|
indifference curve
between two goods is
X
i
Y
i
=
β
1
+
β
2
X
i
How would you estimate the parameters of this model? Apply the preceding model
to the data in Table 5.8 and comment on your results.
guj75772_ch05.qxd 07/08/2008 12:46 PM Page 138
Chapter 5
Two-Variable Regression: Interval Estimation and Hypothesis Testing
139
5.16. Since 1986 the
Economist
has been publishing the Big Mac Index as a crude, and hi-
larious, measure of whether international currencies are at their “correct” exchange
rate, as judged by the theory of
purchasing power parity (PPP).
The PPP holds that
a unit of currency should be able to buy the same bundle of goods in all countries.
The proponents of PPP argue that, in the long run, currencies tend to move toward
their PPP. The
Economist
uses McDonald’s Big Mac as a representative bundle and
gives the information in Table 5.9.
Consider the following regression model:
Y
i
=
β
1
+
β
2
X
i
+
u
i
where
Y
=
actual exchange rate and
X
=
implied PPP of the dollar.
a.
If the PPP holds, what values of
β
1
and
β
2
would you expect a priori?
Money Stock Measure, $ billion
GNP
Year
$ billion
M
1
M
2
M
3
L
1970
992.70
216.6
628.2
677.5
816.3
1971
1,077.6
230.8
712.8
776.2
903.1
1972
1,185.9
252.0
805.2
886.0
1,023.0
1973
1,326.4
265.9
861.0
985.0
1,141.7
1974
1,434.2
277.6
908.5
1,070.5
1,249.3
1975
1,549.2
291.2
1,023.3
1,174.2
1,367.9
1976
1,718.0
310.4
1,163.6
1,311.9
1,516.6
1977
1,918.3
335.4
1,286.7
1,472.9
1,704.7
1978
2,163.9
363.1
1,389.1
1,647.1
1,910.6
1979
2,417.8
389.1
1,498.5
1,804.8
2,117.1
1980
2,631.7
414.9
1,632.6
1,990.0
2,326.2
1981
2,957.8
441.9
1,796.6
2,238.2
2,599.8
1982
3,069.3
480.5
1,965.4
2,462.5
2,870.8
1983
3,304.8
525.4
2,196.3
2,710.4
3,183.1
Definitions:
M
1
=
Currency
+
Demand deposits
+
Travelers checks and other checkable deposits (OCDs).
M
2
=
M
1
+
Overnight RPs and Eurodollars
+
MMMF (Money market mutual fund) balances
+
MMDAs (Money market
deposit accounts)
+
Savings and small deposits.
M
3
=
M
2
+
Large time deposits
+
Term RPs
+
Institutional MMMF.
L
=
M
3
+
Other liquid assets.
TABLE 5.6
GNP and Four
Measures of Money
Stock
Source:
Economic Report of the
President,
1985, GNP data from
Table B-1, p. 232; money stock
data from Table B-61, p. 303.
1)
GNP
t
= −
787.4723
+
8.0863
M
1
t
r
2
=
0.9912
(77.9664)
(0.2197)
2)
GNP
t
=
−
44.0626
+
1.5875
M
2
t
r
2
=
0.9905
(61.0134)
(0.0448)
3)
GNP
t
=
159.1366
+
1.2034
M
3
t
r
2
=
0.9943
(42.9882)
(0.0262)
4)
GNP
t
=
164.2071
+
1.0290
L
t
r
2
=
0.9938
(44.7658)
(0.0234)
Note:
The figures in parentheses are the estimated standard errors.
Consumption of good
X
: 1
2
3
4
5
Consumption of good
Y
: 4
3.5
2.8
1.9
0.8
TABLE 5.8
TABLE 5.7
GNP–Money Stock
Regressions,
1970–1983
guj75772_ch05.qxd 07/08/2008 12:46 PM Page 139
TABLE 5.9
The Hamburger
Standard
Source: McDonald’s;
The
Economist,
February 1, 2007.
*Purchasing power parity: local price divided by price in the United States.
** Dollars per euro.
†
Average of New York, Chicago, San Francisco, and Atlanta.
‡
Dollars per pound.
§
Weighted average of prices in euro area.
