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

TABLE 3.4
What Happens If the
Assumptions of
CLRM Are Violated?
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Part One
Single-Equation Regression Models
3.4. Consider the sample regression
Y
i
= ˆ
β
1
+ ˆ
β
2
X
i
+ ˆ
u
i
Imposing the restrictions (i)
ˆ
u
i
=
0 and (ii)
ˆ
u
i
X
i
=
0, obtain the estimators
ˆ
β
1
and
ˆ
β
2
and show that they are identical with the least-squares estimators given in
Eqs. (3.1.6) and (3.1.7). This method of obtaining estimators is called the
analogy
principle.
Give an intuitive justification for imposing restrictions (i) and (ii).
(
Hint:
Recall the CLRM assumptions about
u
i
.
) In passing, note that the analogy prin-
ciple of estimating unknown parameters is also known as the
method of moments
in
which sample moments (e.g., sample mean) are used to estimate population moments
(e.g., the population mean). As noted in
Appendix A
, a
moment
is a summary statis-
tic of a probability distribution, such as the expected value and variance.
3.5. Show that 
r
2
defined in (3.5.5) ranges between 0 and 1. You may use the
Cauchy–Schwarz inequality, which states that for any random variables 
X
and 
Y
the
following relationship holds true:
[
E
(
X Y
)]
2
≤
E
(
X
2
)
E
(
Y
2
)
3.6. Let 
ˆ
β
Y X
and 
ˆ
β
X Y
represent the slopes in the regression of 
Y
on 
X
and 
X
on 
Y
,
respectively. Show that
ˆ
β
Y X
ˆ
β
X Y
=
r
2
where 
r
is the coefficient of correlation between 
X
and 
Y
.
3.7. Suppose in Exercise 3.6 that 
ˆ
β
Y X
ˆ
β
X Y
=
1
.
Does it matter then if we regress 
Y
on 
X
or 
X
on 
Y
? Explain carefully.
3.8. Spearman’s rank correlation coefficient 
r
s
is defined as follows:
r
s
=
1
−
6
d
2
n
(
n
2
−
1)
where 
d
=
difference in the ranks assigned to the same individual or phenomenon
and 
n
= 
number of individuals or phenomena ranked. Derive 
r
s
from 
r
defined in
Eq. (3.5.13). 
Hint:
Rank the 
X
and 
Y
values from 1 to 
n
. Note that the sum of 
X
and
Y
ranks is 
n
(
n
+
1)
/
2 each and therefore their means are (
n
+
1)
/
2
.
3.9. Consider the following formulations of the two-variable PRF:
Model I: