In class problems – 3-28-13 – some solutions
Directions: You are to work with in pairs. Each pair of midshipmen is to do the following book problems
in class, the corresponding data for which are attached. You should type all your responses in a word
document. Each time you run a regression, you should copy the table of results (as a picture) and paste
the table in your document. Answer all questions posed in the book. When you are done, be sure your
names are on the document. Email me the document as an attachment before the end of class time
(one email per team please).
7.11) This uses data from table 7-5. The data consists of Real Gross Product,
Labor Days, and Real Capital Input in the Manufacturing Sector of Taiwan, 1958 to
1972.
Y = Real gross product (new Taiwan $, in millions)
X2 = Labor input, per thousand persons
X3 = Real capital input (new Taiwan $, in millions)
X4 = Time or trend variable
a)
. reg lny lnl lnk
Source
SS
df
MS
Model
Residual
4.41639958
.073127514
2
12
2.20819979
.006093959
Total
4.4895271
14
.320680507
lny
Coef.
lnl
lnk
_cons
.7147795
1.113473
-7.843845
Std. Err.
.1532679
.2991549
2.67984
t
4.66
3.72
-2.93
Number of obs
F( 2,
12)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.001
0.003
0.013
=
=
=
=
=
=
15
362.36
0.0000
0.9837
0.9810
.07806
[95% Conf. Interval]
.3808375
.4616705
-13.68271
1.048722
1.765276
-2.004975
The output-labor and output-capital elasticities are, 0.7148 and 1.1135, respectively,
and both are individually statistically significant at the 0.005level (one-tail test).
b)
. reg lny lnl
Source
SS
df
MS
Model
Residual
4.33197542
.157551678
1
13
4.33197542
.01211936
Total
4.4895271
14
.320680507
lny
Coef.
lnl
_cons
1.257567
2.069561
Std. Err.
.0665163
.4177431
t
18.91
4.95
Number of obs
F( 1,
13)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
15
357.44
0.0000
0.9649
0.9622
.11009
P>|t|
[95% Conf. Interval]
0.000
0.000
1.113867
1.167082
1.401267
2.97204
Since we have excluded the capital input variable from this model, the estimated outputlabor elasticity of 1.2576 is a biased estimate of the true elasticity; in (a), the true model,
this estimate was 0.7148, which is much smaller than 1.2576.
As noted in the chapter, E(a2) = B2 + B3b32, where b32 is the slope in the regression of
lnX3 on lnX2, which in the present example is 0.488. Using the estimated values in part
a, we therefore see that: E(a2) = 1.2576. Therefore a2 is biased upward.
c)
. reg lny lnk
Source
SS
df
MS
Model
Residual
4.28386128
.205665816
1
13
4.28386128
.015820447
Total
4.4895271
14
.320680507
lny
Coef.
lnk
_cons
2.440904
-19.238
Std. Err.
.1483346
1.774012
t
16.46
-10.84
Number of obs
F( 1,
13)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
=
=
=
=
=
=
15
270.78
0.0000
0.9542
0.9507
.12578
[95% Conf. Interval]
2.120447
-23.07052
By excluding the relevant variable, labor, we are again committing a
2.761362
-15.40548
specification error. By the procedure outlined in (b), it is easy to show that: E(a3) =
1.1135 + 0.7148(1.857) = 2.441. This shows that the estimated elasticity is biased
upward.
---------------------------------------------------------------------------------------------------------
7.14) This uses data from table 7-6.
a)
. reg Y X3
Source
SS
df
MS
Model
Residual
4874.71397
6917.31853
1
26
4874.71397
266.050713
Total
11792.0325
27
436.741944
Y
Coef.
X3
_cons
-4.37562
23.98694
Std. Err.
t
1.022227
5.235037
-4.28
4.58
Number of obs
F( 1,
26)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.000
=
=
=
=
=
=
28
18.32
0.0002
0.4134
0.3908
16.311
[95% Conf. Interval]
-6.476837
13.22617
-2.274403
34.74771
b)
. reg Y X2 X3
Source
SS
df
MS
Model
Residual
6749.45065
5042.58185
2
25
3374.72533
201.703274
Total
11792.0325
27
436.741944
Y
Coef.
X2
X3
_cons
3.943315
-2.499426
3.531812
Std. Err.
1.293445
1.082101
8.111369
t
3.05
-2.31
0.44
Number of obs
F( 2,
25)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.005
0.029
0.667
=
=
=
=
=
=
28
16.73
0.0000
0.5724
0.5382
14.202
[95% Conf. Interval]
1.279416
-4.728055
-13.17387
6.607214
-.2707959
20.23749
c) That Fama is correct in his statement can be seen from the regression results in part
a, and from looking at regression results for X2 regressed on X3.
d)
command for this: drop if Year == 1954 | Year == 1955
. reg Y X2 X3
Source
SS
df
MS
Model
Residual
4834.73239
3893.29415
2
23
2417.36619
169.273659
Total
8728.02654
25
349.121062
Y
Coef.
