Bootstrap Determination of the Co-Integration Rank in Heteroskedastic VAR Models

Abstract

In a recent paper Cavaliere et al. (2012) develop bootstrap implementations of the (pseudo-) likelihood ratio (PLR) co-integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model which obtain under the reduced rank null hypothesis. They propose methods based on an independent and individual distributed (i.i.d.) bootstrap resampling scheme and establish the validity of their proposed bootstrap procedures in the context of a co-integrated VAR model with i.i.d. innovations. In this paper we investigate the properties of their bootstrap procedures, together with analogous procedures based on a wild bootstrap resampling scheme, when time-varying behavior is present in either the conditional or unconditional variance of the innovations. We show that the bootstrap PLR tests are asymptotically correctly sized and, moreover, that the probability that the associated bootstrap sequential procedures select a rank smaller than the true rank converges to zero. This result is shown to hold for both the i.i.d. and wild bootstrap variants under conditional heteroskedasticity but only for the latter under unconditional heteroskedasticity. Monte Carlo evidence is reported which suggests that the bootstrap approach of Cavaliere et al. (2012) significantly improves upon the finite sample performance of corresponding procedures based on either the asymptotic PLR test or an alternative bootstrap method (where the short run dynamics in the VAR model are estimated unrestrictedly) for a variety of conditionally and unconditionally heteroskedastic innovation processes.

Keywords:

• Bootstrap
• Co-integration
• Heteroskedasticity
• Rank determination
• Trace statistic

JEL Classification:

• C30
• C32

ACKNOWLEDGMENTS

We thank the editor and two anonymous referees for their helpful and constructive comments on earlier versions of this paper. Rahbek is also affiliated with CREATES. The authors also 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. Parts of this paper were written while Cavaliere was affiliated with the University of Copenhagen, Department of Economics. Parts of this paper were written while Taylor visited the Economics Department at Queen's University, Canada, whose hospitality is gratefully acknowledged, as a John Weatherall Distinguished Fellow.

We dedicate this paper to the enormous contribution that Les Godfrey has made to the discipline of econometrics.

Notes

In fact, Johansen (1996) does not impose the fourth moment condition in Assumption 𝒱.

By the PLR test of Johansen (1996) we mean the test based on the likelihood which obtains under the assumption that ϵ _t in (2.1) are Gaussian i.i.d. disturbances. The associated estimators from (2.1) under this assumption will, correspondingly, be referred to as pseudo maximum likelihood estimators.

Notice that if the p-value of a test converges in large samples to a uniform distribution on [0, 1] under the null hypothesis, then for any chosen significance level η, as the sample size diverges the probability of rejecting the null hypothesis converges to η; i.e., the test has asymptotic size η, as required.

It is worth noting that the large sample results that we establish for the wild bootstrap version of in this section are obtained under weaker conditions than were required in Cavaliere et al. (2010b) who, in deriving the large sample properties of their proposed wild bootstrap test, additionally required the innovations, z _t , in Assumption 𝒱″ to be symmetrically distributed.

The Gauss procedure for computing the bootstrap algorithms is available from the authors upon request.

Notes: ‘ERF' denotes the empirical rejection rates; ‘RC' denotes the percentage of times the bootstrap algorithm generates explosive samples.

Notes: see Table 1.1.

Notes: ‘Restricted' denotes Algorithm 2 of Section 3, ‘Unrestricted' denotes Algorithm 2 of Swensen (2006) [i.i.d. bootstrap] and Cavaliere et al. (2010a,b) [Wild Bootstrap].

Entries denote the frequency with which each value of r is selected by the given algorithm.

Notes: see Table 1.2.

Notes: see Table 1.1.

Notes: see Table 1.1.

Notes: see Table 1.2.

Notes: see Table 1.2.

Notes: see Table 1.1.

Notes: see Table 1.2.

Since steps (i) and (ii) of Algorithm 1 do not depend on the method used to resample the residuals (i.i.d. or wild bootstrap resampling), the number of root check violations associated with and will be identical and are therefore reported only once. The same equivalence applies to the tests and of Swensen (2006) and Cavaliere et al. (2010a,b). This feature does not hold for the sequential procedures, however, since algorithms that tend to select higher values of the co-integration rank will necessarily perform a larger number of root checks.

Notes: see Table 1.1.

Notes: see Table 1.1.

Notes: see Table 1.2.

Notes: see Table 1.2.

Notes: see Table 1.1.

Notes: see Table 1.2.

Notes: see Table 1.1.

Notes: see Table 1.2.

Notes: see Table 1.1.

Notes: see Table 1.2.