|
89
Output per Hour of All
Real Compensation per
Persons
1
Hour
2,3
Nonfarm
Nonfarm
Business Business Business
Business
Year
Sector
Sector Sector Sector
1960
48.9
51.9
60.8
63.3
1961
50.6
53.5
62.5
64.8
1962
52.9
55.9
64.6
66.7
1963
55.0
57.8
66.1
68.1
1964
56.8
59.6
67.7
69.3
1965
58.8
61.4
69.1
70.5
1966
61.2
63.6
71.7
72.6
1967
62.5
64.7
73.5
74.5
1968
64.7
66.9
76.2
77.1
1969
65.0
67.0
77.3
78.1
1970
66.3
68.0
78.8
79.2
1971
69.0
70.7
80.2
80.7
1972
71.2
73.1
82.6
83.2
1973
73.4
75.3
84.3
84.7
1974
72.3
74.2
83.3
83.8
1975
74.8
76.2
84.1
84.5
1976
77.1
78.7
86.4
86.6
1977
78.5
80.0
87.6
88.0
1978
79.3
81.0
89.1
89.6
1979
79.3
80.7
89.3
89.7
1980
79.2
80.6
89.1
89.6
1981
80.8
81.7
89.3
89.8
1982
80.1
80.8
90.4
90.8
1983
83.0
84.5
90.3
90.9
1984
85.2
86.1
90.7
91.1
1985
87.1
87.5
92.0
92.2
1986
89.7
90.2
94.9
95.2
1987
90.1
90.6
95.2
95.5
1988
91.5
92.1
96.5
96.7
1989
92.4
92.8
95.0
95.1
1990
94.4
94.5
96.2
96.1
1991
95.9
96.1
97.4
97.4
1992
100.0
100.0
100.0
100.0
1993
100.4
100.4
99.7
99.5
1994
101.3
101.5
99.0
99.1
1995
101.5
102.0
98.7
98.8
1996
104.5
104.7
99.4
99.4
1997
106.5
106.4
100.5
100.3
1998
109.5
109.4
105.2
104.9
1999
112.8
112.5
108.0
107.5
2000
116.1
115.7
112.0
111.5
2001
119.1
118.6
113.5
112.8
2002
124.0
123.5
115.7
115.1
2003
128.7
128.0
117.7
117.1
2004
132.7
131.8
119.0
118.2
2005
135.7
134.9
120.2
119.3
1
Output refers to real gross domestic product in the sector.
2
Wages and salaries of employees plus employers’ contributions for social insurance and private benefit plans.
3
Hourly compensation divided by the consumer price index for all urban consumers for recent quarters.
TABLE 3.6
Productivity and
Related Data,
Business Sector
1960–2005
(Index numbers,
1992
=
100; quarterly
data seasonally
adjusted)
Source:
Economic Report of the
President,
2007, Table 49.
guj75772_ch03.qxd 23/08/2008 02:34 PM Page 89
90
Part One
Single-Equation Regression Models
3.23. Table 3.8 gives data on gross domestic product (GDP) for the United States for the
years 1959–2005.
a
. Plot the GDP data in current and constant (i.e., 2000) dollars against time.
b
. Letting
Y
denote GDP and
X
time (measured chronologically starting with 1 for
1959, 2 for 1960, through 47 for 2005), see if the following model fits the GDP
data:
Y
t
=
β
1
+
β
2
X
t
+
u
t
Estimate this model for both current and constant-dollar GDP.
c
. How would you interpret
β
2
?
d
. If there is a difference between
β
2
estimated for current-dollar GDP and that
estimated for constant-dollar GDP, what explains the difference?
e
. From your results what can you say about the nature of inflation in the United
States over the sample period?
