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
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).
Since we have excluded the capital input variable from this model, the estimated output-labor 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.
By excluding the relevant variable, labor, we are again committing a
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.
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.
command for this: drop if Year == 1954 | Year == 1955
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.
Commands for this are:
gen D = 0
replace D = 1 if Year > 1976
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)
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.
Now individually both slope coefficients are statistically significant and they each have the correct sign.
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.