6. Heteroskedasticity (Violation of Assumption #B2)

6. Heteroskedasticity
(Violation of Assumption #B2)
Assumption #B2:
• Error term ui has constant variance for i = 1, . . . , N , i.e.
Var(ui) = σ 2
Terminology:
• Homoskedasticity: constant variances of the ui
• Heteroskedasticity: non-constant variances of the ui
100
Typical situations of heteroskedasticity:
• Cross-section data sets covering household and regional data
• Data with measurement errors following a trend
• Financial market data (exchange-rate, asset-price returns)
Example: [I]
• Rents for (business) real-estate in distinct town quarters
101
Example: [II]
• Variables:
yi = rent for real-estate in quarter i (EUR/m2)
xi = distance to city center (in km)
• Single-regressor model:
yi = α + β · xi + ui,
RENT
16.8
16.2
15.9
15.4
16.4
13.2
12.8
12.2
15.0
13.6
14.1
13.3
i = 1, . . . , 12
DISTANCE
05
14
11
22
13
32
31
44
37
30
35
41
102
17
16
Rent
15
14
13
12
0
1
2
3
4
5
Distance
Dependent Variable: RENT
Method: Least Squares
Date: 11/06/04 Time: 14:25
Sample: 1 12
Included observations: 12
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DISTANCE
17.39303
-1.073535
0.527137
0.181780
32.99528
-5.905673
0.0000
0.0001
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.777169
0.754885
0.775981
6.021462
-12.88980
2.183344
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
14.57500
1.567352
2.481634
2.562452
34.87697
0.000150
Example: [III]
• Residual variation increases with the regressor
−→ indication of heteroskedasticity
2
Residuals
1
0
-1
-2
0
1
2
3
4
5
Distance
104
Issues:
• Consequences of heteroskedasticity
• Diagnostics
(Tests for heteroskedasticity)
• Estimation and hypothesis-testing in the presence of heteroskedasticity
(weighted OLS estimation, Aitken-estimator)
105
6.1 Consequences
Homoskedasticity vs. heteroskedasticity: [I]
• Consider the linear regression model
y = Xβ + u
• Homoskedasticity means that



2
Cov(u) = σ IN = 


(validity of Assumption #B3)
σ2
0
...
0
0
σ2
...
0
··· 0
··· 0
· · · ...
· · · σ2






106
Homoskedasticity vs. heteroskedasticity: [II]
• Heteroskedasticity means that

σ12 0 · · ·

 0 σ2 · · ·
Cov(u) = 
...2 · · ·
 ...

0
where
Ω
0

0
0
...
2
· · · σN
σ12/σ 2
0
···

2/σ 2 · · ·

0
σ
2
=
..
...

.
···

0
0



 ≡ σ 2Ω


0
0
...
2 /σ 2
· · · σN






107
Example:
• For the real-estate data set we could assume
σi2 = σ 2xi,
that is



Ω=

x1 0
0 x2
...
...
0 0
··· 0
··· 0
· · · ...
· · · xN





−→ Ω can be determined directly from the data
108
Now:
• Central result
(proof and derivations are given below)
Theorem 6.1: (Consequences of heteroskedasticity)
In the presence of heteroskedasticity the OLS estimator
b = (X0X)−1X0y
β
b is no longer BLUE,
is unbiased. However, the OLS estimator β
i.e., there is another linear and unbiased estimator of β with a
”smaller” covariance matrix.
109
Proof of unbiasedness:
• We have
b = (X0X)−1X0y
β
= (X0X)−1X0 (Xβ + u)
= β + (X0X)−1X0u
• It follows that

