Econ 620 Midterm + Key 2006 - Econ 620 Spring 2005...

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Econ 620 Spring 2005 Professor N. Kiefer TA H. Choi Suggested Solutions to the midterm exam 1. We have Z = XR, where R = 1 1 1 1 - 1 1 0 0 1 . Therefore we have = β from Ey = = XRα = Xβ. This implies R ˆ a LS = ˆ β LS . 2. We have ˆ β 2 for a) ( X 0 2 M 1 X 2 ) - 1 X 0 2 M 1 y, b) ( X 0 2 X 2 ) - 1 X 0 2 P 1 y, c) ( X 0 2 P 1 X 2 ) - 1 X 0 2 P 1 y, d) ( X 0 2 X 2 ) - 1 X 0 2 M 1 y, e) ( X 0 2 M 1 X 2 ) - 1 X 0 2 M 1 y, f) ( X 0 2 M 1 X 2 ) - 1 X 0 2 M 1 y, g) ( X 0 2 M 1 X 2 ) - 1 X 0 2 M 1 y, h) ( X 0 2 M 1 X 2 ) - 1 X 0 2 M 1 y. Hence we have 4 different estimators. (a)-(d) are all different and (e)-(h) are the same as (a). 3. The prediction y p is 10 × 1 vector and has the form y p = Cy, where y = ( y 1 ,...,y n ) 0 . Under LS assumption, y p is given by the relationship Cy = X p ˆ β, where X p is known and ˆ β is OLS estimator. We have C = X p ( X 0 X ) - 1 X 0 . The variance of the prediction errors is Var( - ε p ) = C Var( ε ) C 0 + V ar ( ε p ) = σ 2 ( CC 0 + I ) = σ 2 X p ( X 0 X ) - 1 X
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