Overview — Page 1
Overview
Linear regression model.
Y
X
X
X
X
E Y
X
j
j
j
k
kj
j
i
ij
j
i
k
j
j
1
1
2
2
1
...
(

)
i
partial slope coefficient (also called partial regression coefficient, metric coefficient, or just
regression coefficient).
It represents the change in E(Y) associated with a oneunit increase in X
i
when all other IVs are held constant.
the intercept.
Geometrically, it represents the value of E(Y) where the regression surface (or
plane) crosses the Y axis.
Substantively, it is the expected value of Y when all the IVs equal 0.
the deviation of the value Y
j
from the mean value of the distribution given X.
This error
term may be conceived as representing (1) the effects on Y of variables not explicitly included in
the equation, and (2) a residual random element in the dependent variable.
Parameter estimation.
In most situations, we are not in a position to determine the
population parameters directly.
Instead, we must estimate their values from a finite sample from
the population.
The sample regression model is written as
Y
a
b X
b X
b X
e
a
b X
e
Y
e
j
j
j
k
kj
j
i
ij
j
i
k
j
j
1
1
2
2
1
...
where a is the sample estimate of
and b
k
is the sample estimate of
k
.
Hypothesis testing.
We use the sample to test hypotheses about the population parameters.
Some typical hypotheses:
H
0
:
1
=
0
H
A
:
1
<>
0
One of the Betas = 0.
Use a T test or an F test.
H
0
:
1
=
2
=
3
... =
k
= 0
H
A
:
At least one beta <> 0
Test of whether any Betas differ from 0.
Equivalent to a test of R
2
= 0.
F test
H
0
:
2
=
3
... =
k
= 0
H
A
:
At least one of the above betas <> 0
Test whether a subset of the betas = 0.
Same
as a test of R
2
constrained
= R
2
unconstrained
.
F test
H
0
:
1
=
2
H
A
:
1
<>
2
Test whether two betas are =.
F test, or maybe
a ttest.
H
0
:
(1)
=
(2)
H
A
:
(1)
<>
(2)
Test whether the betas are the same in two
different populations, e.g. is the model for men
the same as the model for women?
F test
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Assumptions and violations of assumptions
Assumptions concerning correct model specification
.
One of the most critical
assumptions you make is that the model is correctly specified, i.e. it really is the case that
Y
X
j
i
ij
j
i
k
1
.
Incorrect model specification can result in biased parameter estimates
and violations of other assumptions.
The following assumptions all follow from the requirement
that the model be correctly specified.
A good theory, and an accurate understanding of what a
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 Spring '11
 RichardWilliams
 Regression Analysis

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