# 程序代写代做代考 ECON 61001: Lecture 6

ECON 61001: Lecture 6
Alastair R. Hall
The University of Manchester
Alastair R. Hall ECON 61001: Lecture 6 1 / 22

Outline of this lecture
More on GLS
Cross-section data with heteroscedasticity
OLS-based inference GLS/WLS
testing for heteroscedasticity
Alastair R. Hall ECON 61001: Lecture 6 2 / 22

GLS
Recall that our model is:
y = Xβ0 + u
where
CA1: true model is: y = Xβ0 + u.
CA2: X is fixed in repeated samples.
CA3: X is rank k.
CA4: E[u] = 0.
CA5-NS Var [u] = Σ where Σ is a T × T positive definite matrix.
CA6: u ∼ Normal.
Alastair R. Hall ECON 61001: Lecture 6 3 / 22

GLS
The Generalized Least Squares estimator of β0 is: βˆGLS = (X′Σ−1X)−1X′Σ−1y
If Assumptions CA1-CA4, CA5-NS and CA6 hold then: βˆGLS ∼N􏰚β0,(X′Σ−1X)−1􏰛.
But need Σ in order to calculate βˆGLS.
If Σ unknown then GLS is an infeasible estimator.
Alastair R. Hall ECON 61001: Lecture 6 4 / 22

GLS
Solution assume t − sth element of Σ is given by: σt,s = ht,s (zt, zs, α)
where
ht,s is some specified function,
zt is a vector of observable variables, α is a p × 1 vector of parameters.
So Σ = Σ(α).
Estimate α from the sample {yt,xt′,zt′; t = 1,2,…,T} → αˆ. Then set Σˆ = Σ(αˆ).
Alastair R. Hall ECON 61001: Lecture 6 5 / 22

GLS
→ Feasible Generalized Least Squares (FGLS) estimator, βˆFGLS = (X′Σˆ−1X)−1X′Σˆ−1y.
Does FGLS have finite sample properties of GLS? Maybe: consider E[βˆFGLS] = β0 +E􏰉(X′Σˆ−1X)−1X′Σˆ−1u􏰊=β0?
FGLS does inherit large sample properties of GLS.
Alastair R. Hall ECON 61001: Lecture 6 6 / 22

Cross-section data with heteroscedasticity
Assumptions:
CS1: yi = xi′β0 + ui
CS2: { (ui,xi′), i = 1,2,…N} forms an independent and identically distributed sequence.
CS3: E[xi xi′] = Q, finite, p.d. CS4: E[ui|xi]=0.
CS5-H: Var [ui |xi ] = h(xi ), positive, and h(xi ) ̸= h(xj ) for some i ̸= j.
Note:
cross-section data with random sample from homogenous population
have conditional heteroscedasticity but not unconditional heteroscedasticity
Alastair R. Hall ECON 61001: Lecture 6 7 / 22

OLS
Recall:
ˆ 􏰂 N 􏰃−1 N
N1/2(βN − β0) = N−1 􏰈xixi′ N−1/2 􏰈xiui. i=1 i=1
As in Lecture 4: WLLN & Slutsky’s Theorem →
and CLT →
N
and
N−1/2􏰈xiui →d N(0,Ω), whereΩ=limN→∞ΩN i=1
􏰂 􏰈N 􏰃−1 N − 1 x i x i ′
i=1
→p Q − 1 ,
􏰄−1/2􏰈N 􏰅 ΩN=Var N xiui .
i=1
Alastair R. Hall ECON 61001: Lecture 6 8 / 22

OLS
What is Ω here?
Assumption CS2 ⇒ {xiui; i = 1,2,…N} are i.i.d. and so
Cov [xi ui , xj uj ] = 0 (i ̸= j ). ⇒ΩN =Var[xiui].
Using E [xi ui ] = 0,
Var[xiui] = E[ui2xixi′]=E􏰜E[ui2|xi]xixi′]􏰝=E[h(xi)xixi′]=Ωh, say.
Alastair R. Hall ECON 61001: Lecture 6 9 / 22

OLS
Therefore, under our assumptions, we have:
N
N−1/2􏰈xiui →d N(0,Ωh).
i=1
Under Assumptions CS1-CS4 and CS5-H: N1/2(βˆN−β0)→d N(0,Vh).
whereVh =Q−1ΩhQ−1.
Alastair R. Hall ECON 61001: Lecture 6 10 / 22

OLS
To use this result as basis for inference, need a consistent estimator of Vh and so Ωh.
Using WLLN we have:
N
N−1 􏰈ui2xixi′ →p E[ui2xixi′] = Ωh.
i=1
Also it can be shown that
and so
NN N−1􏰈ei2xixi′ −N−1􏰈ui2xixi′ →p 0.
i=1 i=1
N Ωˆh=N−1􏰈ei2xixi′→p Ωh.
i=1
Alastair R. Hall ECON 61001: Lecture 6 11 / 22

OLS
S e t Qˆ = N − 1 X ′ X t h e n
Vˆh = Qˆ−1ΩˆhQˆ−1 →p Vh.
Let Vˆh,l,l is (l, l) element of Vˆh then:
􏰟Vˆh,l,l/N ∼ “White standard errors”
Can then perform inference using same techniques as in Lecture 4 provided we use modified variance estimator.
For example, an approximate 100(1 − α)% confidence interval for β0,l is given by,
􏰘 βˆ N , l ± z 1 − α / 2 􏰟 Vˆ h , l , l / N 􏰙 .
For other inference procedures see Lecture Notes Section 4.3.1.
Alastair R. Hall ECON 61001: Lecture 6 12 / 22