b = β + (X0X)−1X0E (u)
E β

b =β
• From Assumption #B1 we have E (u) = 0N ×1 and E β
(Assumption #B2 is not needed)
110
Now:
• Construction of a linear estimator of β that, in the presence
of heteroskedasticity, is (1) unbiased and (2) more efficient
b = (X0X)−1X0y
than the OLS estimator β
Procedure: [I]
• We transform the heteroskedastic model y = Xβ + u such
that
the parameter vector β remaines unchanged
heteroskedasticity vanishes
the transformed model y∗ = X∗β + u∗ satisfies all #A-,
#B-, #C-assumptions
111
Procedure: [II]
• New estimator of β :
OLS estimator of the transformed model
h
i
GLS
0 ∗ −1 ∗ 0 ∗
∗
b
= X X
β
X y
Example: [I]
• Consider the single-regressor model
where
yi = α + β · xi + ui
(i = 1, . . . , N )
Var(ui) = σi2 = σ 2xi
(cf. our real-estate example)
112
Example: [II]
• Transformation:
• Define
α
xi
ui
yi
√ = √ +β·√ +√
xi
xi
xi
xi
yi
∗
yi ≡ √ ,
xi
1
∗
zi ≡ √ ,
xi
xi
∗
xi ≡ √ ,
xi
ui
∗
ui ≡ √
xi
−→ transformed model:
yi∗ = α · zi∗ + β · x∗i + ui∗
(multiple regression without intercept)
113
Example: [III]
• We have
E(u∗i ) =
Var(u∗i ) =
1
√ E(ui) = 0
xi
1
√
xi
!2
Var(ui) =
1 2
σ xi = σ 2
xi
−→ model is homoskedastic
114
Summary:
• Transformed model is homoskedastic
• Transformed model satisfies all #A-, #B-, #C-assumptions
• yi∗, zi∗, xi∗ can be obtained from the original (xi, yi)-values
−→ OLS estimator of the transformed model is obtainable
115
Generalization: [I]
• Consider the heteroskedastic model
y = Xβ + u
where
(cf. Slide 107)
i
0
Cov(u) = E uu = σ 2Ω
h
• All elements of the diagonal matrix
Ω

σ12/σ 2
···
0


σ22/σ 2 · · ·
0

=
...
...
···

0
are positive
0
0
0
...
2 /σ 2
· · · σN






116
Generalization: [II]
−→ Ω is a positively definite matrix
−→ there is at least one regular (N × N ) matrix P such that
P0P = Ω−1
(vgl. Econometrics I, Slide 49)
117
Generalization: [III]
• We transform the heteroskedastic model
y = Xβ + u
via the matrix P into
Py = PXβ + Pu
• Using the notation
y∗ ≡ Py,
X∗ ≡ PX,
u∗ ≡ Pu,
we obtain
y ∗ = X∗ β + u ∗
118
Generalization: [IV]
• For the vector u∗ of the transformed model we have

E u∗

= E (Pu) = PE (u) = 0N ×1
n
o
∗
0
= E [Pu − E(Pu)] [Pu − E(Pu)]
Cov u
h
i
0
0
= E Puu P )
h
i
0
= PE uu P0
= σ 2PΩP0 = σ 2IN
−→ transformed model is homoskedastic
119
Remark:
• The validity of
PΩP0 = IN
follows from the equation
P0P = Ω−1
after left-hand and right-hand-side multiplication with PΩ
and P−1:
PΩP0PP−1 = PΩΩ−1P−1
120
Summary:
• There always exists a transformation matrix P that transforms the heteroskedastic model
y = Xβ + u
with Cov(u) = σ 2Ω
into the homoskedastic model
y∗ = X∗β + u∗
with Cov(u∗) = σ 2IN
• The homoskedastic model satisfies all #A-, #B-, #C-assumptions
• β remains unaffected by the transformation
121
Now:
• Potential estimator of β : (cf. Slide 112)
OLS estimator of the transformed model
b GLS
β
=
=
=
=
h
i
0 ∗ −1 ∗ 0 ∗
∗
X X
X y
i−1
0
(PX)0Py
(PX) PX
h
h
i−1
0
0
X P PX
X0P0Py
h
i−1
−1
0
X0Ω−1y
XΩ X
122
Definition 6.2: (Generalized Least Squares estimator)
The OLS estimator of β obtained from the transformed homoskedastic model
b GLS
β
i−1
0
−1
X0Ω−1y
= XΩ X
h
is called the Generalized Least Squares (GLS) estimator (also:
Aitken-estimator).
Theorem 6.3: (Properties of the GLS estimator)
The GLS estimator
h
i−1
GLS
0
−1
b
β
= XΩ X
X0Ω−1y
is linear and unbiased. Its covariance matrix is given by

b GLS
Cov β
(Proof: see class)

