Lecture 29-2007 - Distribution of Estimates and...

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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

AEB 6933 – Mathematical Statistics for Food and Resource Economics Lecture XXIX
Professor Charles Moss Fall 2007
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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 Gauss-Markov
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)