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

Capital Input
Output
Labor Input
Capital
Value Added
Worker Hrs
Expenditure
(thousands of $)
(thousands)
(thousands of $)
Area
Y
X
2
X
3
Alabama
38,372,840
424,471
2,689,076
Alaska
1,805,427
19,895
57,997
Arizona
23,736,129
206,893
2,308,272
Arkansas
26,981,983
304,055
1,376,235
California
217,546,032
1,809,756
13,554,116
Colorado
19,462,751
180,366
1,790,751
Connecticut
28,972,772
224,267
1,210,229
Delaware
14,313,157
54,455
421,064
District of Columbia
159,921
2,029
7,188
Florida
47,289,846
471,211
2,761,281
Georgia
63,015,125
659,379
3,540,475
Hawaii
1,809,052
17,528
146,371
Idaho
10,511,786
75,414
848,220
Illinois
105,324,866
963,156
5,870,409
Indiana
90,120,459
835,083
5,832,503
Iowa
39,079,550
336,159
1,795,976
Kansas
22,826,760
246,144
1,595,118
Kentucky
38,686,340
384,484
2,503,693
Louisiana
69,910,555
216,149
4,726,625
TABLE 7.3 
Value Added, Labor
Hours, and Capital
Input in the Manu-
facturing Sector of
the U.S., 2005
Source:
2005 Annual Survey
of Manufacturers
, Sector 31:
Supplemental Statistics
for U.S.
16
To see this, differentiate Eq. (7.9.3) partially with respect to the log of each 
X
variable. Therefore,
∂
ln
Y
/∂
ln
X
2
=
(
∂
Y
/∂
X
2
)(
X
2
/
Y
)
=
β
2
, which, by definition, is the elasticity of 
Y
with respect to 
X
2
and 
∂
ln
Y
/∂
ln
X
3
=
(
∂
Y
/∂
X
3
)(
X
3
/
Y
)
=
β
3
, which is the elasticity of 
Y
with respect to 
X
3
, and so on.
17
Notice that in the Cobb–Douglas production function (7.9.1) we have introduced the stochastic
error term in a special way so that in the resulting logarithmic transformation it enters in the usual
linear form. On this, see Section 6.9.
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Chapter 7
Multiple Regression Analysis: The Problem of Estimation
209
Maine
7,856,947
82,021
415,131
Maryland
21,352,966
174,855
1,729,116
Massachusetts
46,044,292
355,701
2,706,065
Michigan
92,335,528
943,298
5,294,356
Minnesota
48,304,274
456,553
2,833,525
Mississippi
17,207,903
267,806
1,212,281
Missouri
47,340,157
439,427
2,404,122
Montana
2,644,567
24,167
334,008
Nebraska
14,650,080
163,637
627,806
Nevada
7,290,360
59,737
522,335
New Hampshire
9,188,322
96,106
507,488
New Jersey
51,298,516
407,076
3,295,056
New Mexico
20,401,410
43,079
404,749
New York
87,756,129
727,177
4,260,353
North Carolina
101,268,432
820,013
4,086,558
North Dakota
3,556,025
34,723
184,700
Ohio
124,986,166
1,174,540
6,301,421
Oklahoma
20,451,196
201,284
1,327,353
Oregon
34,808,109
257,820
1,456,683
Pennsylvania
104,858,322
944,998
5,896,392
Rhode Island
6,541,356
68,987
297,618
South Carolina
37,668,126
400,317
2,500,071
South Dakota
4,988,905
56,524
311,251
Tennessee
62,828,100
582,241
4,126,465
Texas
172,960,157
1,120,382
11,588,283
Utah
15,702,637
150,030
762,671
Vermont
5,418,786
48,134
276,293
Virginia
49,166,991
425,346
2,731,669
Washington
46,164,427
313,279
1,945,860
West Virginia
9,185,967
89,639
685,587
Wisconsin
66,964,978
694,628
3,902,823
Wyoming
2,979,475
15,221
361,536
EXAMPLE 7.3
(
Continued
)
ln
Y
i
=
3.8876 
0.4683ln
X
2
i 
0.5213ln
X
3
i
(0.3962)
(0.0989)
(0.0969)
t
=
(9.8115)
(4.7342)
(5.3803)
(7.9.4)
R
2
=
0.9642
df 
=
48
¯
R
2
=
0
.
