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1、School of Economics and Management Aarhus University Bartholins All 10, Building 1322, DK-8000 Aarhus C Denmark CREATES Research Paper 2010-75 A Bootstrap Cointegration Rank Test for Panels of VAR Models Laurent A.F. Callot A BOOTSTRAP COINTEGRATION RANK TEST FOR PANELS OF VAR MODELSLAURENT A. F. CA
2、LLOT, CREATES AARHUS UNIVERSITY LCALLOTCREATES.AU.DKDecember 6, 2010Abstract. This paper proposes a sequential procedure to de- termine the common cointegration rank of panels of cointegrated VARs. It shows how a panel of cointegrated VARs can be trans- formed in a set of independent individual mode
3、ls. The likelihood function of the transformed panel is the sum of the likelihood func- tions of the individual Cointegrated VARs (CVAR) models.A bootstrap based procedure is used to compute empirical distribu- tions of the trace test statistics for these individual models. From these empirical dist
4、ributions two panel trace test statistics are con- structed. The satisfying small sample properties of these tests are documented by means of Monte Carlo. An empirical application illustrates the usefullness of this tests.Key words: Rank test, Panel data, Cointegration, Bootstrap, Cross section depe
5、ndence.JEL classification: C12, C32, C33I would like to thank Niels Haldrup, M Hashem Pesaran and Michael Jansson for valuable suggestions, and Anders Bredahl Kock for comments and helpfull discus- sion. All remaining errors and shortcommings are mine. Financial support by the Center for Research in
6、 Econometric Analysis of Time Series, CREATES, funded by the Danish National Research Foundation, is gratefully acknowledged. 12LAURENT A. F. CALLOT1. IntroductionThis paper proposes a panel rank test statistic based on a bootstrap procedure for determination of the cointegration rank in a Panel Coi
7、n- tegrated Vector Autoregressivve (PCVAR) models. The cointegration rank is an important element to understand the dynamics of a sys- tem of variables.The sequential likelihood-based procedure for the determination of the cointegration rank in a system of variables which are at most I(1) (see Johan
8、sen (1995) is frequently used in empirical research. The trace test is a likelihood ratio test for the hypothesis that the true rank of the system (noted r0throughout) is equal to r (hereafter H(r) against the hypothesis that the system has full rank: H(p) in a system with p endogenous variables. Th
9、e sequential procedure uses ttrace test to first test H(r) for r = 0. If this is rejected r is incremented by 1 and H(1) is tested, and so on until it is not possible to reject or that H(p 1) is rejected, in which case the rank is set to p. The poor small sample performances of this procedure have b
10、een documented (see among others Reinsel and Ahn (1992) and Johansen (2002b). Johansen (2002b) proposes a small sample Bartlett correction based onfinding the expectation of the likelihood ratio test statistics to correct it. However this correction doesnt always produce sizes close to their nominal
11、 values. Bootstrap methods are increasingly used to compute empirical test statistic distributions that are more accurate than the asymptotic dis- tribution, thus yielding tests with small sample sizes closer to their nominal values. Swensen (2006) proposes a bootstrap algorithm to es- timate an emp
12、irical distribution of the trace test statistics. He shows that this algorithm is valid only for a restricted class of data generat- ing processes (DGP). Cavaliere, Rahbek, and Taylor (2010) (hereafterCRT) proposes a different bootstrap algorithm valid for any DGP with variables at most I(1). They o
13、btain sizes very close to their nominal level even for samples of only 100 observations. The main aim of this paper is to extend the procedure of CRT to multivariate panel models. Panels are increasingly used in empirical economics to analyze data set composed of many countries, regions, industries
14、or markets. One of their advantages is that they can theoretically be used to improve inference on the parameters of a model by exploiting the informationcontained in several series representing the same quantities for different individuals.Two difficulties arise when working with panels.(1) Panels
15、where the number of N is small relative to T can be es- timated as a single model. However the number of parameters in such a model grows quadratically with the number of indi- viduals. This is often referred to as the curse of dimensionality.A BOOTSTRAP COINTEGRATION RANK TEST FOR PANELS OF VAR MOD
16、ELS 3In order to estimates panels with large N and T, one has to control the number of parameters. (2) Many economical series exhibit common patterns across indi- viduals. When ignored in the modelling of the panel, this com- mon patterns translate in cross section dependence of the resid- uals, leading to biased inference. Breitung and Pesaran (2008) review the literature on cointegration and rank test in panels. Only a handful of procedures exist to test for multiple cointegration