X2
X3
_cons
4.280255
-1.457611
-3.797987
Std. Err.
1.239397
1.078517
8.128288
t
3.45
-1.35
-0.47
Number of obs
F( 2,
23)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.002
0.190
0.645
=
=
=
=
=
=
26
14.28
0.0001
0.5539
0.5151
13.011
[95% Conf. Interval]
1.716367
-3.688693
-20.61263
6.844144
.7734714
13.01666
As a result of omitting just two observations, the regression results have changed
dramatically. Inflation now has no statistically discernible effect on real rate of return on
common stocks.
e)
Commands for this are:
gen D = 0
replace D = 1 if Year > 1976
. reg Y X2 X3 D
Source
SS
df
MS
Model
Residual
4840.1879
3887.83864
3
22
1613.39597
176.719938
Total
8728.02654
25
349.121062
Y
Coef.
X2
X3
D
_cons
4.253064
-1.602367
1.51557
-3.359139
Std. Err.
t
1.275786
1.375914
8.62583
8.672594
3.33
-1.16
0.18
-0.39
Number of obs
F( 3,
22)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.003
0.257
0.862
0.702
=
=
=
=
=
=
26
9.13
0.0004
0.5546
0.4938
13.294
[95% Conf. Interval]
1.607246
-4.455837
-16.37331
-21.345
6.898881
1.251103
19.40445
14.62672
Since the dummy coefficient is not statistically significant, there does not seem to be
any difference in the behavior of real stock returns between the two periods. Of course,
we are assuming that only intercepts differ between the two periods, but not the slopes.
But this assumption can be tested by introducing a multiplicative dummy variable.
8.22) This uses data from table 8-6. Hypothetical Data on Consumption.
Expenditure (Y), Weekly Income (X2), and Wealth (X3)
a)
. reg Y X2 X3
Source
SS
df
MS
Model
Residual
8607.14137
282.858634
2
7
4303.57068
40.4083763
Total
8890
9
987.777778
Y
Coef.
X2
X3
_cons
.8716397
-.0349521
24.33698
Std. Err.
.314379
.0301199
6.280051
t
2.77
-1.16
3.88
Number of obs
F( 2,
7)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.028
0.284
0.006
=
=
=
=
=
=
10
106.50
0.0000
0.9682
0.9591
6.3568
[95% Conf. Interval]
.1282514
-.1061744
9.487018
1.615028
.0362701
39.18694
b) Collinearity may be present in the data, because despite the high R2 value, only the
coefficient of the income variable is statistically significant. In addition, the wealth
coefficient has the wrong sign.
c)
. reg Y X2
Source
SS
df
MS
Model
Residual
8552.72727
337.272727
1
8
8552.72727
42.1590909
Total
8890
9
987.777778
Y
Coef.
X2
_cons
.5090909
24.45455
Std. Err.
.0357428
6.413817
t
14.24
3.81
Number of obs
F( 1,
8)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
10
202.87
0.0000
0.9621
0.9573
6.493
P>|t|
[95% Conf. Interval]
0.000
0.005
.4266678
9.664256
.591514
39.24483
. reg Y X3
Source
SS
df
MS
Model
Residual
8296.51507
593.484933
1
8
8296.51507
74.1856167
Total
8890
9
987.777778
Y
Coef.
X3
_cons
.0480387
26.45198
Std. Err.
.0045426
8.446165
t
10.58
3.13
Number of obs
F( 1,
8)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
10
111.83
0.0000
0.9332
0.9249
8.6131
P>|t|
[95% Conf. Interval]
0.000
0.014
.0375634
6.975084
.0585139
45.92887
Now individually both slope coefficients are statistically significant and they each have
the correct sign.
d)
. reg X3 X2
Source
SS
df
MS
Model
Residual
3550584.55
44541.4545
1
8
3550584.55
5567.68182
Total
3595126
9
399458.444
X3
Coef.
X2
_cons
10.37273
-3.363636
Std. Err.
.4107525
73.7069
t
25.25
-0.05
Number of obs
F( 1,
8)
Prob > F
R-squared
Adj R-squared
Root MSE
P>|t|
0.000
0.965
=
=
=
=
=
=
10
637.71
0.0000
0.9876
0.9861
74.617
[95% Conf. Interval]
9.42553
-173.3321
11.31992
166.6048
This regression shows that the two variables are highly collinear.
e) We can drop either X2 or X3 from the model. But keep in mind that in that case we will
be committing a specification error. The problem here is that our sample is too small to
isolate the individual impact of income and wealth on consumption expenditure.