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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 lefthandside variable)
X
i
is the independent variable (or the regressor 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)
Figure 4.1
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.
Table 4.1 and Figure 4.2
districtwide average test scores for 5
th
graders 420 California districts (1999)
districtwide studentteacher ratio
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
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The OLS estimators of the slope β
0
and the intercept β
1
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 observations of and , i= 1,…,n.
These are estimates of the unknown true
population intercept (β
0
), slope (β
1
), and error term (u
i
).
2
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 Spring '08
 ABDUS,S.

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