Gauss-Markov theorem

A Gauss-Markov linear model is a linear statistical model that satisfies all the conditions of a general linear model except the normality of the error terms. Formally, if 𝒀 is an m-dimensional response variable vector, and 𝒁𝒊=zi(𝑿), i=1,,k are the m-dimensional functions of the explanatory variable vector 𝑿, a Gauss-Markov linear model has the form:


with ϵ the error vector such that

  1. 1.

    E[ϵ]=𝟎, and

  2. 2.


In other words, the observed responses Yi, i=1,,m are not assumed to be normally distributed, are not correlated with one another, and have a common varianceMathworldPlanetmath Var[Yi]=σ2.

Gauss-Markov Theorem. Suppose the response variable 𝒀=(Y1,,Ym) and the explanatory variables 𝑿 satisfy a Gauss-Markov linear model as described above. Consider any linear combinationMathworldPlanetmath of the responses

Y=i=1mciYi, (1)

where ci. If each μi is an estimatorMathworldPlanetmath for response Yi, parameter θ of the form

θ=i=1mciμi, (2)

can be used as an estimator for Y. Then, among all unbiased estimatorsMathworldPlanetmath for Y having form (2), the ordinary least square estimator (OLS)

θ^=i=1mciμi^, (3)

yields the smallest variance. In other words, the OLS estimator is the uniformly minimum variance unbiased estimator.

Remark. θ^ in equation (3) above is more popularly known as the BLUE, or the best linear unbiased estimator for a linear combination of the responses in a Gauss-Markov linear model.

Title Gauss-Markov theorem
Canonical name GaussMarkovTheorem
Date of creation 2013-03-22 15:02:53
Last modified on 2013-03-22 15:02:53
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 8
Author CWoo (3771)
Entry type Theorem
Classification msc 62J05
Synonym BLUE
Related topic LinearLeastSquaresFit
Defines Gauss-Markov linear model
Defines best linear unbiased estimator