Let 0 and 1 be the OLS intercept and slope estimators, respectively, and let u
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Let β̂0 and β̂1 be the OLS intercept and slope estimators, respectively, and let u be the sample average of the errors (not the residuals!).
(i) Show that β̂2 can be written as β̂1 = β1 + Σni = 1wiui, where wi = di/SSTx and di = xi – x̅.
(ii) Use part (i), along with Σni = 1wi = 0, to show that β̂1 and u̅ are uncorrelated.
(iii) Show that β̂0 can be written as β̂0 = β0 + u̅ – (β̂1 = β1) x̅.
(iv) Use parts (ii) and (iii) to show that Var (β̂0) = s2/n + s2(x̅)2/SSTx.
(v) Do the algebra to simplify the expression in part (iv) to equation (2.58).
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Related Book For
Introductory Econometrics A Modern Approach
ISBN: 9781337558860
7th Edition
Authors: Jeffrey Wooldridge
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