h
i−1
2
0
−1
=σ XΩ X
.
123
Remark:
• Under homoskedasticity it follows that
Ω = IN
and thus
b GLS =
β
=
=
h
i−1
0
−1
X0Ω−1y
XΩ X
i−1
0
X0 I N y
X IN X
h
h
i−1
0
XX
X0y
b
= β
(GLS and OLS estimators coincide)
124
Obviously:
• Both, the GLS estimator
h
i−1
GLS
0
−1
b
= XΩ X
X0Ω−1y
β
and the OLS estimator
i−1
h
0
b
X0 y
β= XX
are linear and unbiased estimators
Question:
• Which estimator is more efficient?
125
Answer:
• The transformed model
y∗ = X∗β + u∗
satisfies all #A-, #B-, #C-assumptions
−→ The GLS estimator
h
i−1
GLS
0
−1
b
β
= XΩ X
X0Ω−1y
is BLUE of β
(Gauß-Markov-Theorem)
−→ The OLS estimator
cannot be efficient
h
i−1
0
b
β= XX
X0y
126
Question:
• What are the consequences of erroneously using the OLS
estimator and its associated formulae for the standard errors
in the presence of heteroskedasticity?
Answer:
b ) under heteroskedasticity (the true situa• We compare Cov(β
b ) as computed under homoskedasticity (the
tion) with Cov(β
untrue situation)
127
Comparison: [I]
• Under heteroskedasticity we have
h
i0

i h
b −E β
b
b
b
b −E β
= E β
β
Cov β
= E
= E
= E
=
h
ih
b −β β
b −β
β
(

−1
X0 X

−1
X0 X
X0 u
i0

X0X

−1
X0uu0X X0X
0)
X0 u
−1
−1
−1
h
i
0
0
0
0
XX
X E uu X X X

−1

0
0
0
2
= σ XX
X ΩX X X
128
Comparison: [II]
• If we neglect heteroskedasticity we have

−1
2
0
b
Cov β = σ X X
Similarly, in the estimation of σ 2 we have:
• Under heteroskedasticity:
σ
ˆ2 =
b
b 0 Pu
b∗
b ∗0u
(Pu)
u
=
N −K−1
N −K−1
• Under the neglection of heteroskedasticity:
b 0u
b
u
2
σ
ˆ =
N −K−1
129
Summary:
• Under the neglection of heteroskedasticity, estimation of β

