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

85
6. A concept related to the coefficient of determination is the 
coefficient of correlation,
r
.
It is a measure of 
linear association
between two variables and it lies between 
−
1
and
+
1
.
7. The CLRM is a theoretical construct or abstraction because it is based on a set of
assumptions that may be stringent or “unrealistic.” But such abstraction is often neces-
sary in the initial stages of studying any field of knowledge. Once the CLRM is mastered,
one can find out what happens if one or more of its assumptions are not satisfied. The first
part of this book is devoted to studying the CLRM. The other parts of the book consider
the refinements of the CLRM. Table 3.4 gives the road map ahead.
Questions
3.1. Given the assumptions in column 1 of the table, show that the assumptions in column
2 are equivalent to them.
Assumptions of the Classical Model
(1)
(2)
E
(
u
i
|
X
i
) 
=
0
E
(
Y
i
|
X
i
) 
=
β
2
+
β
2
X
cov (
u
i
, 
u
j
) 
=
0 
i
=
j
cov (
Y
i
, 
Y
j
) 
=
0 
i
=
j
var (
u
i
|
X
i
) 
=
σ
2
var (
Y
i
|
X
i
) 
=
σ
2
3.2. Show that the estimates 
ˆ
β
1
=
1
.
572 and 
ˆ
β
2
=
1
.
357 used in the first experiment of
Table 3.1 are in fact the OLS estimators.
3.3. According to Malinvaud (see footnote 11), the assumption that
E
(
u
i
|
X
i
)
=
0 is quite
important. To see this, consider the PRF:
Y
=
β
1
+
β
2
X
i
+
u
i
.
Now consider
two situations: (i)
β
1
=
0,
β
2
=
1, and
E
(
u
i
)
=
0; and (ii)
β
1
=
1,
β
2
=
0, and
E
(
u
i
)
=
(
X
i
−
1)
.
Now take the expectation of the PRF conditional upon
X
in the
two preceding cases and see if you agree with Malinvaud about the significance of
the assumption
E
(
u
i
|
X
i
)
=
0
.
EXERCISES
Assumption 
Number
Type of Violation
Where to Study?
1
Nonlinearity in parameters
Chapter 14
2
Stochastic regressor(s)
Chapter 13
3
Nonzero mean of 
u
i
Introduction to Part II
4
Heteroscedasticity
Chapter 11
5
Autocorrelated disturbances
Chapter 12
6
Sample observations less
Chapter 10
than the number of regressors
7
Insufficient variability in regressors
Chapter 10
8
Multicollinearity*
Chapter 10
9
Specification bias*
Chapters 13, 14
1 0**
Nonnormality of disturbances
Chapter 13
*These assumptions will be introduced in Chapter 7, when we discuss the multiple regression model. 
**Note:
The assumption that the disturbances 
u
i
are normally distributed is not a part of the CLRM. But more on this in Chapter 4.