1
Distribution of Estimates and
Multivariate Regression
Lecture XXVII
I.
Models and Distributional Assumptions
A.
Conditional Normal Model
1.
The conditional normal model assumes that the observed random
variables are distributed
2
~
,
i
i
y
N
x
Thus,
i
i
i
E
y x
x
and the variance of
i
y
equals
2
. The conditional normal can be
expressed as
2
~
0,
i
i
i
i
y
x
N
Further, the
i
are independently and identically distributed
(consistent with our BLUE proof).
2.
Given this formulation, the likelihood function for the simple
linear model can be written:
2
2
2
1
1
, ,
exp
2
2
n
i
i
i
y
x
L
x
Taking the log of this likelihood function yields:
2
2
2
1
1
ln
ln 2
ln
2
2
2
n
i
i
i
n
n
L
y
x
As discussed in Lecture XVII, this likelihood function can be
concentrated in such a way so that
2
2
2
1
ˆ
ln
ln
2
2
1
ˆ
n
i
i
i
n
n
L
y
x
n
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AEB 6933
–
Mathematical Statistics for Food and Resource Economics
Lecture XXIX
Professor Charles Moss
Fall 2007
2
So that the least squares estimator are also maximum likelihood
estimators if the error terms are normal.
3.
Proof of the variance of
can be derived from the GaussMarkov
results.
a)
Note from last lecture:
1
1
1
1
1
ˆ
n
n
i
i
i
i
i
i
i
xx
n
n
n
i
i
i
i
i
i
i
i
x
x
d y
x
S
d
d
x
d
(1)
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 Fall '09
 CARRIKER
 Normal Distribution, xx xx xx, Professor Charles Moss

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