Estimation in a Multivariate "Errors in Variables" Regression Model: Large Sample Results

Abstract

In a multivariate "errors in variables" regression model, the unknown mean vectors $\mathbf{u}_{1i}: p \times 1, \mathbf{u}_{2i}: r \times 1$ of the vector observations $\mathbf{x}_{1i}, \mathbf{x}_{2i}$, rather than the observations themselves, are assumed to follow the linear relation: $\mathbf{u}_{2i} = \alpha + B\mathbf{u}_{1i}, i = 1,2,\cdots, n$. It is further assumed that the random errors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{u}_i, \mathbf{x}'_i = (\mathbf{x}'_{1i}, \mathbf{x}'_{2i}), \mathbf{u}'_i = (\mathbf{u}'_{1i}, \mathbf{u}'_{2i})$, are i.i.d. random vectors with common covariance matrix $\Sigma_e$. Such a model is a generalization of the univariate $(r = 1)$ "errors in variables" regression model which has been of interest to statisticians for over a century. In the present paper, it is shown that when $\Sigma_e = \sigma^2I_{p+r}$, a wide class of least squares approaches to estimation of the intercept vector $\alpha$ and slope matrix $B$ all lead to identical estimators $\hat{\alpha}$ and $\hat{B}$ of these respective parameters, and that $\hat{\alpha}$ and $\hat{B}$ are also the maximum likelihood estimators (MLE's) of $\alpha$ and $B$ under the assumption of normally distributed errors $\mathbf{e}_i$. Formulas for $\hat{\alpha}, \hat{B}$ and also the MLE's $\hat{U}_1$ and $\hat{\sigma}^2$ of the parameters $U_1 = (\mathbf{u}_{11}, \cdots, \mathbf{u}_{1n})$ and $\sigma^2$ are given. Under reasonable assumptions concerning the unknown sequence $\{\mathbf{u}_{1i}, i = 1,2,\cdots\}, \hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are shown to be strongly (with probability one) consistent estimators of $\alpha, B$ and $\sigma^2$, respectively, as $n \rightarrow \infty$, regardless of the common distribution of the errors $\mathbf{e}_i$. When this common error distribution has finite fourth moments, $\hat{\alpha}, \hat{B}$ and $r^{-1}(r + p)\hat{\sigma}^2$ are also shown to be asymptotically normally distributed. Finally large-sample approximate $100(1 - \nu){\tt\%}$ confidence regions for $\alpha, B$ and $\sigma^2$ are constructed.