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 is an -dimensional response variable vector, and , are the -dimensional functions of the explanatory variable vector , a Gauss-Markov linear model has the form:
with the error vector such that
In other words, the observed responses , are not assumed to be normally distributed, are not correlated with one another, and have a common variance .
Gauss-Markov Theorem. Suppose the response variable and the explanatory variables satisfy a Gauss-Markov linear model as described above. Consider any linear combination of the responses
can be used as an estimator for . Then, among all unbiased estimators for having form (2), the ordinary least square estimator (OLS)
yields the smallest variance. In other words, the OLS estimator is the uniformly minimum variance unbiased estimator.
Remark. 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.
|Date of creation||2013-03-22 15:02:53|
|Last modified on||2013-03-22 15:02:53|
|Last modified by||CWoo (3771)|
|Defines||Gauss-Markov linear model|
|Defines||best linear unbiased estimator|