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

In a multivariate "errors in variables" regression model, the unknown mean vectors u₁ᵢ: p × 1, u₂ᵢ: r × 1 of the vector observations x₁ᵢ, x₂ᵢ, rather than the observations themselves, are assumed to follow the linear relation: u₂ᵢ = α + Bu₁ᵢ, i = 1,2,⋯, n. It is further assumed that the random errors eᵢ = xᵢ - uᵢ, x'ᵢ = (x'₁ᵢ, x'₂ᵢ), u'ᵢ = (u'₁ᵢ, u'₂ᵢ), are i.i.d. random vectors with common covariance matrix Σₑ. 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 Σₑ = σ²Iₚ₊ᵣ, a wide class of least squares approaches to estimation of the intercept vector α and slope matrix B all lead to identical estimators α̂ and B̂ of these respective parameters, and that α̂ and B̂ are also the maximum likelihood estimators (MLE's) of α and B under the assumption of normally distributed errors eᵢ. Formulas for α̂, B̂ and also the MLE's Û₁ and σ̂² of the parameters U₁ = (u₁₁, ⋯, u₁ₙ) and σ² are given. Under reasonable assumptions concerning the unknown sequence ₁ᵢ, i = 1,2,⋯\, α̂, B̂ and r⁻¹(r + p)σ̂² are shown to be strongly (with probability one) consistent estimators of α, B and σ², respectively, as n → ∞, regardless of the common distribution of the errors eᵢ. When this common error distribution has finite fourth moments, α̂, B̂ and r⁻¹(r + p)σ̂² are also shown to be asymptotically normally distributed. Finally large-sample approximate 100(1 - ν)% confidence regions for α, B and σ² are constructed.