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.

中文翻译：

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秩错误的协整估计量的渐近

协整分析是为在坐标之间表现出平稳关系的非平稳线性过程开发的。多维协整过程中协整关系的估计通常分两个步骤进行。首先估计秩，然后估计协整矩阵，条件是估计的秩（降秩回归）。估计量的渐近线通常是在知道真实秩的假设下推导出来的。在本文中，我们量化了偏差，并在错误指定等级的情况下找到协整估计量的渐近分布。我们发现估计量是无偏的，但当排名被高估时方差增加，而对低估的排名引入偏差，通常方差较小。如果某个特征值问题对应于被低估的秩的特征值很小，则偏差很小，并且由于方差减小，它可能比高估的秩更可取。结果显示在模拟数据上。