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334
Part Two
Relaxing the Assumptions of the Classical Model
FIGURE 10.3
Individual confidence intervals for
β
2
and
β
3
and joint confidence
interval (ellipse) for
β
2
and
β
3
.
β
3
2
β
0
0.1484
–1.004
2.887
– 0.2332
95% confidence
interval for
Joint 95% confidence
interval for
and
95% confidence
interval for
β
3
β
3
2
β
2
β
EXAMPLE 10.1
(
Continued
)
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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?
335
EXAMPLE 10.2
Consumption
Function for
United States,
1947–2000
We now consider a concrete set of data on real consumption expenditure (C), real dis-
posable personal income (Yd), real wealth (W), and real interest rate (I) for the United
States for the period 1947–2000. The raw data are given in Table 10.7.
TABLE 10.7
U.S. Consumption Expenditure for the Period 1947–2000
Year
C
Yd
W
I
1947
976.4
1035.2
5166.815
10.35094
1948
998.1
1090
5280.757
4.719804
1949
1025.3
1095.6
5607.351
1.044063
1950
1090.9
1192.7
5759.515
0.407346
1951
1107.1
1227
6086.056
5.283152
1952
1142.4
1266.8
6243.864
0.277011
1953
1197.2
1327.5
6355.613
0.561137
1954
1221.9
1344
6797.027
0.138476
1955
1310.4
1433.8
7172.242
0.261997
1956
1348.8
1502.3
7375.18
0.736124
1957
1381.8
1539.5
7315.286
0.260683
1958
1393
1553.7
7869.975
0.57463
1959
1470.7
1623.8
8188.054
2.295943
1960
1510.8
1664.8
8351.757
1.511181
1961
1541.2
1720
8971.872
1.296432
1962
1617.3
1803.5
9091.545
1.395922
1963
1684
1871.5
9436.097
2.057616
1964
1784.8
2006.9
10003.4
2.026599
1965
1897.6
2131
10562.81
2.111669
1966
2006.1
2244.6
10522.04
2.020251
1967
2066.2
2340.5
11312.07
1.212616
1968
2184.2
2448.2
12145.41
1.054986
1969
2264.8
2524.3
11672.25
1.732154
1970
2317.5
2630
11650.04
1.166228
1971
2405.2
2745.3
12312.92
0.712241
1972
2550.5
2874.3
13499.92
0.155737
1973
2675.9
3072.3
13080.96
1.413839
1974
2653.7
3051.9
11868.79
1.042571
1975
2710.9
3108.5
12634.36
3.533585
1976
2868.9
3243.5
13456.78
0.656766
1977
2992.1
3360.7
13786.31
1.190427
1978
3124.7
3527.5
14450.5
0.113048
1979
3203.2
3628.6
15340
1.70421
1980
3193
3658
15964.95
2.298496
1981
3236
3741.1
15964.99
4.703847
1982
3275.5
3791.7
16312.51
4.449027
1983
3454.3
3906.9
16944.85
4.690972
1984
3640.6
4207.6
17526.75
5.848332
1985
3820.9
4347.8
19068.35
4.330504
1986
3981.2
4486.6
20530.04
3.768031
1987
4113.4
4582.5
21235.69
2.819469
1988
4279.5
4784.1
22331.99
3.287061
(
Continued
)
Source: See Table 7.12.
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336
Part Two
Relaxing the Assumptions of the Classical Model
TABLE 10.7
Continued
Year
C
Yd
W
I
1989
4393.7
4906.5
23659.8
4.317956
1990
4474.5
5014.2
23105.13
3.595025
1991
4466.6
5033
24050.21
1.802757
1992
4594.5
5189.3
24418.2
1.007439
1993
4748.9
5261.3
25092.33
0.62479
1994
4928.1
5397.2
25218.6
2.206002
1995
5075.6
5539.1
27439.73
3.333143
1996
5237.5
5677.7
29448.19
3.083201
1997
5423.9
5854.5
32664.07
3.12
1998
5683.7
6168.6
35587.02
3.583909
1999
5968.4
6320
39591.26
3.245271
2000
6257.8
6539.2
38167.72
3.57597
We use the following for analysis
ln C
t
=
β
1
+
β
2
ln Y
d
t
+
β
3
ln W
t
+
β
4
I
t
+
u
t
(10.6.6)
where ln stands for logarithm.
In this model the coefficients
β
2
and
β
3
give income and wealth elasticities, respectively
(why?) and
β
4
gives semielasticity (why?). The results of regression (10.6.6) are given in
the following table.
