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Determining the cointegration rank in heteroskedastic VAR models of unknown order. (English) Zbl 1441.62228

Summary: We investigate the asymptotic and finite sample properties of a number of methods for estimating the cointegration rank in integrated vector autoregressive systems of unknown autoregressive order driven by heteroskedastic shocks. We allow for both conditional and unconditional heteroskedasticity of a very general form. We establish the conditions required on the penalty functions such that standard information criterion-based methods, such as the Bayesian information criterion [BIC], when employed either sequentially or jointly, can be used to consistently estimate both the cointegration rank and the autoregressive lag order. In doing so we also correct errors which appear in the proofs provided for the consistency of information-based estimators in the homoskedastic case by A. Aznar and M. Salvador [Econom. Theory 18, No. 4, 926–947 (2002; Zbl 1109.62333)]. We also extend the corpus of available large sample theory for the conventional sequential approach of S. Johansen [Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford Univ. Press (1995; Zbl 0928.62069)] and the associated wild bootstrap implementation thereof of G. Cavaliere et al. [“Bootstrap determination of the co-integration rank in heteroskedastic VAR models”, Econ. Rev. 33, No. 5–6, 606-650 (2014; doi:10.1080/07474938.2013.825175)] to the case where the lag order is unknown. In particular, we show that these methods remain valid under heteroskedasticity and an unknown lag length provided the lag length is first chosen by a consistent method, again such as the BIC. The relative finite sample properties of the different methods discussed are investigated in a Monte Carlo simulation study. The two best performing methods in this study are a wild bootstrap implementation of the Johansen [loc. cit.] procedure implemented with BIC selection of the lag length and joint IC approach [cf. P. C. B. Phillips, Econometrica 64, No. 4, 763–812 (1996; Zbl 0899.62144), ref32] which uses the BIC to jointly select the lag order and the cointegration rank.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62E20 Asymptotic distribution theory in statistics
62P20 Applications of statistics to economics
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