Estimation of long-run parameters in unbalanced cointegration

This paper analyses the asymptotic properties of nonlinear least squares estimators of the long run parameters in a bivariate unbalanced cointegration framework. Unbalanced cointegration refers to the situation where the integration orders of the observables are different, but their corresponding balanced versions (with equal integration orders after filtering) are cointegrated in the usual sense. Within this setting, the long run linkage between the observables is driven by both the cointegrating parameter and the difference between the integration orders of the observables, which we consider to be unknown. Our results reveal three noticeable features. First, superconsistent (faster than n-consistent) estimators of the difference between memory parameters are achievable. Next, the joint limiting distribution of the estimators of both parameters is singular, and, finally, a modified version of the “Type II” fractional Brownian motion arises in the limiting theory. A Monte Carlo experiment and the discussion of an economic example are included.