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TA session 6
Econ. 103, winter 2010
Wed., Feb. 10, 2010, 10:00 a.m. and 1:00 p.m. in PP2400E.
2 Verbose discussion of certain true/false
2.1 In a singlevariable OLS regression, if
ˆ
β
= 0
then
R
2
= 0
This is
true
actually, and it’s easy to see why intuitively. If the coeﬃcient
ˆ
β
equals zero
then
ˆ
β
·
x
equals zero, and the data explain nothing about the dependent variable.
Let’s show it mathematically. If
β
= 0.
..
ˆ
y
= ˆ
c
+ 0
The constant
c
is the only parameter left that OLS may use to minimize the SSE. The only
c
that minimizes the SSE is
c
=
Y
. (Think about it.) So
ˆ
y
=
Y
⇒
RSS
=
X
(
y

ˆ
y
)
2
=
X
(
y

y
)
2
that means
RSS
=
TSS
=
X
(
y

y
)
2
also
Since
R
2
= 1

RSS
TSS
and
RSS
=
TSS
, then
R
2
= 1

1 = 0.
Let’s have a little discussion about the meaning of
R
2
. The purpose of a statistical model
is to explain the variation in a dependent variable (
y
i
) using several explanatory variables
(the
x
’s). The variation in the sample, before using any model to explain it, is
TSS
=
X
(
y

y
)
2
Then
TSS
is just the sum of the errors of the sample average. One way to judge an estimator
is by the amount of variation in the sample it leaves. So, try to think of
y
as the “baseline”
or “default” statistical model.
If none of the independent variables
x
explain anything about the data, the worst you can
do is regress
y
on a constant
c
. But if a statistical model succeeds in explaining something
about the data, it should be able to form better predictions than the sample average. That
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This note was uploaded on 03/15/2010 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.
 Winter '07
 SandraBlack
 Econometrics

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