Narrow-band Weighted Nonlinear Least Squares Estimation of Unbalanced Cointegration Systems

We discuss cointegration relationships when covariance stationary observables exhibit unbalanced integration orders. Least squares type estimates of the long run coefficient are expected to converge either to 0 or to infinity if one does not account for the true unknown unbalance parameter. We propose a class of narrow-band weighted non-linear least squares estimators of these two parameters and analyze its asymptotic properties. The limit distribution is shown to be Gaussian, albeit singular, and it covers the entire stationary region in the particular case of the generalized non-linear least squares estimator, thereby allowing for straightforward statistical inference. A Monte Carlo study documents the good finite sample properties of our class of estimators. They are further used to provide new perspectives on the risk-return relationship on financial stock markets. In particular, we find that the variance risk premium estimated in an appropriately rebalanced cointegration system is a better return predictor than existing risk premia measures.