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

sampling or probability distributions
of the estimators are known, one
can make confidence interval statements such as Eq. (5.2.1). In Chapter 4 we saw that
under the assumption of normality of the disturbances 
u
i
the OLS estimators 
ˆ
β
1
and 
ˆ
β
2
are
themselves normally distributed and that the OLS estimator
ˆ
σ
2
is related to the 
χ
2
(chi-
square) distribution. It would then seem that the task of constructing confidence intervals is
a simple one. And it is!
5.3
Confidence Intervals for Regression Coefficients 
β
1 
and 
β
2
Confidence Interval for 
β
2
It was shown in Chapter 4, Section 4.3, that, with the normality assumption for 
u
i
, the OLS
estimators 
ˆ
β
1
and 
ˆ
β
2
are themselves normally distributed with means and variances given
therein. Therefore, for example, the variable
Z
=
ˆ
β
2
−
β
2
se (
ˆ
β
2
)
=
(
ˆ
β
2
−
β
2
)
x
2
i
σ
(5.3.1)
as noted in Eq. (4.3.6), is a standardized normal variable. It therefore seems that we can use
the normal distribution to make probabilistic statements about
β
2
provided the true popula-
tion variance
σ
2
is known. If
σ
2
is known, an important property of a normally distributed
variable with mean
µ
and variance
σ
2
is that the area under the normal curve between
µ
±
σ
is about 68 percent, that between the limits
µ
±
2
σ
is about 95 percent, and that between
µ
±
3
σ
is about 99.7 percent.
But 
σ
2
is rarely known, and in practice it is determined by the unbiased estimator 
ˆ
σ
2
. If
we replace 
σ
by
ˆ
σ
, Equation 5.3.1 may be written as
t
=
ˆ
β
2
−
β
2
se (
ˆ
β
2
)
=
Estimator
−
Parameter
Estimated standard error of estimator
=
(
ˆ
β
2
−
β
2
)
x
2
i
ˆ
σ