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

In a multivariate "errors in variables" regression model, the unknown mean vectors u₁₈: p 1, u₂₈: r 1 of the vector observations x₁₈, x₂₈, rather than the observations themselves, are assumed to follow the linear relation: u₂₈ = + Bu₁₈, i = 1, 2, , n. It is further assumed that the random errors eᵢ = xᵢ - uᵢ, x'ᵢ = (x'₁₈, x'₂₈), u'ᵢ = (u'₁₈, u'₂₈), are i. i. d. random vectors with common covariance matrix 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 ₑ = ²I₏+ₑ, a wide class of least squares approaches to estimation of the intercept vector and slope matrix B all lead to identical estimators and B of these respective parameters, and that and B are also the maximum likelihood estimators (MLE's) of and B under the assumption of normally distributed errors eᵢ. Formulas for, B and also the MLE's U₁ and ² of the parameters U₁ = (u₁₁, , u₁₍) and ² are given. Under reasonable assumptions concerning the unknown sequence \u₁₈, i = 1, 2, \, , B and r^-1 (r + p) ² are shown to be strongly (with probability one) consistent estimators of, B and ², respectively, as n, regardless of the common distribution of the errors eᵢ. When this common error distribution has finite fourth moments, , B and r^-1 (r + p) ² are also shown to be asymptotically normally distributed. Finally large-sample approximate 100 (1 -) \% confidence regions for, B and ² are constructed.