ECON 206
January 03, 2011
METU- Department of Economics
Instructor: H. Ozan ERUYGUR
e-mail: [email protected]
1
LECTURE 11
SIMPLE REGRESSION MODEL - III
Outline of today’s lecture:
I. Variance of the random variable
u
.....................................................................................
1
II. A Measure of Goodness of Fit: Coefficient of Determination (
2
R
)
...............................
2
A. Relationship between
2
R
and Correlation Coefficient
...............................................
7
III. Sampling Distribution of the Least Squares Estimates
....................................................
9
IV. Multiple Regression Analysis
........................................................................................
10
V. Two Explanatory Variable Regression
...........................................................................
11
VI. Testing Hypotheses
........................................................................................................
12
A. Testing Individual Coefficients
...................................................................................
12
One-Tailed Test
..........................................................................................................
12
Two-Tailed Test
...........................................................................................................
13
B. Testing Several Coefficients Jointly
...........................................................................
14
VII. Confidence Intervals for
β
k
...........................................................................................
16
I. Variance of the random variable
u
The formulae of the variance of
0
β
±
and
1
β
±
involve the variance of
the random term
u
, which we have denoted by
2
σ
.
However, the true variance of
t
u
can not be computed since the
values of
t
u
are not observable. But we may obtain an
unbiased
estimate of
2
σ
from the expression
2
2
2
1
2
T
t
t
u
S
T
σ
=
=
=
−
∑
±
±
where
t
u
±
is the OLS residual, and hence it is given by
t
t
t
u
Y
Y
=
−
±
±
.

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ECON 206
January 03, 2011
METU- Department of Economics
Instructor: H. Ozan ERUYGUR
e-mail: [email protected]
2
II. A Measure of Goodness of Fit: Coefficient of
Determination (
2
R
)
We now consider the goodness of fit of the fitted regression line to a
set of data; that is, we shall find out how “well” the sample
regression line fits the data.
The coefficient of determination
2
r
(two-variable case) or
2
R
(multiple regression) is a summary measure that tells how well the
sample regression line fits the data.
We need to know how “good” is the fit of the regression line to the
sample observations of Y and X, that is to say we need to measure
the dispersion around the regression line.
This knowledge is essential, because the closer the observations to
the line, the better the goodness of fit, that is better is the
explanation of the variations of
Y
by the changes in the explanatory
variables.
Below, we will show prove that a measure of the goodness of fit is
the square of the correlation coefficient,
2
r
, which shows the
percentage of the total variation of the dependent variable that can
be explained by the independent variable X.