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1 September 2004
The Classical Linear Regression Model
A.
A brief review of some basic concepts associated with vector random variables
12
n
Let y denote an n x l vector of random variables, i.e., y = (y , y , . . ., y )'.
1.
The expected value of y is defined by
This is sometimes denoted by the vector
:
, so that
:
= E(y).
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2.
The variance of the vector y can be defined as follows:
3
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3.
The n x l vector of random variables, y, is said to be distributed as a multivariate normal with mean
vector
:
and variance covariance matrix
3
(denoted y

N(
:
,
3
)) if the density of y is given by
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Consider the special case where n = 1:
y = y ,
:
=
:
,
3
=
F
.
2
is just the normal density for a single random variable.
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B.
The Linear Regression Model
1.
structure
Consider the model defined by
tl
If we want to include an intercept, define x = 1 for all t and obtain
2.
intuition for the model
t1
a.
graphical analysis
–
Consider a linear model of the relationship between y and x where x = 1
2t
and the only other variable is x .
Assume that there are n observations on each variable and that
t
g
is not a constant but a random error that varies with t.
We then have
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nn
The n observations (y , x ), . . ., (y , x ) are graphically depicted as
While a straight line obviously does not go through all the points, one could eyeball an upward
sloping line that would seem to catch the gist of the relationship.
b. population regression line
The linear model postulates that the points y depend on x in a linear fashion, but that actual
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observations will deviate from this linear relationship to random disturbances (errors).
Thus the
population regression line might fit through the scatter of points as follows.
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The true population regression line is unknown because
$
and
$
are unknown parameters.
The observations don't lie on the population line because of the random error term. If
g
were a
1
nonzero constant, it would be like shifting the intercept (
$
) of the line.
It is usually assumed
that the error terms have an expected value of zero or
so that the "average" value of y corresponding to any given x lies on the population regression
line.
c.
sample regression line
A variety of techniques can be used to estimate the unknown parameters (
$
,
$
) and thus
estimate the unknown population regression line.
The most common technique for a linear
model such as this is “ordinary” least squares.
The estimated regression line is referred to as the
sample regression line. It is given by
t
where e is now an estimated disturbance or error term.
The estimated error is also sometimes
tt
denoted
. The residual e is the vertical distance from the y to the sample regression line.
The
sample regression line is not synonymous with the population regression line due to differences
in
$
and
. The residual e is also different from
g
and they may have different properties.
In
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the same way that
are estimators of the
$
's, the sample regression line is an estimator
of the population regression line.
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This note was uploaded on 05/04/2008 for the course ECON 300 taught by Professor Gang during the Spring '06 term at Rutgers.
 Spring '06
 Gang

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