Weak exogeneity in I(2) VAR systems

This paper defines parametric conditions under which a subset of variables is weakly exogenous with respect to the (multi)-cointegration parameters in I(2) VAR systems. The weak exogeneity conditions can be interpreted in terms of common trends, corresponding to the cumulation of the errors from the marginal equations into the I(2) trends, or in terms of ‘no levels and difference feedback’ into the marginal model equations. A modified version of the two-stage procedure proposed in Johansen (1995) is adopted for conditional statistical inference. Corresponding tests for the above restrictions are derived and discussed. Asymptotic properties of the tests and of the conditional estimators are analyzed. It is shown that if the conditions of weak exogeneity do not apply, the conditional estimators of the long-run parameters can be inconsistent and/or present limit distributions with nuisance parameters, according to which part of the conditions fails to hold. A test for weak exogeneity restrictions as a routine check before any analysis of conditional models is strongly recommended.