Year
Gold Price
NYSE
CPI
1974
159.2600
463.5400
49.30000
1975
161.0200
483.5500
53.80000
1976
124.8400
575.8500
56.90000
1977
157.7100
567.6600
60.60000
1978
193.2200
567.8100
65.20000
1979
306.6800
616.6800
72.60000
1980
612.5600
720.1500
82.40000
1981
460.0300
782.6200
90.90000
1982
375.6700
728.8400
96.50000
1983
424.3500
979.5200
99.60000
1984
360.4800
977.3300
103.9000
1985
317.2600
1142.970
107.6000
1986
367.6600
1438.020
109.6000
1987
446.4600
1709.790
113.6000
1988
436.9400
1585.140
118.3000
1989
381.4400
1903.360
124.0000
1990
383.5100
1939.470
130.7000
1991
362.1100
2181.720
136.2000
1992
343.8200
2421.510
140.3000
1993
359.7700
2638.960
144.5000
1994
384.0000
2687.020
148.2000
1995
384.1700
3078.560
152.4000
1996
387.7700
3787.200
156.9000
1997
331.0200
4827.350
160.5000
1998
294.2400
5818.260
163.0000
1999
278.8800
6546.810
166.6000
2000
279.1100
6805.890
172.2000
2001
274.0400
6397.850
177.1000
2002
309.7300
5578.890
179.9000
2003
363.3800
5447.460
184.0000
2004
409.7200
6612.620
188.9000
2005
444.7400
7349.000
195.3000
2006
603.4600
8357.990
201.6000
TABLE 3.7
Gold Prices, New
York Stock Exchange
Index, and Consumer
Price Index for U.S.
for 1974–2006
guj75772_ch03.qxd 23/08/2008 02:34 PM Page 90
Chapter 3
Two-Variable Regression Model: The Problem of Estimation
91
3.24. Using the data given in Table I.1 of the Introduction, verify Eq. (3.7.1).
3.25. For the SAT example given in Exercise 2.16 do the following:
a
. Plot the female reading score against the male reading score.
b
. If the scatterplot suggests that a linear relationship between the two seems
appropriate, obtain the regression of female reading score on male reading score.
c
. If there is a relationship between the two reading scores, is the relationship
causal?
3.26. Repeat Exercise 3.25, replacing math scores for reading scores.
3.27. Monte Carlo study
classroom assignment:
Refer to the 10
X
values given in
Table 2.4. Let
β
1
=
25 and
β
2
=
0
.
5
.
Assume
u
i
≈
N
(0, 9), that is,
u
i
are normally
distributed with mean 0 and variance 9. Generate 100 samples using these values,
obtaining 100 estimates of
β
1
and
β
2
.
Graph these estimates. What conclusions can
you draw from the Monte Carlo study?
Note:
Most statistical packages now can gen-
erate random variables from most well-known probability distributions. Ask your in-
structor for help, in case you have difficulty generating such variables.
3.28. Using the data given in Table 3.3, plot the number of cell phone subscribers against
the number of personal computers in use. Is there any discernible relationship be-
tween the two? If so, how do you rationalize the relationship?
Year
NGDP
RGDP
Year
NGDP
RGDP
1959
506.6
2,441.3
1983
3,536.7
5,423.8
1960
526.4
2,501.8
1984
3,933.2
5,813.6
1961
544.7
2,560.0
1985
4,220.3
6,053.7
1962
585.6
2,715.2
1986
4,462.8
6,263.6
1963
617.7
2,834.0
1987
4,739.5
6,475.1
1964
663.6
2,998.6
1988
5,103.8
6,742.7
1965
719.1
3,191.1
1989
5,484.4
6,981.4
1966
787.8
3,399.1
1990
5,803.1
7,112.5
1967
832.6
3,484.6
1991
5,995.9
7,100.5
1968
910.0
3,652.7
1992
6,337.7
7,336.6
1969
984.6
3,765.4
1993
6,657.4
7,532.7
1970
1,038.5
3,771.9
1994
7,072.2
7,835.5
1971
1,127.1
3,898.6
1995
7,397.7
8,031.7
1972
1,238.3
4,105.0
1996
7,816.9
8,328.9
1973
1,382.7
4,341.5
1997
8,304.3
8,703.5
1974
1,500.0
4,319.6
1998
8,747.0
9,066.9
1975
1,638.3
4,311.2
1999
9,268.4
9,470.3
1976
1,825.3
4,540.9
2000
9,817.0
9,817.0
1977
2,030.9
4,750.5
2001
10,128.0
9,890.7
1978
2,294.7
5,015.0
2002
10,469.6
10,048.8
1979
2,563.3
5,173.4
2003
10,960.8
10,301.0
1980
2,789.5
5,161.7
2004
11,712.5
10,703.5
1981
3,128.4
5,291.7
2005
12,455.8
11,048.6
1982
3,255.0
5,189.3
Source:
