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Unformatted text preview: 2.5 Variances of the OLS EstimatorsWe have proven that the sampling distribution of OLS estimates (B hat and B 1 hat) are centered around the true valueHow FAR are they distributed around the true values?The best estimator will be most narrowly distributed around the true valuesTo determine the best estimator, we must calculate OLS variance or standard deviationrecall that standard deviation is the square root of the variance. GaussMarkov Assumption SLR.5 (Homoskedasticity) The error u has the same variance given any value of the explanatory variable. In other words, 2 x)  ar(u σ = V GaussMarkov Assumption SLR.5 (Homoskedasticity)While variance can be calculated via assumptions SLR.1 to SLR.4, this is very complicatedThe traditional simplifying assumption is homoskedasticity, or that the unobservable error, u, has a CONSTANT VARIANCENote that SLR.5 has no impact on unbiasnessSLR.5 simply simplifies variance calculations and gives OLS certain efficiency properties GaussMarkov Assumption SLR.5 (Homoskedasticity)While assuming x and u are independent will also simplify matters, independence is too strong an assumptionNote that: )  ( E )  ( E )]  ( [ )  ( x)  ar(u 2 2 2 2 2 x u x u x u E x u E V = = = σ GaussMarkov Assumption SLR.5 (Homoskedasticity)if the variance is constant given x, it is always constantTherefore: ) ( ) ( E 2 2 u Var u = = σ σ 2 is also called the ERROR VARIANCE or DISTURBANCE VARIANCE GaussMarkov Assumption SLR.5 (Homoskedasticity)SLR.4 and SLR.5 can be rewritten as conditions of y (using the fact we expect the error to be zero and y only varies due to the error) (2.56) )  ( (2.55) )  ( 2 1 σ β β = + = x y Var x x y E Heteroskedastistic Example • Consider the following model:...
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 Spring '09
 Priemaza
 Econometrics, Variance, Trigraph, error variance, yi −β1 xi

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