GLS
Impose Assumptions CS1-CS4, & CS5-H: ⇒ E[u|X] = 0 & (with σ i2 = h ( x i ) )

.. . 0 0 … 0 σN2
 σ 12 0 0 . . . 0   0 σ2 0 … 0 
2
Var[u|X] = Σ =  . . . …….
σ00…0 100…0 1 σ1
 0 σ2 0 … 0   0 1 0 … 0   σ2 
. ……. …….
D= . .
0 0 … 0 σN 0 0 … 0
. .. ,andS= . . and transformed regression model is
. ..
1
σN
yi = 􏰘 1 􏰙β0,1 +􏰘xi,2 􏰙β0,2 +…+􏰘xi,k 􏰙β0,k +􏰘 1 􏰙ui σi σi σi σi σi
Alastair R. Hall ECON 61001: Lecture 6 13 / 22

Asymptotic properties of GLS
Theorem: If Assumptions CS1 – CS4 hold then βˆGLS is a consistent estimator for β0.
Theorem: If Assumptions CS1 – CS4 and CS5-H hold then: N1/2(βˆGLS−β0)→d N(0,VGLS),
where VGLS = {E[σ−2xi x′]}−1. ii
Alastair R. Hall ECON 61001: Lecture 6 14 / 22

FGLS
Need to assume functional form for Σ(α): σ i 2 = σ 02 ∗ v ( x i )
suffices to divide variables by 􏰞v(xi) (see example later)
σi2 = h(xi,α)
E [ u i2 | x i ] = h ( x i , α ) ⇒
ui2 = h(xi,α) + ai, where E[ai|xi] = 0.
So estimate α using model via Nonlinear LS (NLS) or OLS
ei2 =h(xi,α)+“error”.
Alastair R. Hall ECON 61001: Lecture 6 15 / 22

Weighted Least Squares
Let {wi; i = 1,2,…,N} be a set of positive constants and use to weight observations in regression model:
wi∗yi =(wi∗xi)′β0+wi∗ui. Weighted Least Squares (WLS) is OLS estimator based on
weighted regression model:
βˆWLS = (X′W2X)−1X′W2y.
where W2 = diag(w12, w2, . . ., wN2 ).
It can be shown that under Assumptions CS1-CS4, & CS5-H that
N1/2(βˆWLS−β0)→d N(0,VWLS)
where V = Q−1Ω Q−1, Q = plim N−1X′W X,
WLSwwwwT→∞ 2 Ωw = plimT→∞ N−1X′W2ΣW2X.
Alastair R. Hall ECON 61001: Lecture 6 16 / 22

OLS, GLS & WLS
wi =1foralli ⇒WLS=OLS
wi = 1/σi for all i ⇒ WLS=GLS
Re-weighting of observations is source of efficiency gains from GLS
over OLS.
What happens if assume incorrect model for σi2?
GLS is WLS and may or may not be more efficient than OLS;
inferences based on “GLS” valid but must be performed with heteroscedasticity robust estimator of WLS variance formula (see Lecture Notes).
Alastair R. Hall ECON 61001: Lecture 6 17 / 22

Breusch-Pagan test for heteroscedasticity
Assume:
σi2 = h(δ + zi′α)
h( · ) is (a twice continuously differentiable) function,
independent of i, and h( · ) > 0,
zi is a (p × 1) vector of observable variables, (δ, α′)′ is a vector of (p + 1) × 1 of parameters.
Test:
H0 : α = 0 vs HA : αl ̸= 0foratleastonel = 1,2,…,p
Test stat is BPN = NR2 where R2 is from regression of ei2 on (1,zi′).
Under H0: BPN →d χ2p – see Lecture Notes for regularity conditions.
where
Alastair R. Hall ECON 61001: Lecture 6 18 / 22

Breusch-Pagan test for heteroscedasticity
Note:
statistic does not depend on h( · ).
depends on zi
White’s (1980) direct test for heteroscedasticity (see Lecture Notes)
Alastair R. Hall ECON 61001: Lecture 6 19 / 22

Empirical example
Suppose a researcher is interested in studying the savings behaviour of households:
yi = xi′β0 +ui = β0,1 +β0,2mi +ui,
where
yi is the level of savings of household i
mi is household income.
Var [ui |xi ] arguably depends on mi .
OLS estimation results:
yi = 124.84
(655.39) [522.91]
+
0.147 mi . (0.058) [0.061]
where(·)=conventionalOLSs.e.’s, [·]=Whites.e.’s
Alastair R. Hall ECON 61001: Lecture 6 20 / 22

Empirical example
Now suppose Var[ui |xi] = σ02mi. WLS estimation results:
yi = −124.95 + 0.172 mi. (480.86) (0.057) [266.59] [0.050]
The number in parentheses are the standard errors assuming have correct model for heteroscedasticity; the number in brackets are heteroscedastcity robust standard errors.
Alastair R. Hall ECON 61001: Lecture 6 21 / 22