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Stock and Watson Chapter 4:
Linear Regression with One Regressor
The linear regression model is:
Y
i
= β
0
+ β
1
X
i
+
u
i
The subscript
i
runs over observations,
i
= 1,…,
n
;
Y
i
is the dependent variable
(or the regressand or the response variable or the lefthandside variable)
X
i
is the independent variable
(or the regressor or the explanatory variable or the righthandside variable)
β
0
+ β
1
X
i
is the population regression line (or population regression function)
β
0
is the intercept of the population regression line
β
1
is the slope of the population regression line
u
i
is the error term (the error term contains all the other factors besides
X
the determine
the value of the dependent variable,
Y
, for a specific observation
i
)
In practice, we don’t know the intercept β
0
and the slope β
1
.
We use the sample
regression line to estimate the population regression line.
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View Full Document Table 4.1
Summary of the Distribution of StudentTeacher Ratios and FifthGrade
Test Scores for 420 K8 Districts in California in 1998
x
s
10%
25%
40%
50%
60%
75%
90%
STR
19.6
1.9
17.3
18.6
19.3
19.7
20.1
20.9
21.9
TS
665.2
19.1
630.4
640.0
649.1
654.5
659.4
666.7
679.1
4
Mechanics of OLS
The population regression line:
Test Score
=
β
0
+
1
STR
1
=
Tes
S
∆
∆
= ??
Ordinary Least Squares (OLS) Estimator
OLS regression line minimizes:
•the sum of the squares of the vertical distances between the data points and the regression line
•the sum of the squared mistakes made in predicting
Y
given
X
let b
0
and b
1
be some estimators of β
0
+ β
1
the value of
Y
i
predicted using this line is b
0
+ b
1
X
i
2
The OLS estimators of the slope β
1
and the intercept β
0
are:
1
ˆ
β
=
∑
∑



2
)
(
)
)(
(
X
X
Y
Y
X
X
i
i
i
0
ˆ
=
Y

1
ˆ
X
The OLS predicted values
i
Y
ˆ
and residuals
i
u
ˆ
are
i
Y
ˆ
=
0
ˆ
+
1
ˆ
X
i
,
i
= 1,…,
n
i
u
ˆ
=
Y
i

i
Y
ˆ
,
i
= 1,…,
n
The estimated slope intercept (
0
ˆ
), slope (
1
ˆ
), and residual (
i
u
ˆ
) are computed from a
sample of
n
1
), and error term (
u
i
).
3
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This note was uploaded on 10/17/2010 for the course PAM 2100 taught by Professor Lewis during the Spring '10 term at Adelphi.
 Spring '10
 lewis

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