Lecture "Advanced Econometrics (Part II) - Chapter 7: Greneralized linear regression model" presentation of content: Model, properties of ols estimators, white's heteroscedascity consistent estimator, greneralized least squares estimation.
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Nội dung Text: Lecture Advanced Econometrics (Part II) - Chapter 7: Greneralized linear regression model
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
Chapter 7
GENERALIZED LINEAR REGRESSION MODEL
I. MODEL:
Our basic model:
Y = X.β + ε with ε ~ N [0, σ 2 I ]
We will now generalize the specification of the error term.
E(ε) = 0, E(εε') = σ 2 Ω = Σ .
n ×n
This allows for one or both of:
1. Heteroskedasticity.
2. Autocorrelation.
The model now is:
(1) Y = X β + ε
n ×k
(2) X is non-stochastic and Rank ( X ) = k .
(3) E(ε) = 0
n ×1
(4) E(εε') = Σ = σ ε2 Ω
n× n n× n
Heteroskedasticity case:
σ 12 0 0
0 σ 22 0
Σ=
0 0 σ n2
Nam T. Hoang
University of New England - Australia 1 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
Autocorrelation case:
1 ρ1 ρ n −1
ρ 1 ρ n −2
2
Σ = σε 1
ρ n −1 ρ n −2 1
ρ i = Corr (ε t , ε t −i ) = correlation between errors that are i periods apart.
II. PROPERTIES OF OLS ESTIMATORS:
1. βˆ = ( X ′X ) −1 X ′Y = ( X ′X ) −1 X ′( Xβ + ε )
βˆ = β + ( X ′X ) −1 X ′ε
E ( βˆ ) = β + ( X ′X ) −1 X ′E (ε ) = β
βˆ is still an unbiased estimator
2. VarCov( βˆ ) = E [( βˆ − β )( βˆ − β )' ]
= E[( X ′X ) −1 X ′ε )(( X ′X ) −1 X ′ε )' ]
= E [( X ′X ) X ′εε ' X ( X ′X ) ]
−1 −1
= ( X ′X ) −1 X ′E (εε ' ) X ( X ′X ) −1
= ( X ′X ) −1 X ′(σ 2 Ω) X ( X ′X ) −1
≠ σ 2 ( X ′X ) −1
so standard formula for σˆ βˆ no longer holds and σˆ βˆ is a biased estimator of true σˆ βˆ .
→ βˆ ~ N[β, σ 2 ( X ′X ) −1 X ′Ω) X ( X ′X ) −1 ]
so the usual OLS output will be misleading, the std error, t-statistics, etc will be based on
σˆ ε2 ( X ' X ) −1 not on the correct formula.
3. OLS estimators are no longer best (inefficient).
Nam T. Hoang
University of New England - Australia 2 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
Note: for non-stochastic X, we care about the efficient of βˆ . Because we know if n↑ →
Var( βˆ j ) ↓ → plim βˆ = β, βˆ is consistent.
4. If X is stochastic:
- OLS estimators are still consistent (when E(ε|X) = 0.
- IV estimators are still consistent (when E(ε|X) ≠ 0).
- The usual covariance matrix estimator of VarCov( βˆ ) which is σˆ ε2 ( X ' X ) −1 will be
inconsistent (n →∞) for the true VarCov( βˆ ).
We need to know how to deal with these issues. This will lead us to some generalized
estimator.
III. WHITE'S HETEROSCEDASCITY CONSISTENT ESTIMATOR OF VarCov( β ).
ˆ
(Or Robust estimator of VarCov( βˆ )
If we knew σ2Ω then the estimator of the VarCov( βˆ ) would be:
V = ( X ′X ) −1 X ′(σ 2 Ω) X ( X ′X ) −1
−1 −1
= X ′X X ′(σ 2 Ω) X X ′X
1 1 1 1
nn n n
−1 −1
= X ′X X ′Σ) X X ′X
1 1 1 1
nn n n
1
If Σ is unknown, we need a consistent estimator of X ′Σ) X (Note that the number of
n
unknowns is Σ grows one-for-one with n, but [X ′Σ) X ] is k×k matrix it does not grow
with n).