9627
From Eq. (7.9.4) we see that in the U.S. manufacturing sector for 2005, the output elas-
ticities of labor and capital were 0.4683 and 0.5213, respectively. In other words, over the
50 U.S. states and the District of Columbia, holding the capital input constant, a 1 percent
increase in the labor input led on the average to about a 0.47 percent increase in the out-
put. Similarly, holding the labor input constant, a 1 percent increase in the capital input
led on the average to about a 0.52 percent increase in the output. Adding the two output
elasticities, we obtain 0.99, which gives the value of the returns to scale parameter. As is
evident, the manufacturing sector for the 50 United States and the District of Columbia
was characterized by constant returns to scale.
From a purely statistics viewpoint, the estimated regression line fits the data quite well.
The
R
2
value of 0.9642 means that about 96 percent of the variation in the (log of) output is
explained by the (logs of) labor and capital. In Chapter 8, we shall see how the estimated
standard errors can be used to test hypotheses about the “true” values of the parameters of
the Cobb–Douglas production function for the U.S. manufacturing sector of the economy.
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210
Part One
Single-Equation Regression Models
7.10
Polynomial Regression Models
We now consider a class of multiple regression models, the 
polynomial regression
models,
that have found extensive use in econometric research relating to cost and produc-
tion functions. In introducing these models, we further extend the range of models to which
the classical linear regression model can easily be applied.
To fix the ideas, consider Figure 7.1, which relates the short-run marginal cost (MC) of
production (
Y
) of a commodity to the level of its output (
X
). The visually-drawn MC curve
in the figure, the textbook U-shaped curve, shows that the relationship between MC and
output is nonlinear. If we were to quantify this relationship from the given scatterpoints,
how would we go about it? In other words, what type of econometric model would capture
first the declining and then the increasing nature of marginal cost?
Geometrically, the MC curve depicted in Figure 7.1 represents a 
parabola
. Mathemati-
cally, the parabola is represented by the following equation:
Y
=
β
0
+
β
1
X
+
β
2
X
2
(7.10.1)
which is called a 
quadratic function,
or more generally, a 
second-degree polynomial
in the
variable 
X
—the highest power of 
X
represents the degree of the polynomial (if 
X
3
were
added to the preceding function, it would be a third-degree polynomial, and so on).
The stochastic version of Eq. (7.10.1) may be written as
Y
i
=
β
0
+
β
1
X
i
+
β
2
X
2
i
+
u
i
(7.10.2)
which is called a 
second-degree polynomial
regression.
The general 
kth degree polynomial regression
may be written as
Y
i
=
β
0
+
β
1
X
i
+
β
2
X
2
i
+ · · · +
β
k
X
k
i
+
u
i
(7.10.3)
Notice that in these types of polynomial regressions there is only one explanatory variable
on the right-hand side but it appears with various powers, thus making them multiple re-
gression models. Incidentally, note that if 
X
i
is assumed to be fixed or nonstochastic, the
powered terms of 
X
i
also become fixed or nonstochastic.
Do these models present any special estimation problems? Since the second-degree
polynomial (7.10.2) or the 
k
th degree polynomial (7.10.13) is linear in the parameters, the
β
’s, they can be estimated by the usual OLS or ML methodology. But what about the
MC
Output
X
Y
Marginal cost
FIGURE 7.1
The U-shaped
marginal cost curve.
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Chapter 7
Multiple Regression Analysis: The Problem of Estimation
211
EXAMPLE 7.4
Estimating the
Total Cost
Function
As an example of the polynomial regression, consider the data on output and total cost of
production of a commodity in the short run given in Table 7.4. What type of regression
model will fit these data? For this purpose, let us first draw the scattergram, which is
shown in Figure 7.2.
From this figure it is clear that the relationship between total cost and output resem-
bles the elongated S curve; notice how the total cost curve first increases gradually and
then rapidly, as predicted by the celebrated law of 
diminishing returns
. This S shape of the
total cost curve can be captured by the following cubic or 
third-degree polynomial:
Y
i
=
β
0
+
β
1
X
i
+
β
2
X
2
i
+
β
3
X
3
i
+
u
i
(7.10.4)
where 
Y
=
total cost and 
X
=
output.
Given the data of Table 7.4, we can apply the OLS method to estimate the parameters
of Eq. (7.10.4). But before we do that, let us find out what economic theory has to say
about the short-run cubic cost function (7.10.4). Elementary price theory shows that in
the short run the marginal cost (MC) and average cost (AC) curves of production are
typically U-shaped—initially, as output increases both MC and AC decline, but after a
certain level of output they both turn upward, again the consequence of the law of di-
minishing return. This can be seen in Figure 7.3 (see also Figure 7.1). And since the MC
and AC curves are derived from the total cost curve, the U-shaped nature of these curves
puts some restrictions on the parameters of the total cost curve (7.10.4). As a matter of