Determining the Cointegration Rank in Heteroskedastic Var Models of Unknown Order

Abstract

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 Aznar and Salvador (2002). We also extend the corpus of available large sample theory for the conventional sequential approach of Johansen (1995) and the associated wild bootstrap implementation thereof of Cavaliere, 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 (1995) procedure implemented with BIC selection of the lag length and joint IC approach (cf. Phillips, 1996) which uses the BIC to jointly select the lag order and the cointegration rank.

Co‐integration Rank Determination in Partial Systems Using Information Criteria

“…(), CDRT and Cavaliere et al . () to the case of larger dimensional VARs. Moreover, in order to fully exploit the potential of the conditional analysis we specify a ‘full’ (and rather arbitrary) structure for the covariance matrix Σ such that the resultant ω matrix in equation is non‐zero for all DGPs considered .…”

“…While we do not consider the determination of the system lag order and the choice of the form of the deterministic component in this paper, the strategies outlined by Cavaliere et al . () and Aznar and Salvador (), respectively, can also be extended to the present framework.…”

“…Remark Theorem assumes that the system lag length k in equation is known. This assumption, which is often unreasonable in practice, can be relaxed and information criteria can be used to determine both the autoregressive lag length and the co‐integration rank following either a jointly or a sequential procedure (see Cavaliere et al ., ). Moreover, Aznar and Salvador () provide an approach based on information criteria to also determine the form of the deterministic component.…”

The Relation between Monetary Policy and the Stock Market in Europe

“…Thus, there may be cointegration among the variables, which is worth taking into account in our structural analysis. Since the VAR(2)-MSH(2) model, and thus a model with conditional heteroscedasticity, was favoured in the previous analysis, we base tests for the cointegration rank on Johansen's (1995) cointegration rank tests robustified for conditional heteroscedasticity by generating the p-values with a wild bootstrap algorithm, as proposed by Cavaliere et al (2010) and further investigated by Cavaliere et al (2018). The results are presented in Table 2 and suggest a cointegration rank of r = 2 if a 5% significance level is used.…”

A unifying theory of tests of rank

“…Another consequence, is that little is known about the behaviour of tests of rank under local alternatives or under misspecification. Local power is considered in Cragg & Donald (1997) and a handful of papers surveyed by Hubrich et al (2001), while misspecification is considered in Robin & Smith (2000), Caner (1998), Cavaliere et al (2010b), Aznar & Salvador (2002), and Cavaliere et al (2014). All of these results relate to specific tests and there is as of yet no known general principle that unifies all of these results.…”