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

consistency,
which is dis-
cussed more fully in 
Appendix A.
For the two-variable model we have already shown that the OLS
estimator 
ˆ
β
2
is an unbiased estimator of the true 
β
2
. Now we show that 
ˆ
β
2
is also a consistent esti-
mator of 
β
2
. As shown in 
Appendix A,
a sufficient condition for consistency is that 
ˆ
β
2
is unbiased
and that its variance tends to zero as the sample size 
n
tends to infinity.
Since we have already proved the unbiasedness property, we need only show that the variance of
ˆ
β
2
tends to zero as 
n
increases indefinitely. We know that
var (
ˆ
β
2
)
=
σ
2
x
2
i
=
σ
2
/
n
x
2
i
/
n
(26)
By dividing the numerator and denominator by 
n
, we do not change the equality.
Now
lim var (
ˆ
β
2
)
=
lim
σ
2
/
n
x
2
i
/
n
=
0
(27)
n
→ ∞
n
→ ∞
where use is made of the facts that (1) the limit of a ratio quantity is the limit of the quantity in the
numerator to the limit of the quantity in the denominator (refer to any calculus book); (2) as 
n
tends
to infinity, 
σ
2
/
n
tends to zero because 
σ
2
is a finite number; and [(
x
2
i
)
/
n
]
=
0 because the vari-
ance of 
X
has a finite limit because of Assumption 7 of CLRM.
The upshot of the preceding discussion is that the OLS estimator 
ˆ
β
2
is a consistent estimator of
true 
β
2
. In like fashion, we can establish that 
ˆ
β
1
is also a consistent estimator. Thus, in repeated
(small) samples, the OLS estimators are unbiased and as the sample size increases indefinitely the
OLS estimators are consistent. As we shall see later, even if some of the assumptions of CLRM are
not satisfied, we may be able to obtain consistent estimators of the regression coefficients in several
situations.
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97
Chapter
4
Classical Normal 
Linear Regression
Model (CNLRM)
What is known as the