1
Exceptions to Ordinary Least Squares
Lecture XXXI
I.
Heteroscedasticity
A.
Using the derivation of the variance of ordinary least squares estimator
1
1
1
1
1
ˆ
ˆ
ˆ
:
X X
X
V
X X
X
X
X X
V
X X
X SX
X X
S
E
under the GaussMarkov assumptions
2
T T
S
E
I
.
B.
However, if we assume that
2
T T
S
E
I
the ordinary least squares
estimator is still unbiased, but is no longer efficient. In this case, we use the
generalized least squares estimator
1
X AX
X Ay
1.
The variance of this estimator is then
1
1
1
1
1
1
1
1
X AX
X AX
X A
X AX
X AX
X AX
X A
X AX
X A
V
X AX
X A
A X
X AX
X AX
X ASA X
X AX
2.
Setting
1
A
S
1
1
1
:
V
X AX
X A X
X AX
A
A
X AX
C.
Seemingly Unrelated Regressions
1.
One of the uses of generalized least squares is the estimation of
simultaneous systems of equations without endogeneity.
a)
Derived input demand equations derived from cost minimization
implies relationship between the parameters
1
1
11
1
12
2
11
1
2
2
21
1
22
2
21
2
x
A w
A w
y
x
A w
A w
y
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AEB 6933
–
Mathematical Statistics for Food and Resource Economics
Lecture XXXI
Professor Charles Moss
Fall 2007
2
where
1
x
and
2
x
are input levels,
1
w
and
2
w
are the respective
input prices,
y
is the level of output, and
1
,
2
,
11
A
,
12
A
,
21
A
,
22
A
,
11
and
21
are estimated parameters.
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 Fall '09
 CARRIKER
 Least Squares, Regression Analysis, Professor Charles Moss, Economics Professor Charles, Resource Economics Professor, Lecture XXXI Fall

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