1
Let: Σ* = X ′ΣX
n
1 n n
Σ* = ∑∑σ ij Xk ×1i X1×k′j
n i =1 j =1
Nam T. Hoang
University of New England - Australia 3 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
σ 12 0 0
0 σ 22 0
In the case of heteroskedasticity Σ =
0 0 σ n2
1 n 2
Σ* = ∑σ i X i X i′
n i =1
White (1980) showed that if:
1 n 2
Σ0 = ∑ ei X i X i′
n i =1
then plim(Σ0) = plim(Σ*)
so we can estimate by OLS and then a consistent estimator of V will be:
−1 −1
11 1 n 1
Vˆ = X ′X ∑ ei2 X i X i′ X ′X
nn n i =1 n
Vˆ = n ( X ′X ) Σ 0 ( X ′X )
−1 −1
Vˆ is consistent estimator for V, so White's estimator for VarCov( βˆ ) is:
VarCov ( βˆ ) = ( X ′X ) X ' Σˆ X ( X ′X ) = Vˆ
−1 −1
e12 0 0
0 e22 0 1
where: Σ =
ˆ (Note Σ 0 = Σˆ )
n
2
0 0 en
Vˆ is consistent for V = n ( X ′X )−1 σ 2 Ω( X ′X )−1 regardless of the (unknown) form of the
heteroskedasticity (only for heteroskedasticity).
Newey - West produced a corresponding consistent estimator of V when there is
autocorrelation and/or heteroskedasticity.
Note that White's estimator is only for the case of heteroskedasticity and autocorrelation.
White's estimator just modifies the covariance matrix estimator, not βˆ . The t-statistics,
F-statistics, etc will be modified, but only in a manner that is appropriate asymptotically.
So if we have heteroskedasticity or autocorrelation, whether we modify the covariance
matrix estimator or not, the usual t-statistics will be unreliable in finite samples (the
Nam T. Hoang
University of New England - Australia 4 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
white's estimator of VarCov( βˆ ) only useful when n is very large, n → ∞ the Vˆ →
VarCov( βˆ ).
→ βˆ is still inefficient.
→ To obtain efficient estimators, use generalized lest squares - GLS
A good practical solution is to use White's adjustment, then use Wald test, rather than the
F-test for exact linear restrictions. Now let's turn to the estimation of β, taking account of
the full process for the error term.
IV. GENERALIZED LEAST SQUARES ESTIMATION (GLS):
OLS estimator will be inefficient in finite samples.
1. Assume E(εε') = n×Σn is known, positive definite.
→ there exists C j and λ j j = 1,2, ... ,n such that
n ×1 n ×1
Σ Cj = Cj λj (characteristic vector C, Eigen-value λ).
n× n
n ×1 n ×1 n ×1
→ before C'ΣC = Λ where C = [C1 C 2 C n ]
n ×1
λ1 0 0 λ1 0 0
0 λ 0 0 λ2 0
Λ= Λ1 / 2 =
2
0 0 λn 0 0 λn
C ' ΣC = Λ = ( Λ1 / 2 )' ( Λ1 / 2 )
→ ( Λ−1 / 2 )C 'ΣC ( Λ−1 / 2 ) = ( Λ−1 / 2 )( Λ1 / 2 )( Λ1 / 2 )( Λ−1 / 2 ) = I
H' H'
→ HΣ H ' = I
→ Σ = H −1 IH ' −1 = H −1 H ' −1
→ Σ = H'H H = Λ−1 / 2 C '
Nam T. Hoang
University of New England - Australia 5 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
Our model: Y = Xβ + ε
Pre-multiply by H:
HX β + H
HY =
ε
Y* X* ε*
→ Y * = X *β + ε *
ε* will satisfy all classical assumption because:
E(ε*ε*') = E[H(εε')H'] = HΣH' = I.
Since transformed model meets classical assumptions, application of OLS to (Y*, X*) data
yields BLUE.
