Gauss–Markov theorem

This article is not about Gauss-Markov processes.

In statistics, the Gauss-Markov theorem states that in a linear model in which the errors have expecation zero and are uncorrelated and homoscedastic, the best linear unbiased estimators of the coefficients are the least-squares estimators. The errors are not assumed to be normally distributed, nor are they assumed to be independent (but only uncorrelated --- a weaker condition), nor are they assumed to be identically distributed (but only homoscedastic --- a weaker condition).

More explicitly, and more concretely, suppose we have

$Y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}$

for i = 1, . . . , n, where β₀ and β₁ are non-random but unobservable parameters, x_i are non-random and observable, ε_i are random, and so Y_i are random. (We set x in lower-case because it is not random, and Y in capital because it is random.) The random variables x_i are called the "errors". The Gauss-Markov assumptions state that

• $E\left(\varepsilon _{i}\right)=0,$
• $var\left(\varepsilon _{i}\right)=\sigma ^{2}<\infty ,$

(i.e., all errors have the same variance; that is "homoscedasticity"), and

• $cov(\left(\varepsilon _{i},\varepsilon _{j}\right)=0$

for $i\not =j$, that is "uncorrelatedness."