计量经济学 - Best Linear Unbiased Estimator 最佳线性无偏估计计量经济学 - Best Linear Unbiased Estimator 最佳线性无偏估计.pdf

Proof that OLS is BLUE Best Linear Unbiased Estimator 1 • A K variable General Linear Model (GLM) is: Y X U (1) 1 b (X X ) X Y (2) ols We want to prove ‘bols’ is Best Linear and Unbiased estimator (BLUE) amongst all the linear unbiased estimators. This requires comparing the variance of ‘bols’ with other competing estimator. Let us define a class of linear estimator as:  AY (3) Where A is KxT matrix; it does not depend on Y or  . -1  Note if A=(X’X) X’ then is OLS. The idea is to find a matrix A that generates an estimator  from within the whole class of linear unbiased estimators. A may take any form across a whole class of possible matrices giving rise to different parameter vectors. BEST implies parameter with lowest variance. 2 • (i) when comparing the efficiency of two estimators of a single parameter, compare their variances. • (ii) when comparing the efficiency of two estimators for a vector of parameter, compare their covariance matrices. • Therefore define: • OLS parameter covariance matrix: b • Any other linear parameter covariance matrix:   • Note that 2 1 b  (X X ) • If bols is better than  then that implies:   b  • In order to work out which of the above two covariance matrices are smaller, let us define a vector (a) which picks any linear combination of the elements in the parameter vector , as: • a’ = (a , a , …., a ) 1 2 k