (pseudo-) like-
lihood ratio (PLR) test of Johansen (1996) for determining the co-integration rank of a
vector autoregressive (VAR) system of variables integrated of order one [denoted I(1)]
can have quite poor finite sample properties; see, in particular, Johansen (2002) and the
references therein. As a consequence, a number of recent articles have investigated the use
of alternative methods for determining the co-integration rank.
Information-based methods are widely used for model selection and therefore provide
a well-established alternative to approaches based on Neyman–Pearson type tests for
determining the co-integration rank. A number of recent articles have focused on the issue
of co-integration rank estimation using standard information criteria. Under conditions
which, among other things, rule out conditional or unconditional heteroskedasticity in the
shocks, Aznar and Salvador (2002) and Kapetanios (2000) demonstrate the weak consis-
tency (formally defined in section III) of approaches based on the familiar BIC (Schwarz,
1978) and Hanna-Quinn Information Criterion (HQC) (Hannan and Quinn, 1979), outlin-
ing the conditions which need to hold on the associated penalty functions for this to obtain.
Using Monte Carlo simulation, Wang and Bessler (2005) present results which suggest
that the performance of a BIC-based approach is close to that of the approach based on
PLR (trace) tests and tends to outperform the corresponding approach based on the AIC
(Akaike, 1974). Indeed, Kapetanios (2004) establishes the inconsistency of theAIC-based
approach, deriving its asymptotic distribution and showing that the resulting estimate of
the co-integration rank displays a severe upward bias. Baltagi andWang (2007) revisit 165
data sets used in published studies. They find that the percentage of agreement between
procedures based on AIC, HQC and BIC is quite lowat below 60%, suggesting an apparent
divergence in the co-integration rank suggested by the different information criteria.
Many key macroeconomic and financial variablesappear to be characterized by perma-
nent changes in unconditional volatility (see, e.g. McConnell and Perez Quiros, 2000; Sen-
sier and van Dijk, 2004) and/or the presence of conditionally heteroskedastic shocks (see,
e.g. Gon¸calves and Kilian, 2004). Several authors have shown that traditional
co-integration tests can display significant upward size distortions in the presence of con-
ditionally heteroskedasticity (Lee and Tse, 1996; Cavaliere, Rahbek and Taylor, 2010a
[CRT2010a, hereafter]) or non-stationary heteroskedasticity (Cavaliere, Rahbek andTay-
lor, 2010b [CRT2010b, hereafter]). Specifically, CRT2010b show that the sequential PLR
method of Johansen (1996) is no longer valid, even asymptotically, in the presence of per-
manent changes in the error variance. Cavaliere,Rahbek and Taylor(2012, 2014) [CRT2012
and CRT2014, respectively, hereafter] show that this can be rectified via a bootstrap im-
plementation of Johansen’s sequential method. Specifically, they show that when (i) the
bootstrap samples are constructed using the restricted (under the reduced rank null) pa-
rameter estimates of the underlying VAR model and (ii) the wild bootstrap re-sampling
scheme is used, then the bootstrap sequential procedure is consistent, in the sense that
the probability of selecting a rank smaller than the true rank converges to zero and the
probability of overestimating the true rank is bounded by the chosen significance level.
The asymptotic validity of the information criteria based approaches to selecting the
co-integration rank has only been established for the case of i.i.d. shocks. Cheng and
©2014 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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