BLUE - LAURIER Business & Economics Wilfrid Laurier...

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LAURIER Wilfrid Laurier University School of Business and Economics (Econometrics) Instructor: Jean Eid Lecture Notes: Gauss-Markov Theorem Assume that the following hold MLR-1 Linearity in the parameter MLR-2 Independence of the error term u and random sampling MLR-3 No perfect Collinearity MLR-4 Zero conditional mean MLR-5 Homoskedasticity i.e var ( u i | x ) = σ 2 Also assume that y i = β o + β 1 x i + u i We know that the OLS estimator for β 1 is b β 1 = n X i =1 x i - x y i n X i =1 x i - x 2 THE ROAD AHEAD We need to come up with another estimator that is linear, but also we need to make sure that this estimator is unbiassed. We need to do so because it is unfair to compare the OLS to any other estimator since biased estimators are not the best by construction. So in order to compare apples to apples we need to compare the OLS estimator to other linear unbiassed estimators, and show that OLS is best. Now let there be any other estimators that are unbiassed and linear. For an estimator to be linear, it should be a linear function of the observable values. Denote this estimator by e β 1 , and e β 1 = n X i =1 c i y i where c i = g x i.e c i is a function of x. For example c i = x i - x n X i =1 x i - x 2 and in this case e β 1 = b β 1 . In general c i can be any function of the independent variable (linear or nonlinear function). As long as we can compute each c i for
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PAGE 2 each data point i we multiply that by y i and sum over all these to get e β 1 . Note that e β 1 = n X i =1 c i y i = n X i =1 c i β o + β 1 x i + u i = β o n X i =1 c i + β 1 n X i =1 c i x i + n X i =1 c i u i Recall that the Gauss-Markov theorem states that OLS BLUE i.e B est L inear U nbiassed E stimator. So in order to compare apples to apples, we need e β 1 to be unbiassed i.e. E e β 1 | x = β 1 E e β 1 | x = E β o n X i =1 c i + β 1 n X i =1 c i x i + n X i =1 c i u i | x = β o n X i =1 c i + β 1 n X i =1 c i x i + n X i =1 c i E u i | x the reason why c i came out of the expectation is because c i are only a function of x and since we are conditioning on x any function of just x is constant with respect to the expectation. In addition, E u i | x =0 = β o n X i =1 c i + β 1 n X i =1 c i x i But in order for e β 1 to be unbiassed we need to have β o n X i =1 c i + β 1 n X i =1 c i x i = β 1 (1) How can we make sure that ∀ { β o 1 } ∈ R 2 , (1) above always hold. This means that for any values from
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This note was uploaded on 10/16/2011 for the course ECONOMICS 655 taught by Professor Jeaneid during the Fall '11 term at Wilfred Laurier University .

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BLUE - LAURIER Business & Economics Wilfrid Laurier...

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