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 asymptotic bias and find the asymptotic distributions of the cointegration estimator in case of misspecified rank. Furthermore, we suggest a new class of weighted reduced rank estimators that allow for more flexibility in settings where rank selection is hard. We show empirically that a proper choice of weights can lead to increased predictive performance when there is rank uncertainty. Finally, we illustrate the estimators on empirical EEG data from a psychological experiment on visual processing.

翻译：协整分析是针对表现出坐标间平稳关系的非平稳线性过程而发展的。在多维协整过程中，协整关系的估计通常分两步进行：首先估计秩，然后在给定估计秩的条件下估计协整矩阵（降秩回归）。该估计量的渐近性质通常基于已知真实秩的假设推导得出。本文量化了秩误设情况下协整估计量的渐近偏差，并推导了其渐近分布。此外，我们提出了一类新的加权降秩估计量，在秩选择困难的情况下提供了更大灵活性。实证表明，当存在秩不确定性时，适当的权重选择能够提升预测性能。最后，我们通过一项视觉处理心理实验的脑电图数据对估计量进行了实例验证。