The McGraw-Hill Series Economics essentials of economics brue, McConnell, and Flynn Essentials of Economics

Appendix A.
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70
Part One
Single-Equation Regression Models
where var 
=
variance and se 
=
standard error and where 
σ
2
is the constant or
homoscedastic variance of 
u
i
of Assumption 4.
All the quantities entering into the preceding equations except 
σ
2
can be estimated from
the data. As shown in Appendix 3A, Section 3A.5, 
σ
2
itself is estimated by the following
formula:
ˆ
σ
2
=
ˆ
u
2
i
n
−
2
(3.3.5)
where 
ˆ
σ
2
is the OLS estimator of the true but unknown 
σ
2
and where the expression 
n
−
2
is known as the 
number of degrees of freedom (df),
ˆ
u
2
i
being the sum of the residuals
squared or the 
residual sum of squares (RSS).
16
Once 
ˆ
u
2
i
is known, 
ˆ
σ
2
can be easily computed. 
ˆ
u
2
i
itself can be computed either
from Eq. (3.1.2) or from the following expression (see Section 3.5 for the proof ):
ˆ
u
2
i
=
y
2
i
− ˆ
β
2
2
x
2
i
(3.3.6)
Compared with Eq. (3.1.2), Equation 3.3.6 is easy to use, for it does not require computing
ˆ
u
i
for each observation although such a computation will be useful in its own right (as we
shall see in Chapters 11 and 12).
Since
ˆ
β
2
=
x
i
y
i
x
2
i
an alternative expression for computing 
ˆ
u
2
i
is