ˆ = X0X −1 X0y is unbiased, but invia the OLS estimator β
efficient
• Under the neglection of heteroskedasticity, the ordinary estimators of
b)
the covariance matrix Cov(β
the variance σ 2 of the error term
are biased
−→ statistics of t- and F -tests are based on biased estimators
−→ t- and F -tests are very likely to be misleading
130
6.2 Diagnostics
Now:
• Statistical tests for heteroskedasticity
Basic structure of all tests:
• H0: Homoskedasticity
vs.
H1: Heteroskedasticity
131
Consequence:
• Non-rejection of H0
−→ OLS results are ”unsuspicious”
• Rejection of H0
−→ problematic OLS results
(cf. Section 6.1)
−→ application of alternative estimation procedures
(cf. Section 6.3)
132
Problem of all tests:
• Tests are based on different patterns of heteroskedasticity
(e.g. σi2 = σ 2xki, σi2 = σ 2x2
ki etc.)
−→ alternative tests for heteroskedasticity
1. The Goldfeld-Quandt test (special case)
Assumed pattern of heteroskedasticity:
• Variances of the ui are split into two groups:
2
σi2 = σA
2
σi2 = σB
for all i belonging to group A (i ∈ A)
for all i belonging to group B (i ∈ B)
133
Hypothesis test:
2 = σ2
• H0 : σA
B
versus
2 =
2
6 σB
H1 : σA
Test statistic: [I]
• Notation
NA is the number of observations in group A
NB is the number of observations in group B
P
A
ˆi2 is the sum of squared residuals in group A
Suˆuˆ = i∈A u
P
B
Suˆuˆ = i∈B u
ˆ2
i is the sum of squared residuals in group B
134
Test statistic: [II]
• Under the Assumption #B4 it follows that
1
2
SuˆAuˆ/σA
NA − K − 1
∼ FNA−K−1,NB −K−1
1
2
SuˆBuˆ/σB
NB − K − 1
2 = σ 2 we have
• Under H0 : σA
B
T =
SuˆAuˆ/(NA − K − 1)
SuˆBuˆ/(NB − K − 1)
∼ FNA−K−1,NB −K−1
−→ Reject H0 at the significance level α if
T ∈ [0, FNA−K−1,NB −K−1;α/2]∪[FNA−K−1,NB −K−1;1−α/2, +∞]
135
Remarks: [I]
• We can also test the one-sided alternative
2 > σ2
H1 : σA
B
via the statistic T
• The critical region of this test at the α-level is given by
[FNA−K−1,NB −K−1;1−α, +∞]
• We test the reverse alternative
2 < σ2
H1 : σA
B
by interchanging the role of the groups A and B
136
Remarks: [II]
• The general Goldfeld-Quandt test can be used to test wether
the σi2-values depend in a monotone way on a single exogenous variable xki
(cf. Gujarati, 2003)
137
2. The Breusch-Pagan test
Assumed pattern of heteroskedasticity: [I]
• Consider J ≤ K exogenous variables z1, . . . , zJ and J coefficients α1, . . . , αJ
• For i = 1, . . . , N consider the transformation
h(α1 · z1i + α2 · z2i + . . . + αJ · zJi)
with h : R → R+ satisfying the following properties:
h is continuously differentiable
h(0) = 1
138
Assumed pattern of heteroskedasticity: [II]
• We assume that the variances of the ui are given by
σi2 = σ 2 · h(α1 · z1i + α2 · z2i + . . . + αJ · zJi)
Example:
• For h : R → R+ mit h(x) = exp(x) we have
σi2 = σ 2 · exp(α1 · z1i + α2 · z2i + . . . + αJ · zJi)
(multiplicative heteroskedasticity)
139
Heteroskedasticity test:
• Defining α ≡ [α1 . . . αJ ]0, the testing problem is
H0 : α = 0J×1
versus
H1 : α =
6 0J×1
Test statistic: [I]
• There is a test that does not depend on the function h
(Breusch-Pagan test)
140
Test statistic: [II]
• Derivation (in its simplest form):
Estimate the model
y = Xβ + u
by OLS
Compute the residuals
b
b =y−u
b = y − Xβ
u
Estimate the model
∗
ˆ2
(i = 1, . . . , N )
u
i = α0 + α1 · z1i + . . . + αJ · zJi + ui
and find the coefficient of determination R2
141
Test statistic: [III]
−→ Test statistic:
T = N R2
We have
T
∼
asmp.
χ2
J
(chisquare distribution with J degrees-of-freedom)
• Under H0 : α = 0J×1 the impact of z1, . . . , zJ on u
ˆ2
i should
be equal to zero
−→ decision rule:
Reject H0 at the α-level if
2
T = N R2 > χJ;1−α
142
Remark:
• The Breusch-Pagan test is a Lagrange-Multiplier test
(cf. the lecture ’Advanced Statistics’)
143
3. The White test
Special feature of previous tests:
• Explicit structural form of heteroskedasticity
White test:
• Allows for entirely unknown patterns of heteroskedasticity
• Best-known heteroskedasticity test
• Theoretical foundation:
Eicker (1967)
White (1980)
144
Preliminaries: [I]
• Covariance matrix of OLS estimator under heteroskedasticity

−1
−1

0
0
2
0
b
X ΩX X X
Cov β = σ X X
(cf. Slide 128)
• Question:

b without any strucCan we consistently estimate Cov β
tural assumption on the σi2?
(i.e. in the presence of heteroskedasticity of unknown form)
• Answer:
Yes, see White (1980)
145
Preliminaries: [II]
• Consider the following partitioning of the X matrix:

where


X=

1 x11 x21
1 x12 x22
...
...
...
1 x1N x2N
xi0 =
h
· · · xK1
· · · xK2
...
···
· · · xKN






0
x1
 x0

=  ..2
 .
x0N
1 x1i x2i · · · xKi





i
• Estimate the model
y = Xβ + u
by OLS
146
Preliminaries: [III]
• Compute the residuals
b
b = y − Xβ
b =y−y
u

b under heteroskedasticity
• A consistent estimator of Cov β
of unknown form is given by
N
−1 X

−1
0
0
2
0
b = XX
d β
Cov
u
ˆi xixi X X
i=1
147
Definition 6.4: (Heteroskedasticity-robust standard errors)
The standard errors of the OLS estimators

−1
0
b
β= XX
X0y,
which are given by the square root of the diagonal elements of
the estimated covariance matrix
N