Dependent Variable: LOG (C)
Method: Least Squares
Sample: 1947–2000
Included observations: 54
Coefficient
Std. Error
t
-Statistic
Prob.
C
-0.467711
0.042778
-10.93343
0.0000
LOG (YD)
0.804873
0.017498
45.99836
0.0000
LOG (WEALTH)
0.201270
0.017593
11.44060
0.0000
INTEREST
-0.002689
0.000762
-3.529265
0.0009
R
-squared
0.999560
Mean dependent var.
7.826093
Adjusted
R
-squared
0.999533
S.D. dependent var.
0.552368
S.E. of regression
0.011934
Akaike info criterion
-5.947703
Sum squared resid.
0.007121
Schwarz criterion
-5.800371
Log likelihood
164.5880
Hannan-Quinn cariter.
-5.890883
F
-statistic
37832.59
Durbin-Watson stat.
1.289219
Prob(
F
-statistic)
0.000000
Note:
LOG stands for natural log.
The results show that all the estimated coefficients are highly statistically significant, for
their
p
values are extremely small. The estimated coefficients are interpreted as follows.
The income elasticity is
≈
0
.
80, suggesting that, holding other variables constant, if
income goes up by 1 percent, the mean consumption expenditure goes up by about
EXAMPLE 10.2
(
Continued
)
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Chapter 10
Multicollinearity: What Happens If the Regressors Are Correlated?
337
0.8 percent. The wealth coefficient is
≈
0
.
20, meaning that if wealth goes up by 1 percent,
mean consumption goes up by only 0.2 percent, again holding other variables constant.
The coefficient of the interest rate variable tells us that as the interest rate goes up by
one
percentage point
, consumption expenditure goes
down
by 0.26 percent, ceteris paribus.
All the regressors have signs that accord with prior expectations, that is, income and
wealth both have a positive impact on consumption but interest rate has a negative
impact.
Do we have to worry about the problem of multicollinearity in the present case? Ap-
parently not, because all the coefficients have the right signs, each coefficient is individu-
ally statistically significant, and the
F
value is also statistically highly significant, suggesting
that, collectively, all the variables have a significant impact on consumption expenditure.
The
R
2
value is also quite high.
Of course, there is usually some degree of collinearity among economic variables. As
long as it is not exact, we can still estimate the parameters of the model. For now, all we
can say is that, in the present example, collinearity, if any, does not seem to be very severe.
But in Section 10.7 we provide some diagnostic tests to detect collinearity and reexamine
the U.S. consumption function to determine whether it is plagued by the collinearity
problem.
10.7
Detection of Multicollinearity
Having studied the nature and consequences of multicollinearity, the natural question is:
How does one know that collinearity is present in any given situation, especially in models
involving more than two explanatory variables? Here it is useful to bear in mind Kmenta’s
warning:
1.
Multicollinearity is a question of degree and not of kind. The meaningful distinction is
not between the presence and the absence of multicollinearity, but between its various degrees.
2.
Since multicollinearity refers to the condition of the explanatory variables that are as-
sumed to be nonstochastic, it is a feature of the sample and not of the population.
Therefore, we do not “test for multicollinearity” but can, if we wish, measure its degree in
any particular sample.
17
Since multicollinearity is essentially a sample phenomenon, arising out of the largely
nonexperimental data collected in most social sciences, we do not have one unique method
of detecting it or measuring its strength. What we have are some rules of thumb, some in-
formal and some formal, but rules of thumb all the same. We now consider some of these
rules.
1.
High
R
2
but few significant
t
ratios.
As noted, this is the “classic” symptom of mul-
ticollinearity. If
R
2
is high, say, in excess of 0.8, the
F
test in most cases will reject the
hypothesis that the partial slope coefficients are simultaneously equal to zero, but the indi-
vidual
t
tests will show that none or very few of the partial slope coefficients are statistically
different from zero. This fact was clearly demonstrated by our consumption–income–wealth
example.
Although this diagnostic is sensible, its disadvantage is that “it is too strong in the sense
that multicollinearity is considered as harmful only when all of the influences of the
explanatory variables on
Y
cannot be disentangled.”
18
17
Jan Kmenta,
Elements of Econometrics,
2d ed., Macmillan, New York, 1986, p. 431.
18
Ibid., p. 439.
EXAMPLE 10.2
(
Continued
)
guj75772_ch10.qxd 12/08/2008 02:45 PM Page 337
2.
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