计量经济学笔记-------chapter4.pdf

C H A P T E R 4 F I N ITE— S AM PL E PRO PERTIES OF THE LSE 1
C h a p t e r 4 F i n ite— S am pl e P ro perties of the L SE
F i n nit e — s a m p l e th eo r y : n is assu med to b e fi x ed, normal distn assumed
L arg e—sample theory : n is sent to ∞, general distn assumed
4 . 1 U n b i a s e d ness
W rite
b = ( X ′ X ) − 1 X ′ y = ( X ′ X ) − 1 X ′( Xβ+ ε)
= β+ ( X ′ X ) − 1 X ′ε.
T hen


E ( b | X ) = β+ E ( X ′ X ) − 1 X ′ε| X
= β+ ( X ′ X ) − 1 XE ( ε| X )
= β.
Theref ore
E ( b ) = E x { E [ b | X ] } = E x [ β] = β.



c enter of the true parameter
distribution b v ector
T h e v ar ianc e o f t he L S E and the G au ss— M ark ov theorem
The O LS estimator of β is
b = ( X ′ X ) − 1 X ′ y.
( X ′ X ) − 1 X ′ is an k × n vector. Thus each element of b can be w ritten as a linear
combination of y 1 ,
, y n . We call b a linear estimator for this reason.
The covariance matrix of b is
 
V a r ( b | X ) = E ( b −β) ( b −β) ′| X


= E ( X ′ X ) − 1 X ′εε′ X ( X ′ X ) − 1 | X
= ( X ′ X ) − 1 X ′ E ( εε′| X ) X ( X ′ X ) − 1
 
= ( X ′ X ) − 1 X ′σ 2 I X ( X ′ X ) − 1
= σ 2 ( X ′ X ) − 1
C onsider an arbitrary linear estimator of β, b 0 = C y where C is a k × n matrix. For
b 0 to be unbiased, we should have
E ( Cy | X ) = E ( CXβ+ Cε| X )
= β.
C H A P T E R 4 F I N ITE— S AM PL E PRO PERTIES OF THE LSE 2
F o