→ βˆGLS = ( X * ' X * ) −1 X * ' Y *
→ H ' H X ) −1 X '
= (X ' H ' HY
Σ −1 Σ −1
→ βˆGLS = ( X ' Σ −1 X ) −1 X ' Σ −1Y
Moreover:
[ ] [
VarCov ( βˆGLS ) = ( X * ' X * ) −1 X * ' E (ε *ε * ' ) X * ( X * ' X * ) −1 ]
= ( X * ' X * ) − 1 = ( X ' Σ −1 X ) −1
VarCov( βˆGLS ) = ( X ' Σ −1 X ) −1
→ βˆGLS ~ N [β , ( X ' Σ −1 X )]
Note that: βˆGLS is BLUE of βˆ → E ( βˆGLS ) = β
GLS estimator is just OLS, applied to the transformed model → satisfy all assumptions.
Gauss - Markov theorem can be applied
→ βˆGLS is BLUE of βˆ .
→ βˆOLS must be inefficient in this case.
→ Var ( βˆ j GLS ) ≤ Var ( βˆ j OLS ) .
Nam T. Hoang
University of New England - Australia 6 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
Example:
σ 12 0 0 1 / σ 12 0 0
0 σ 22 0 0 1 / σ 22 0
Σ = → Σ −1 =
known
0 0 σ n2 0 0 1 / σ n2
1 / σ 12 0 0
0 1 / σ 22 0
H'H = Σ-1 →H =
0 0 1 / σ n2
1 / σ 12 0 0 Y1 Y1 / σ 12
0 1 / σ 22 0 Y2 Y2 / σ 22
HY = = =Y*
0 0 1 / σ n2 Yn Yn / σ n2
1 / σ 1 X 12 / σ 1 X 1k / σ 1
1/ σ 2 X 22 / σ 2 X 2k / σ 2
X = HX =
*
1 / σ n X n2 / σ n X nk / σ n
Transformed model has each observations divided by σi:
Yi 1 X X εi
= β1 + β 2 i 2 + + β k ik +
σi σi σi σi σi
Apply OLS to this transformed equation → "Weighted Least Squares":
Let: βˆ = GSL estimator.
εˆ = Y * − X * βˆGLS
εˆ' εˆ
σˆ =
n−k
Then to test: H0: Rβ = q (F Wald test).
[ Rβˆ − q ]' [ R ( X * ' X * ) −1 R ' ]−1 [ Rβˆ − q ]
Fnr− k = r ~F if H0 is true.
( r ,n − k )
σˆ 2
Nam T. Hoang
University of New England - Australia 7 University of Economics - HCMC - Vietnam
- Advanced Econometrics Chapter 7: Generalized Linear Regression Model
[εˆc′εˆc − εˆ' εˆ ]
and F r
= r
n −k
εˆ' εˆ
(n − k )
where: εˆc = Y * − X * βˆc GLS
βˆc GLS = βˆGLS − ( X ' Σ −1 X ) −1 R ' [ R ( X ' Σ −1 X ) −1 R' ]−1 ( RβˆGLS − q)
is the "constrained" GLS estimator of β.
2. Feasible GLS estimation:
In practice, of course, Σ is usually unknown, and so βˆ cannot be constructed, it is not
feasible. The obvious solution is to estimate Σ, using some Σˆ then construct:
βˆGLS = ( X ' Σˆ −1 X ) −1 X ' Σˆ −1Y
A practical issue: Σ is an (n×n), it has n(n+1)/2 distinct parameters, allowing for
symmetry. But we only have "n" observations → need to constraint Σ. Typically Σ =
Σ(θ) where θ contain a small number of parameters.
Ex: Heteroskedasticity var(εi) = σ2(θ1+θ2Zi).
θ1 + θ 2 z1 0 0
0 θ1 + θ 2 z n 0
Σ=
0 0 θ1 + θ 2 z n
just 2 parameters to be estimated to form Σˆ .
Serial correlation:
1 ρ ρ n−1
ρ 1 ρ n −2
Σ= 1 = Σ( ρ )
n −1
ρ ρ 1
n −2
only one parameter to be estimated.
• If Σˆ is consistent for Σ then β will be asymptotically efficient for β.
• Of course to apply β we want to know the form of Σ → construct tests.
Nam T. Hoang
University of New England - Australia 8 University of Economics - HCMC - Vietnam
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