Economic Report of the President,
2007. Table B-1 and B-2.
TABLE 3.8
Nominal and Real
Gross Domestic
Product, 1959–2005
(billions of dollars,
except as noted;
quarterly data at
seasonally adjusted
annual rates; RGDP
in billions of chained
[2000] dollars)
guj75772_ch03.qxd 23/08/2008 02:34 PM Page 91
92
Part One
Single-Equation Regression Models
Appendix
3A
3A.1
Derivation of Least-Squares Estimates
Differentiating Eq. (3.1.2) partially with respect to
ˆ
β
1
and
ˆ
β
2
, we obtain
∂
ˆ
u
2
i
∂
ˆ
β
1
= −
2
(
Y
i
− ˆ
β
1
− ˆ
β
2
X
i
)
= −
2
ˆ
u
i
(1)
∂
ˆ
u
2
i
∂
ˆ
β
2
= −
2
(
Y
i
− ˆ
β
1
− ˆ
β
2
X
i
)
X
i
= −
2
ˆ
u
i
X
i
(2)
Setting these equations to zero, after algebraic simplification and manipulation, gives the estimators
given in Eqs. (3.1.6) and (3.1.7).
3A.2
Linearity and Unbiasedness Properties
of Least-Squares Estimators
From Eq. (3.1.8) we have
ˆ
β
2
=
x
i
Y
i
x
2
i
=
k
i
Y
i
(3)
where
k
i
=
x
i
x
2
i
which shows that
ˆ
β
2
is a
linear estimator
because it is a linear function of
Y
; actually it is a weighted
average of
Y
i
with
k
i
serving as the weights. It can similarly be shown that
ˆ
β
1
too is a linear estimator.
Incidentally, note these properties of the weights
k
i
:
1.
Since the
X
i
are assumed to be nonstochastic, the
k
i
are nonstochastic too.
2.
k
i
=
0
.
3.
k
2
i
=
1
x
2
i
.
4.
k
i
x
i
=
k
i
X
i
=
1
.
These properties can be directly verified from the definition of
k
i
.
For example,
k
i
=
x
i
x
2
i
=
1
x
2
i
x
i
,
since for a given sample
x
2
i
is known
=
0,
since
x
i
, the sum of deviations from the mean value, is
always zero
Now substitute the PRF
Y
i
=
β
1
+
β
2
X
i
+
u
i
into Equation (3) to obtain
ˆ
β
2
=
k
i
(
β
1
+
β
2
X
i
+
u
i
)
=
β
1
k
i
+
β
2
k
i
X
i
+
k
i
u
i
(4)
=
β
2
+
k
i
u
i
where use is made of the properties of
k
i
noted earlier.
guj75772_ch03.qxd 23/08/2008 02:34 PM Page 92
Chapter 3
Two-Variable Regression Model: The Problem of Estimation
93
Now taking expectation of Equation (4) on both sides and noting that
k
i
, being nonstochastic, can
be treated as constants, we obtain
E
(
ˆ
β
2
)
=
β
2
+
k
i
E
(
u
i
)
=
β
2
(5)
since
E
(
u
i
)
=
0 by assumption. Therefore,
ˆ
β
2
is an unbiased estimator of
β
2
.
Likewise, it can be
proved that
ˆ
β
1
is also an unbiased estimator of
β
1
.
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