Gauss-Markov Theorem – Under 4 assumptions + homoskedasticity, OLS is BLUE 1) Linear in Parameters – Population model is 01 y x u β = + + Violations: • Including irrelevant variables does not affect unbiasedness but increases variance if correlated with relevant variable of interest. • Omitting relevant variable will bias estimator and generally decrease variance unless uncorrelated with all included variables. 2) Random Sampling 3) Zero Conditional Mean - E( | ) E( )0 u x u = = Corr( , )0 u x = a Cov( , )0 u x = a Cov( , ) Corr( , ) or sd( )sd( ) u x u x u x u x u x u x σ ρ σ σ = = Represents two assumptions in one: E( | ) E( ) u x u = and E( )0 u = . u and x are assumed to be random variables that follow a joint probability distribution The assumption E( | ) E( ) u x u = is very important; it implies, among other things, that u and x are uncorrelated (i.e., not linearly related). The assumption E( )0 u = is not crucial as long as an intercept term is included in the equation (intuitive explanation). Violations:
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