−1 X
−1

0
2
0
0
b
d β = XX
Cov
ˆi xixi X X
u
,
i=1
are called heteroskedasticity-robust standard errors or White standard errors.
148
Remarks:
• White standard errors are available in EViews
• White standard errors should be reported in empirical studies
149
Dependent Variable: RENT
Method: Least Squares
Date: 11/06/04 Time: 14:25
Sample: 1 12
Included observations: 12
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DISTANCE
17.39303
-1.073535
0.527137
0.181780
32.99528
-5.905673
0.0000
0.0001
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.777169
0.754885
0.775981
6.021462
-12.88980
2.183344
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
14.57500
1.567352
2.481634
2.562452
34.87697
0.000150
Dependent Variable: RENT
Method: Least Squares
Date: 11/12/04 Time: 12:23
Sample: 1 12
Included observations: 12
White Heteroskedasticity-Consistent Standard Errors & Covariance
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DISTANCE
17.39303
-1.073535
0.251847
0.137597
69.06182
-7.802039
0.0000
0.0000
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.777169
0.754885
0.775981
6.021462
-12.88980
2.183344
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
14.57500
1.567352
2.481634
2.562452
34.87697
0.000150
Now:
• White test for heteroskedasticity of unknown form
Basis of the test: [I]
• Comparison of the estimated
covariance matrices of the OLS

b = X0X −1 X0y under
estimator β
homoskedasticity:

−1
0
2
b
d β =σ
ˆ XX
,
Cov
heteroskedasticity:
b 0u
b
u
2
where σ
ˆ =
N −K −1
N

−1

−1 X

2
0
0
0
b
d β = XX
u
ˆi xixi X X
Cov
i=1
(the estimated White covariance matrix)
151
Basis of the test: [III]
• Under homoskedasticity (H0) both estimators should not differ substantially
−→ test statistic of the White test
152
Test statistic of the White test: [I]
• Estimate the model
y = Xβ + u
by OLS and compute the residuals
b
b = y − Xβ
b =y−y
u
ˆ2
• Use the squared residuals u
i , the exogenous variables x1i, . . . , xKi ,
2
their squared values x2
1i , . . . , xKi and all cross products xki xli (k =
1, . . . , K, l = 1, . . . , K, k 6= l) to specify the model
u
ˆ2
i = γ0 + γ1x1i + . . . + γK xKi +
K X
K
X
δkl xkixli + u∗i
k=1 l=1
153
Test statistic of the White test: [II]
• Estimated the model by OLS and find the coefficient of determination R2
• Under H0 we have
T = N R2
∼
asmp.
χ2
K(K+1)
• Reject H0 at the significance level α if
T > χ2
K(K+1);1−α
154
Example:
• Test the data set on Slide 102 for heteroskedasticity via
the Goldfeld-Quandt test
the Breusch-Pagan test
the White test
(see Class)
Interesting question:
• Which test should be preferred?
155
Remarks:
• The White test
is the most general test
often has low power
• Whenever we realistically conjecture an explicit pattern of
2 ), we should use the alterheteroskedasticty (e.g. σi2 = σ 2xki
native tests
(Goldfeld-Quandt test, Breusch-Pagan test)
−→ use graphical tools to analyze residuals in order to detect
potential patterns of heteroskedasticity
156
6.3 Feasible Estimation Procedures
Result from Section 6.1:
• For the heteroskedastic model
y = Xβ + u,
where
i
0
Cov(u) = E uu = σ 2Ω,
the GLS estimator
is BLUE of β
(vgl. Folie 126)
h
i−1
h
GLS
0
−1
b
β
= XΩ X
X0Ω−1y
157
Problem:
• Frequently, the diagonal matrix Ω is not known
b GLS cannot be computed
−→ GLS estimator β
Resort:
• Replace the unknown (true) Ω by an unbiased and/or conb
sistent estimate Ω
−→ feasible GLS estimator (FGLS)
158
Definition 6.5: (FGLS estimator)
b be an unbiased and/or consistent estimator of the unLet Ω
known covariance matrix Ω of the heteroskedastic model
y = Xβ + u,
where
Cov(u) = σ 2Ω.
The estimator
b FGLS
β

