Practice Problems from the Text - “due” 3/26/13
From Chapter 5, work on 5.26 and 5.27. These problems will require you to access data from
tables 5-17 and 4-6, available on the class website.
From Chapter 6, work on 6.8, 6.10, and 6.22.
5.26. Did it in class (1-21-13)
5.27. Problem does not specify what is dependent and what are independent variables.
Here I regress education on GDP and population:
. reg EDUC GDP POP
Source
SS
df
MS
Model
Residual
1.0884e+09
43364140
2
35
544182785
1238975.43
Total
1.1317e+09
37
30587289.5
EDUC
Coef.
GDP
POP
_cons
.0523186
-50.04769
414.4583
Now do it with logs:
Std. Err.
.0018504
9.958169
266.7247
t
28.27
-5.03
1.55
Number of obs
F( 2,
35)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
0.129
=
=
=
=
=
=
38
439.22
0.0000
0.9617
0.9595
1113.1
[95% Conf. Interval]
.0485621
-70.26385
-127.0216
.0560751
-29.83154
955.9383
. reg lneduc lngdp lnpop
Source
SS
df
MS
Model
Residual
52.7592206
6.83479471
2
35
26.3796103
.195279849
Total
59.5940153
37
1.61064906
lneduc
Coef.
lngdp
lnpop
_cons
1.232876
-.1559398
-5.431023
Std. Err.
.0799177
.0786072
.8101842
t
15.43
-1.98
-6.70
Number of obs
F( 2,
35)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.055
0.000
=
=
=
=
=
=
38
135.09
0.0000
0.8853
0.8788
.4419
[95% Conf. Interval]
1.070634
-.3155209
-7.075784
1.395117
.0036413
-3.786262
(c) In the second model (log-linear), the coefficient of ln GDP suggests that for a one
percent increase in GDP, we should expect about a 1.23% increase in the Education
rate. Similarly, a one percent increase in the population should correspond to about a
0.15% decrease in the Education rate.
(d) We should assess scattergrams and residual plots to determine which model is
more appropriate. Also consider what the goal of the analysis is when choosing a model.
Some Answers for Chapter 6 problems:
6.8. (a) The coefficient -0.1647 is the own-price elasticity, 0.5115 is the income elasticity,
and 0.1483 is the cross-price elasticity.
(b) It is inelastic because, in absolute value, the coefficient is less than one.
(c) Since the cross-price elasticity is positive, coffee and tea are substitute products.
(d) and (e) The trend coefficient of -0.0089 suggests that over the sample period coffee
consumption had been declining at the quarterly rate of 0.89 percent. Among other
things, the side effects of caffeine may have something to do with the decline.
(f) 0.5115.
(g) The estimated t value of the income elasticity coefficient is 1.23, which is not
statistically significant. Therefore, it does not make much sense to test the hypothesis
that it is not different from one.
(h) The dummies here perhaps represent seasonal effects, if any.
(i) Each dummy coefficient tells by how much the average value of lnQ is different from
that of the base quarter, which is the fourth quarter. The actual values of the intercepts
in the various quarters are, respectively, 1.1828, 1.1219, 1.2692, and 1.2789. Taking
the antilogs of these values, we obtain: 3.2635, 3.0707, 3.5580, and 3.5927 as the
average pounds of coffee consumed per capita in the first, second, third, and the fourth
quarter, holding the values of the logs of all explanatory variables zero.
(j) The dummy coefficients D1 and D2 are individually statistically significant.
(k) That seems to be the case in quarters one and two. Among other things, coffee
prices and weather may have something to do with the observed seasonal pattern in
these two quarters.
(l) The benchmark is the fourth quarter. If we choose another quarter for the base, the
numerical values of the dummy coefficients will change.
(m) The implicit assumption that is made is that the partial slope coefficients do not
change among quarters.
(n) We can incorporate differential slope dummies as follows:
ln Q =B1 B2 ln P B3 ln I B4 ln P B5 T B6 D1 B7 D2 B8 D3 B9(D1 ln P )
B10(D2lnP ) B11(D3 ln P ) B12(D1 ln I ) B13(D2 ln I ) B14(D3 ln I ) B15(D1 ln P)
B16(D2 ln P) B17(D3 ln P) u
(o) One could estimate the model given in (n). If there are other substitutes for coffee,
they can be brought in the model.
6.10. (a) Since the coefficient of the Dumsex dummy is statistically significant, Model 2
is preferable to Model 1.
(b) The error of omitting a relevant variable.
(c) Ceteris paribus, the average weight of males is greater than that of females.
(d) There is an additional variable, Dumht, in Model 3, which is statistically insignificant.
As shown in Chapter 11, if an “unnecessary” variable is added to a model, the OLS
estimators, while unbiased and consistent, are generally inefficient. This can be seen
from Model 3. In Model 2 the Dumsex variable was statistically significant, but is
insignificant in Model 3 because of the apparently superfluous Dumht variable. Also,
keep in mind the possibility of multicollinearity.
(e) Choose Model 2. Not only is the Dumsex variable statistically significant in this
model, but the coefficient of the height variable is about the same in both Models 2 and
3. On the other hand, neither dummy variable is statistically significant in Model 3.
(f) We observe from the correlation matrix that the coefficient of correlation between
Dumsex and Dumht is very high, almost unity. As we show in the chapter on
multicollinearity, in cases of very high collinearity, OLS estimators, although unbiased,
have relatively large standard errors. Also, the signs and magnitudes of the coefficients
can change with slight alterations in the data or in the specification of the model.
6.22. (a) Since the p value of the dummy coefficient is about 14%, it seems that
product-differentiation does not lead to a higher rate of return.
(b) From (a) it is obvious that there is no statistical difference in the rate of return for
firms that product-differentiate and the firms that do not.
(c) Perhaps. If we had the original data, we could verify this. Product differentiation is
the result of advertising and marketing strategies. For details, see any industrial
organization textbook.
(d) To the equation given, add the product of D with each of the explanatory variables.
Thus, there will be three additional variables in the model.