220B_mid09

# 220B_mid09 - 1 Econ 220B Winter 2009 James Hamilton Midterm...

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–1– Econ 220B, Winter 2009 James Hamilton Midterm Exam DIRECTIONS: No books or notes of any kind are allowed. Answer all questions on separate paper. 150 points are possible on this exam. 1.) (50 points) Consider the following regression model: y = X β + ε ε | X N ( 0 , σ 2 V ( X )) . Here y is a ( T × 1) vector of observations on the dependent variable, X is a ( T × k ) matrix of observations on the explanatory variables with rank k , β is an unknown ( k × 1) vector of parameters that must be estimated from the data, σ 2 is an unknown scalar that must be estimated from the data, and V ( X ) is a known nonsingular ( T × T ) matrix. The regression includes a constant term, so that each element of the fi rst column of X is equal to unity. a.) Give the formula for the OLS estimate of β , denoted ˆ β OLS . b.) Give the formula for the GLS estimate of β , denoted ˆ β GLS . c.) Consider the centered R 2 de fi ned as R 2 = 1 P T t =1 ( y t x 0 t ˆ β ) 2 P T t =1 ( y t y ) 2 where y t denotes the t th element of the original vector y de fi

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