0 b −1
= XΩ
X
−1
b
X0 Ω
−1
y
is called feasible generalized-least-squares estimator of β .
159
Example: [I]
• Consider the data set on Slide 102
• Classify the ui variances for the central (i = 1, . . . , 5) and the
periphery quarters (i = 6, . . . , 12)
• Consider the following model:
yi = α + β · xi + ui,
where
2
σi2 = σA
for i = 1, . . . , 5
2
σi2 = σB
for i = 6, . . . , 12
160
Example: [II]
• Transformation of the model:
yi
1 + β xi + ui
=
α
σA
σA
σA
σA
for i = 1, . . . , 5
yi
1 + β xi + ui
=
α
σB
σB
σB
σB
for i = 6, . . . , 12
−→ variances of the error terms

ui
2 =1
Var σ
= 12 Var(ui) = 12 σA
A
σ
σ
A
ui
= 12 Var(ui) =
Var σ
B
σB

for i = 1, . . . , 5
A
1 σ2 = 1
2 B
σB
for i = 6, . . . , 12
161
Example: [III]
• Summary:
yi∗ = α · zi∗ + β · x∗i + ui∗
for i = 1, . . . , 12
where
y
x
u
yi∗ = σ i , zi∗ = σ1 , x∗i = σ i , u∗i = σ i
A
A
A
A
for i = 1, . . . , 5
y
x
u
yi∗ = σ i , zi∗ = σ1 , x∗i = σ i , u∗i = σ i
B
B
B
B
for i = 6, . . . , 12
162
Example: [IV]
2 , σ 2 are unknown
• Variances σA
B
−→ estimation of the variances via the respective regressions
yi = α + β · xi + ui
for i = 1, . . . 5 and i = 6, . . . 12 with the respective estimators
b 0u
b
u
2
ˆ =
σ
N −K −1
163
Dependent Variable: RENT
Method: Least Squares
Date: 11/15/04 Time: 10:41
Sample: 1 5
Included observations: 5
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DISTANCE
17.12800
-0.760000
0.329560
0.233619
51.97233
-3.253162
0.0000
0.0474
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.779137
0.705516
0.286124
0.245600
0.439030
2.030098
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
16.14000
0.527257
0.624388
0.468163
10.58306
0.047376
Dependent Variable: RENT
Method: Least Squares
Date: 11/15/04 Time: 10:41
Sample: 6 12
Included observations: 7
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
DISTANCE
14.84061
-0.387372
2.691190
0.746566
5.514518
-0.518872
0.0027
0.6260
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.051094
-0.138687
0.966013
4.665904
-8.512870
2.184558
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
13.45714
0.905276
3.003677
2.988223
0.269228
0.625989
Example: [IV]
• Estimated variances and standard deviations
√
0.245600
2
ˆA =
= 0.081867, σ
0.081867 = 0.286124
σ
ˆA =
3
√
2 = 4.665904 = 0.933181, σ
0.933181 = 0.966013
σ
ˆB
ˆA =
5
−→ (estimated) transformation of the model
(cf. Slides 161, 162)
• FGLS estimates of the coefficients:
ˆFGLS = 17.41618,
α
βˆFGLS = −1.013681
• Estimate of the error-term variance of the transformed model:
10.84938
∗
d
Var(ui ) =
= 1.084938
12 − 1 − 1
165
Dependent Variable: RENT_TR
Method: Least Squares
Date: 11/15/04 Time: 10:51
Sample: 1 12
Included observations: 12
Variable
Z_TR
DISTANCE
_TR
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Coefficient
Std. Error
t-Statistic
Prob.
17.41618
-1.013681
0.251950
0.140983
69.12558
-7.190101
0.0000
0.0000
0.997946
0.997740
1.041604
10.84938
-16.42247
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
Durbin-Watson stat
31.62998
21.91254
3.070412
3.151230
2.397311
Example: [V]
• However, we know that Var(u∗i ) = 1
ˆFGLS, βˆFGLS
−→ Corrected standard errors of α
(via covariance matrix σ 2(X0X)−1 of the OLS estimator)
SE(ˆ
αFGLS) = 0.243,
SE(βˆFGLS) = 0.135
−→ corrected t-values of α
ˆFGLS, βˆFGLS:
ˆFGLS : 71.672,
t-value of α
t-value of βˆFGLS := −7.509
167
Properties of FGLS estimators:
• FGLS estimators are unbiased
• Variances of FGLS estimators are lower than the variances
of the OLS estimators
• FGLS estimators are asymptotically efficient
(approximation to the GLS estimator for N → ∞)
168

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