Actual Under
(
−
)
Big Mac Prices
Dollar
Over (
+
)
Implied
Exchange Valuation
In Local
In
PPP
*
of
Rate,
against the
Currency
Dollars
the Dollar
Jan 31st
Dollar, %
United States
†
$3.22
3.22
Argentina
Peso 8.25
2.65
2.56
3.11
−
18
Australia
A$3.45
2.67
1.07
1.29
−
17
Brazil
Real 6.4
3.01
1.99
2.13
−
6
Britain
£1.99
3.90
1.62
‡
1.96
‡
+
21
Canada
C$3.63
3.08
1.13
1.18
−
4
Chile
Peso 1,670
3.07
519
544
−
5
China
Yuan 11.0
1.41
3.42
7.77
−
56
Colombia
Peso 6,900
3.06
2,143
2,254
−
5
Costa Rica
Colones 1,130
2.18
351
519
−
32
Czech Republic
Koruna 52.1
2.41
16.2
21.6
−
25
Denmark
DKr27.75
4.84
8.62
5.74
+
50
Egypt
Pound 9.09
1.60
2.82
5.70
−
50
Estonia
Kroon 30
2.49
9.32
12.0
−
23
Euro area
§
€
2.94
3.82
1.10**
1.30**
+
19
Hong Kong
HK$12.0
1.54
3.73
7.81
−
52
Hungary
Forint 590
3.00
183
197
−
7
Iceland
Kronur 509
7.44
158
68.4
+
131
Indonesia
Rupiah 15,900
1.75
4,938
9,100
−
46
Japan
¥280
2.31
87.0
121
−
28
Latvia
Lats 1.35
2.52
0.42
0.54
−
22
Lithuania
Litas 6.50
2.45
2.02
2.66
−
24
Malaysia
Ringgit 5.50
1.57
1.71
3.50
−
51
Mexico
Peso 29.0
2.66
9.01
10.9
−
17
New Zealand
NZ$4.60
3.16
1.43
1.45
−
2
Norway
Kroner 41.5
6.63
12.9
6.26
+
106
Pakistan
Rupee 140
2.31
43.5
60.7
−
28
Paraguay
Guarani 10,000
1.90
3,106
5,250
−
41
Peru
New Sol 9.50
2.97
2.95
3.20
−
8
Philippines
Peso 85.0
1.74
26.4
48.9
−
46
Poland
Zloty 6.90
2.29
2.14
3.01
−
29
Russia
Rouble 49.0
1.85
15.2
26.5
−
43
Saudi Arabia
Riyal 9.00
2.40
2.80
3.75
−
25
Singapore
S$3.60
2.34
1.12
1.54
−
27
Slovakia
Crown 57.98
2.13
18.0
27.2
−
34
South Africa
Rand 15.5
2.14
4.81
7.25
−
34
South Korea
Won 2,900
3.08
901
942
−
4
Sri Lanka
Rupee 190
1.75
59.0
109
−
46
Sweden
SKr32.0
4.59
9.94
6.97
+
43
Switzerland
SFr6.30
5.05
1.96
1.25
+
57
Taiwan
NT$75.0
2.28
23.3
32.9
−
29
Thailand
Baht 62.0
1.78
19.3
34.7
−
45
Turkey
Lire 4.55
3.22
1.41
1.41
nil
UAE
Dirhams 10.0
2.72
3.11
3.67
−
15
Ukraine
Hryvnia 9.00
1.71
2.80
5.27
−
47
Uruguay
Peso 55.0
2.17
17.1
25.3
−
33
Venezuela
Bolivar 6,800
1.58
2,112
4,307
−
51
140
guj75772_ch05.qxd 23/08/2008 02:55 PM Page 140
Chapter 5
Two-Variable Regression: Interval Estimation and Hypothesis Testing
141
b.
Do the regression results support your expectation? What formal test do you use
to test your hypothesis?
c.
Should the
Economist
continue to publish the Big Mac Index? Why or why not?
5.17. Refer to the SAT data given in Exercise 2.16. Suppose you want to predict the male
math (
Y
) scores on the basis of the female math scores (
X
) by running the following
regression:
Y
t
=
β
1
+
β
2
X
t
+
u
t
a.
Estimate the preceding model.
b.
From the estimated residuals, find out if the normality assumption can be
sustained.
c.
Now test the hypothesis that
β
2
=
1, that is, there is a one-to-one correspondence
between male and female math scores.
d.
Set up the ANOVA table for this problem.
5.18. Repeat the exercise in the preceding problem but let
Y
and
X
denote the male and fe-
male critical reading scores, respectively.
5.19. Table 5.10 gives annual data on the Consumer Price Index (CPI) and the Wholesale
Price Index (WPI), also called Producer Price Index (PPI), for the U.S. economy for
the period 1980–2006.
PPI (Total
CPI Total
Finished Goods)
1980
82.4
88.0
1981
90.9
96.1
1982
96.5
100.0
1983
99.6
101.6
1984
103.9
103.7
1985
107.6
104.7
1986
109.6
103.2
1987
113.6
105.4
1988
118.3
108.0
1989
124.0
113.6
1990
130.7
119.2
1991
136.2
121.7
1992
140.3
123.2
1993
144.5
124.7
1994
148.2
125.5
1995
152.4
127.9
1996
156.9
131.3
1997
160.5
131.8
1998
163.0
130.7
1999
166.6
133.0
2000
172.2
138.0
2001
177.1
140.7
2002
179.9
138.9
2003
184.0
143.3
2004
188.9
148.5
2005
195.3
155.7
2006
201.6
160.3
TABLE 5.10
CPI and PPI, USA,
1980–2006
Source:
Economic Report of the
President,
2007, Tables B-62
and B-65.
guj75772_ch05.qxd 27/08/2008 10:36 AM Page 141
142
Part One
Single-Equation Regression Models
a.
Plot the CPI on the vertical axis and the WPI on the horizontal axis. A priori, what
kind of relationship do you expect between the two indexes? Why?
b.
Suppose you want to predict one of these indexes on the basis of the other index.
Which will you use as the regressand and which as the regressor? Why?
c.
Run the regression you have decided in (
b
). Show the standard output. Test the
hypothesis that there is a one-to-one relationship between the two indexes.
d.
From the residuals obtained from the regression in (
c
), can you entertain the
hypothesis that the true error term is normally distributed? Show the tests you use.
5.20. Table 5.11 provides data on the lung cancer mortality index (100
=
average) and the
smoking index (100
=
average) for 25 occupational groups.
a.
Plot the cancer mortality index against the smoking index. What general pattern
do you observe?
b.
Letting
Y
=
cancer mortality index and
X
=
smoking index, estimate a linear
regression model and obtain the usual regression statistics.
c.
Test the hypothesis that smoking has no influence on lung cancer at
α
=
5%.
d.
Which are the risky occupations in terms of lung cancer mortality? Can you give
some reasons why this might be so?
e.
Is there any way to bring occupation category explicitly into the regression
analysis?
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