Download Advanced Econometric I

Advanced Econometric I
Zhou Yahong
School of Economics
SHUFE
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Part I
Large Sample Results for the Classical Regression
Model
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
1
Convergence in Probability
2
Law of Large Numbers
3
Central Limit Theorem
4
Asymptotic Properties for OLS
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Convergence in Probability
The random variable xn convergence in probability to a
constant if
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Convergence in Probability
The random variable xn convergence in probability to a
constant if
lim Pr(|xn − c| > ε) = 0
n→∞
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Convergence in Probability
The random variable xn convergence in probability to a
constant if
lim Pr(|xn − c| > ε) = 0
n→∞
p lim xn = c quad
Zhou Yahong
SHUFE
p
or xn → x
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
1
Convergence in Probability
2
Law of Large Numbers
3
Central Limit Theorem
4
Asymptotic Properties for OLS
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Law of Large Numbers
Let if x1 , x2 , . . . , xn are independent and identically (i.i.d)
distributed with mean µ, then
p lim x = µ
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
1
Convergence in Probability
2
Law of Large Numbers
3
Central Limit Theorem
4
Asymptotic Properties for OLS
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Central Limit Theorem
Central Limit Theorem (Univariate)–Lindberg-Levy: if
x1 , x2 , . . . , xn are independent and identically (i.i.d) distributed
with mean µ and variance σ 2 , then
√
Zhou Yahong
d
n(x − µ) → N(0, σ 2 )
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Central Limit Theorem
The Central Limit Theorem (Multivariate)–Lindberg-Levy: if
x1 , x2 , . . . , xn are independent and identically (i.i.d)
distributed with mean vector µ and covariance matrix Q, then
√
d
n(x̄ − µ) → N(0, Q)
where Q = E [xx 0 ] = p lim n1 X 0 X , or equivalently
1
d
(x̄ − µ) → N(0, n × p lim X 0 X )
n
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
1
Convergence in Probability
2
Law of Large Numbers
3
Central Limit Theorem
4
Asymptotic Properties for OLS
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Consistency of b—Suppose then
1
1
b = β + ( X 0 X )−1 ( X 0 ε)
n
n
1
0
( X 0 X )−1 → Q = Ex(i) x(i)
n
(the latter, i.i.d sampling). In addition,
n
1 0
1X
X ε=
xi εi = w̄
n
n
i=1
Thus
p lim b = β + Q −1 p lim w̄ = β
p lim β̂ = β + Q −1 0 = β
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Consistency of s 2 =
s2 =
=
e0e
n−K
=
ε0 Mε
n−K ,
1
[ε0 ε − ε0 X (X 0 X )−1 X 0 ε]
n−K
ε0 ε ε0 X X 0 X −1 X 0 ε
n
[
−
(
)
]
n−K n
n
n
n
from above, we have
p lim
ε0 X
= 0, and
n
p lim(
X 0 X −1
) = Q −1
n
Also by LLN.
n
p lim
ε0 ε
1X 2
= p lim
εi = σ 2
n
n
i=1
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Asymptotic normality—
√
1
1
n(β̂ − β) = [ X 0 X ]−1 √ X 0 ε
n
n
if εi are i.i.d, we can show that
n
1
1 X
√ X 0ε = √
x(i) εi
n
n
i=1
and
var (xi ui ) = σ 2 Exi xi0 = σ 2 Q
Thus
1
d
√ X 0 ε → N(0, σ 2 Q)
n
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Asymptotic normality— while
1
p
[ X 0 X ]−1 → Q −1
n
Thus
√
d
n(b − β) → N(0, σ 2 Q −1 )
and
p lim s 2 (X 0 X /n)−1 = σ 2 Q −1
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Asymptotic behavior of the standard test statistics
F = (Rb − q)0 [σ̂ 2 R(X 0 X )−1 R 0 ]−1 (Rb − q)/J ∼ FJ,(n−K )
under normality p lim σ̂ 2 = σ 2 , thus JF has a chi-square with
J degrees of freedom
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
without normality, we have
√
n(Rb − q) ∼ N[0, R(σ 2 Q −1 )R 0 ]
and
n(Rb − q)0 [R(σ 2 Q −1 )R 0 ]−1 (Rb − q)
has a chi-square with J degrees of freedom.
Zhou Yahong
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
Testing nonlinear restriction R(β) = q in the context of linear
models, since
R(b) ≈ R(β) +
∂R
(b − β)
∂β 0
thus under null:
R(b) − q ≈
√
n(R(b) − R(β)) ≈
Zhou Yahong
∂R
(b − β)
∂β 0
√ ∂R
∂R
∂0R
n 0 (b − β) → N(0, σ 2 0 Q −1
)
∂β
∂β
∂β
SHUFE
Advanced Econometric I
Convergence in Probability
Law of Large Numbers
Central Limit Theorem
Asymptotic Properties for OLS
Asymptotic Properties for OLS
n(R(b) − R(β))0 [σ 2
∂R −1 ∂ 0 R −1
Q
)] (R(b) − R(β))
∂β 0
∂β
and
JF = (R(b)−R(β))0 [σ̂ 2
∂R(b) 0 −1 ∂ 0 R(b) −1
(X X )
)] (R(b)−R(β))
∂β 0
∂β
has a chi-square with J degrees of freedom.
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
Part II
Instrumental variable estimation and measurement
error
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
5
Measurement error
6
IV estimator
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
Measurement Error
In this example, using regression without intercept,
y ∗ = x ∗β + ε
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
Case I
case one
y = y∗ + v
and
y = x ∗β + ε − v
measurement error on the dependent variable can be absorbed
in the error term.
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
Case II
case two —least squares attenuation–bias towards zero
x = x∗ + u
then
y = xβ + ε − βu
P
P
1/n (xi∗ + ui )(βxi∗ + εi )
β
1/n xi yi
P ∗
P 2 =
=
p lim b =
2
1/n (xi + ui )
1 + σu2 /Q ∗
1/n xi
P
where Q ∗ = p lim(1/n) xi∗2 , thus it is an inconsistent
estimator.
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
Case II
In general
y = X ∗β + ε
and
X = X∗ + u
thus
p lim X 0 X /n = Q ∗ + Σu
and p lim X 0 y /n = Q ∗ β
p lim b = [Q ∗ + Σu ]−1 Q ∗ β
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
with a single variable measured
 2
σu
 0
Σu = 

