计量经济学笔记-------chapter10.pdf

CHAPTER 10 GENERALIZED LEAST SQUARES ESTIMATION 1
Chapter 10 Generalized Least Squares Estimation
Model
y = Xβ+ ε
E [ε X] = 0
|
E [εε′ X] = σ2
= Σ(
> 0) .
|
1. Heteroskedasticity

σ2 0
w11 0 1 ∼
∼
 w .. 
2 2  22 . 
σ
= σ.. = . 
. .. 
2
0 wnn 0 σ
∼∼ n
2. Autocorrelation

1 ρ1 ρn−1


2 2 β1 1 ρn−2 
σ
= σ.
.

. .. 
ρ− 1
n 1
OLS and IV estimation
OLS estimation
• The OLS estimator can be written as
−1
b = β+ (X′X) X′ε.
1. Unbiasedness
E [b] = E [E [b X]] = β.
X |
2. Variance—Coviance Matrix

′
V ar [b X] = E (b β) (b β) X
|
−−|
−1 −1
= E (X′X) X′εε′X (X′X) X
|
−1
−1
= (X′X) X′σ2
X (X′X) .
The unconditional variance is
E [V ar [b X]] .
X |
If ε is normally distributed,

−1 −1
b X N β, σ2 (X′X) X′
X (X′X) .
| ∼
CHAPTER 10 GENERALIZED LEAST SQUARES ESTIMATION 2
3. Consistency
Suppose that
X′X
P Q > 0
n →
X′
X
P P > 0.
n →
Then

−1
−1
1 X′X X′
X X′X
V ar [b X] = σ2
| n n n n
P 0
→
and
V ar [b] P 0.
→
Using this and Chebyshev’s inequality, we have for and α Rk 0 and ε> 0
∈−{ }
′′
′α E (b β) (b β) α
P [ α(b β) > ε] −−
| −| ≤ε2
α′V ar (b) α
=
ε2
0 as n
→→∞
which implies
b p β.
→
4. Asymptotic distribution of b
Assume (Xi, εi) is a sequence of independen