The theorem states that out of the class of estimators that are linear in Y, OLS is the “Best” where “Best” refers to the smallest variance of the estimated coefficients. In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator. (15) State the Gauss-Markov theorem. It is worthwhile to consider an alternative statement and proof of the theorem, which also considers the variance of an arbitrary linear combination of the elements of ﬂ^. (An Alternative Statement… It is rather surprising that the second algebraic result is usually derived in a differential way. To apply the Gauss-Markov theorem the Wikipedia says you must assume your data has the following properties: E[e(i)] = 0 (lack of structural errors, needed to avoid bias) On best linear estimation and general Gauss--Markov theorem in linear models with arbitrary nonnegative covariance structure. The assumptions under which this statement is true include all but normality; i.e., the statement is still true when Gauss-Markov theorem reduces linear unbiased estimation to the Least Squares Solution of inconsistent linear equations while the normal equations reduce the second one to the usual solution of consistent linear equations. The Gauss-Markov theorem is a very strong statement. To prove this, take an arbitrary linear, unbiased estimator $\bar{\beta}$ of $\beta$. The Gauss-Markov theorem states that, under the usual assumptions, the OLS estimator $\beta_{OLS}$ is BLUE (Best Linear Unbiased Estimator). Wikipedia’s stated pre-conditions of the Gauss-Markov theorem. We are going to read through the Wikipedia statement of the Gauss-Markov theorem in detail. Solution: The G-M theorem states that, among all linear unbiased estimators of the regression parameters, the ordinary least squares estimates have minimum variance. Without algebra, you cannot make a single step further, whether it is the precise theoretical statement or an application. Is it possible to prove this part of the Gauss-Markov Theorem: w'β ̂ is BLUE (best linear unbiased estimator) for w'β, where β ̂ is the OLS estimate of β, and w is a nonzero vector. ↑ Department of Mathematics and Statistics, FI-33014 University of Tampere, Tampere, Finland. The Gauss{Markov Theorem. of the Gauss{Markov Theorem that uses this measure can also be proved. In statistics, the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero, are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator, provided it exists. Earlier, one of the desirable properties of estimators was that the estimator has minimum variance. The Gauss-Markov theorem states that the OLS estimator is the most efficient. Gauss–Markov stochastic processes (named after Carl Friedrich Gauss and Andrey Markov) are stochastic processes that satisfy the requirements for both Gaussian processes and Markov processes. 1. SIAM Journal on Applied Mathematics , 17 , 1190--1202. Then criticize it. Why do we care about linearity? The so-called Gauss-Markov theorem states that under certain conditions, least-squares estimators are “best linear unbiased estimators” (“BLUE”), “best” meaning having minimum variance in the class of unbiased linear estimators. The concept of … The Gauss-Markov Theorem for the transformed model implies that the BLUE of b for the generalized regression model is the OLS estimator applied to (6): bˆ GLS = (X 0X) 1X0y = (X0P0PX) 1X0P0Py = (X0W 1X) 1X0W 1y (7) This is the GLS estimator.

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