Co‐integration Rank Determination in Partial Systems Using Information Criteria
| Author | Luca Fanelli,Giuseppe Cavaliere,Luca De Angelis |
| Date | 01 February 2018 |
| DOI | http://doi.org/10.1111/obes.12195 |
| Published date | 01 February 2018 |
65
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd.
OXFORD BULLETIN OF ECONOMICSAND STATISTICS, 80, 1 (2018) 0305–9049
doi: 10.1111/obes.12195
Co-integration Rank Determination in Partial
Systems Using Information Criteria*
Giuseppe Cavaliere,†Luca De Angelis† and Luca Fanelli†
†Department of Statistical Sciences, University of Bologna, Via Belle Arti, 41, 40124,
Bologna, Italy (e-mails: giuseppe.cavaliere@unibo.it; l.deangelis@unibo.it; luca.fanelli@uni
bo.it)
Abstract
Weinvestigate the asymptotic and finite sample properties of the most widely used informa-
tion criteria for co-integration rank determination in ‘partial’ systems, i.e. in co-integrated
vector autoregressive (VAR) models where a sub-set of variables of interest is modelled
conditional on another sub-set of variables. The asymptotic properties of theAkaike infor-
mation criterion (AIC), the Bayesian information criterion (BIC) and the Hannan–Quinn
information criterion (HQC) are established, and consistency of BIC and HQC is proved.
Notably, the consistency of BIC and HQC is robust to violations of weak exogeneity of the
conditioning variables with respect to the co-integration parameters. More precisely, BIC
and HQC recover the true co-integration rank from the partial system analysis also when
the conditional model does not convey all information about the co-integration parameters.
This result opens up interesting possibilities for practitioners who can now determine the
co-integration rank in partial systems without being concerned about the weak exogeneity
of the conditioning variables.A Monte Carlo experiment based on a large dimensional data
generating process shows that BIC and HQC applied in partial systems perform reasonably
well in small samples and comparativelybetter than ‘traditional’ methods for co-integration
rank determination. Wefur ther show the usefulness of our approach and the benefits of the
conditional system analysis in two empirical illustrations, both based on the estimation of
VAR systems on US quarterly data. Overall, our analysis shows the gains of combining
information criteria with partial system analysis.
JEL Classification numbers: C30, C32, C52.
*After the first draft of this paper was written, Prof. Jean-Pierre Urbain passed away. Jean-Pierre was a leading
expert of weak exogeneity in co-integrated models and this work has largely benefited from his contribution to the
field. Wededicate this paper to his memory.
We are grateful to the Editor, Anindya Banerjee, and two anonymous referees for their helpful comments on the
earlier version of this paper. We thank the conference participants at the ‘Workshop on Time Series Econometrics’
at the Department of Economic Sciences, University of Salento, Lecce, Italy (June 2016), the ‘Econometric Society
Meeting’, Geneve, Switzerland (August 2016), and the ‘Latin American Econometric Society Annual Meeting’,
Medell´
in, Colombia (November2016). We also thankAnders Rahbek, Peter Boswijk and Alain Hecq for their helpful
comments and suggestions and Gunnar B˚ardsen for the stimulating initial discussions about this paper.We are solely
responsible for all errors. Work supported by the Italian Ministry of Education, University and Research (MIUR),
PRIN 2010–11 project ‘Multivariate statistical models for risk assessment’ and RFO grants from the Universityof
Bologna.
66 Bulletin
I. Introduction
Co-integration rank determination in vector autoregressive (VAR) systems where some
variables of interest, Yt, are modelled conditional on some other variables, Zt, has been
addressed in Harbo et al. (1998) [HJNR hereafter], see also Johansen (1992a, b, 1996).
The idea of the conditional analysis is that the dimensionality of the system is reduced,
and, when conditioning is valid, co-integration tests may have better power properties. The
‘partial system’ approach proves to be useful in situations where (i) the dimension of the
whole VAR system p:=dim(Xt), X
t:=(Y
t,Z
t), is large relative to the sample size T; (ii)
the practitioner is essentially interested in modelling Ytand it is known that Ztcontributes
to the long-run equilibrium of the system but either the theory is silent about the short run
properties of Zt, or the scope of the analysis does not require modelling Ztexplicitly; (iii)
the time series used for Ztare considered poor proxies of the corresponding theoretical
variables they should measure, especially as concerns their transitory dynamics.
In general, the parameters of the (conditional) model for Ytgiven Zt(Yt|Zt) and of
the marginal model for Ztare interrelated, which means that a full system analysis is
needed to draw efficient inference about the parameters of the two models. The special
case where the partial (conditional) model for Ytgiven Ztcontains as much information
about the parameters of interest as the full system is when Ztis weakly exogenous, see
Engle, Hendry and Richard (1983). Under the weak exogeneity of Ztfor the co-integrating
parameters, HJNR derive the asymptotic distribution of the likelihood ratio test for co-
integrating rank in the partial model, and provide tables of critical values. When Ztis not
weakly exogenous for the co-integrating parameters, the conditional model for Ytgiven Zt
conveys only partial information about the long-run structure of the system.
Actually, the requirement of weak exogeneityof Ztcan nullify the benefits of the partial
system analysis as suggested in HJNR and limit its usefulness in applied work. On the one
hand, when weakexogeneity of Ztfails, the practitioner cannot apply the asymptotic critical
values tabulated in HJNR and must resort to the full system analysis. On the other hand,
it is difficult to check the hypothesis of weak exogeneity of Ztefficiently without a full
system analysis or disregarding the model for Zt. Urbain (1992) shows that ‘traditional’
orthogonality tests (Pesaran and Smith, 1990) are not sufficient to address the issue of
weak exogeneity in error-correction models, see also Boswijk and Urbain (1997). It is
not surprising, therefore, that despite conditional (structural) error-correction models have
been widely applied in applied work, only seldom the partial system approach has been
used for inference on the co-integration rank, see Doornik, Hendry and Nielsen (1998),
B˚ardsen and Fisher (1999), Bruggeman, Donati and Warne (2003), and Johansen (1992b)
for an early example. In empirical studies where a (structural) conditional system for Yt
given Ztis used, the co-integration rank is typically either assumed to be known, or is
assumed to be inferred from a full system analysis in a previous stage, see e.g. Boswijk
(1995) and Ericsson (1995).
Information-based methods are a well-established alternative to approaches based on
(sequential) Neyman–Pearson type tests for co-integration rank determination. In partic-
ular, Aznar and Salvador (2002) and Cavaliere et al. (2015) [CDRT hereafter], among
others, show that standard information criteria such as the familiar Bayesian information
criterion (BIC) and Hannan–Quinn information criterion (HQC) provide powerfulalter na-
©2017 The Department of Economics, University of Oxford and JohnWiley & Sons Ltd
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