Asymptotics of Cointegration Estimator with Misspecified Rank

Cointegration analysis was developed for non-stationary linear processes that exhibit stationary relationships between coordinates. Estimation of the cointegration relationships in a multi-dimensional cointegrated process typically proceeds in two steps. First the rank is estimated, then the cointegration matrix is estimated, conditionally on the estimated rank (reduced rank regression). The asymptotics of the estimator is usually derived under the assumption of knowing the true rank. In this paper, we quantify the bias and find the asymptotic distributions of the cointegration estimator in case of misspecified rank. We find that the estimator is unbiased but has increased variance when the rank is overestimated, whereas a bias is introduced for underestimated rank, usually with a smaller variance. If the eigenvalues of a certain eigenvalue problem corresponding to the underestimated rank are small, the bias is small, and it might be preferable to an overestimated rank due to the decreased variance. The results are illustrated on simulated data.