0
with error

0
0
0
0 


0
0
For this special case
β1
1 + σu2 q ∗11
σu2 q ∗k1
= βk − β1
1 + σu2 q ∗11
p lim b1 =
p lim bk
for k 6= 1
where q ∗k1 is the (k, 1)th element of (Q ∗ )−1 . (Use (A-66) to
invert [Q ∗ + Σu ] = [Q ∗ + (σu e1 )(σu e1 )0 ]).
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
[Q ∗ +(σu e1 )(σu e1 )0 ]−1 = Q ∗−1 −σ 2 [
1
1+
]Q
σu2 e10 Q ∗−1 e1
∗−1
e1 e10 Q ∗−1
so
[Q ∗ + (σu e1 )(σu e1 )0 ]−1 Q ∗ β = β − σ 2 [
1
]Q ∗−1 e1 β1
1 + σu2 q ∗11
The direction of the bias (except the first one) depends on
several unknowns and cannot be estimated. The coefficient on
the badly measured variable is still biased toward zero. The
other coefficients are all biased as well, although in unknown
directions. If more than one variable is measured with error,
there is very little that can be said.
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
5
Measurement error
6
IV estimator
Zhou Yahong
SHUFE
Advanced Econometric I
Measurement error
IV estimator
IV estimator
y = X β + ε − uβ = X β + v
if
p lim Z 0 v /n = 0
Zhou Yahong
and p lim Z 0 X /n = Qzx nonsingular
SHUFE
Advanced Econometric I