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

FIGURE 2.4
Regression lines based
on two different
samples.
TABLE 2.4
A Random Sample from the
Population of Table 2.1
Y
X
70
80
65
100
90
120
95
140
110
160
115
180
120
200
140
220
155
240
150
260
TABLE 2.5
Another Random Sample from the
Population of Table 2.1
Y
X
55
80
88
100
90
120
80
140
118
160
120
180
145
200
135
220
145
240
175
260
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44
Part One
Single-Equation Regression Models
as the 
sample regression lines.
Supposedly they represent the population regression line,
but because of sampling fluctuations they are at best an approximation of the true PR. In
general, we would get 
N
different SRFs for 
N
different samples, and these SRFs are not
likely to be the same.
Now, analogously to the PRF that underlies the population regression line, we can
develop the concept of the 
sample regression function
(SRF) to represent the sample
regression line. The sample counterpart of Eq. (2.2.2) may be written as 
ˆ
Y
i
= ˆ
β
1
+ ˆ
β
2
X
i
(2.6.1)
where 
ˆ
Y
is read as “
Y
-hat’’ or “
Y
-cap’’
ˆ
Y
i
=
estimator of 
E
(
Y
|
X
i
)
ˆ
β
1
=
estimator of 
β
1
ˆ
β
2
=
estimator of 
β
2
Note that an
estimator,
also known as a (sample)
statistic,
is simply a rule or formula or
method that tells how to estimate the population parameter from the information provided by
the sample at hand. A particular numerical value obtained by the estimator in an application
is known as an
estimate.
13
It should be noted that an estimator is random, but an estimate is
nonrandom. (Why?)
Now just as we expressed the PRF in two equivalent forms, Eq. (2.2.2) and Eq. (2.4.2),
we can express the SRF in Equation 2.6.1 in its stochastic form as follows:
Y
i
= ˆ
β
1
+ ˆ
β
2
X
i
+ ˆ
u
i
(2.6.2)
where, in addition to the symbols already defined, 
ˆ
u
i
denotes the (sample) 
residual
term.
Conceptually 
ˆ
u
i
is analogous to 
u
i
and can be regarded as an
estimate 
of 
u
i
. It is introduced
in the SRF for the same reasons as 
u
i
was introduced in the PRF.
To sum up, then, we find our primary objective in regression analysis is to estimate the
PRF
(2.4.2)
on the basis of the SRF
(2.6.2)
because more often than not our analysis is based upon a single sample from some popula-
tion. But because of sampling fluctuations, our estimate of the PRF based on the SRF is at
best an approximate one. This approximation is shown diagrammatically in Figure 2.5.
Y
i
= ˆ
β
1
+ ˆ
β
x
i
+ ˆ
u
i
Y
i
=
β
1
+
β
2
X
i
+
u
i
13
As noted in the Introduction, a hat above a variable will signify an estimator of the relevant
population value.
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Chapter 2
Two-Variable Regression Analysis: Some Basic Ideas
45
For 
X
=
X
i
, we have one (sample) observation, 
Y
=
Y
i
. In terms of the SRF, the
observed 
Y
i
can be expressed as
Y
i
= ˆ
Y
i
+ ˆ
u
i
(2.6.3)
and in terms of the PRF, it can be expressed as
Y
i
=
E
(
Y
|
X
i
)
+
u
i
(2.6.4)
Now obviously in Figure 2.5 
ˆ
Y
i
overestimates 
the true 
E
(
Y
|
X
i
) for the 
X
i
shown therein.
By the same token, for any 
X
i
to the left of the point 
A
, the SRF will 
underestimate 
the true
PRF. But the reader can readily see that such over- and underestimation is inevitable
because of sampling fluctuations.
The critical question now is: Granted that the SRF is but an approximation of the PRF,
can we devise a rule or a method that will make this approximation as “close” as possible?
In other words, how should the SRF be constructed so that 
ˆ
β
1
is as “close” as possible to
the true 
β
1
and 
ˆ
β
2
is as “close” as possible to the true 
β
2
even though we will never know
the true 
β
1
and 
β
2
?
The answer to this question will occupy much of our attention in Chapter 3. We note
here that we can develop procedures that tell us how to construct the SRF to mirror the PRF
as faithfully as possible. It is fascinating to consider that this can be done even though we
never actually determine the PRF itself.