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Gauss-Markov theorem

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In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimators of the coefficients are the least-squares estimators. More generally, the best linear unbiased estimator of any linear combination of the coefficients is its least-squares estimator. 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, defined below).

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, xi are non-random and observable, εi are random, and so Yi are random. (We set x in lower-case because it is not random, and Y in capital because it is random.) The random variables εi are called the "errors" (not to be confused with "residuals"; see errors and residuals in statistics). The Gauss–Markov assumptions state that

  • {\rm E}\left(\varepsilon_i\right)=0,
  • {\rm var}\left(\varepsilon_i\right)=\sigma^2<\infty,

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

  • {\rm cov}\left(\varepsilon_i,\varepsilon_j\right)=0

for i\not=j; that is "uncorrelatedness." A linear unbiased estimator of β1 is a linear combination

c_1Y_1+\cdots+c_nY_n

in which the coefficients ci are not allowed to depend on the earlier coefficients βi, since those are not observable, but are allowed to depend on xi, since those are observable, and whose expected value remains β1 even if the values of βi change. (The dependence of the coefficients on the xi is typically nonlinear; the estimator is linear in that which is random; that is why this is "linear" regression.) The mean squared error of such an estimator is

E\left((c_1Y_1+\cdots+c_nY_n-\beta_1)^2\right),

i.e., it is the expectation of the square of the difference between the estimator and the parameter to be estimated. (The mean squared error of an estimator coincides with the estimator's variance if the estimator is unbiased; for biased estimators the mean squared error is the sum of the variance and the square of the bias.) The best linear unbiased estimator is the one with the smallest mean squared error. The "least-squares estimators" of β0 and β1 are the functions \widehat{\beta}_0 and \widehat{\beta}_1 of the Ys and the xs that make the sum of squares of residuals

\sum_{i=1}^n\left(Y_i-\widehat{Y}_i\right)^2=\sum_{i=1}^n\left(Y_i-\left(\widehat{\beta}_0+\widehat{\beta}_1 x_i\right)\right)^2

as small as possible. (It is easy to confuse the concept of error introduced early in this article, with this concept of residual. For an account of the differences and the relationship between them, see errors and residuals in statistics.)

The main idea of the proof is that the least-squares estimators are uncorrelated with every linear unbiased estimator of zero, i.e., with every linear combination

a_1Y_1+\cdots+a_nY_n

whose coefficients do not depend upon the unobservable βi but whose expected value remains zero regardless of how the values of β1 and β2 change.

In terms of the matrix algebra formulation, the Gauss–Markov theorem shows that the difference between the parameter covariance matrix of an arbitrary linear unbiased estimator and OLS is positive semi definite (see also proof in external link).

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