1. Consider a multiple regression;
, with 17 observations given in the table:
Y_{} 
x2 
x3 
15 
12 
360 
13 
8 
372 
17 
10 
355 
18 
12 
340 
24 
16 
326 
23 
14 
304 
28 
18 
284 
32 
21 
271 
32 
21 
253 
37 
26 
239 
39 
29 
205 
42 
31 
195 
44 
33 
172 
46 
34 
158 
49 
37 
137 
53 
41 
112 
55 
43 
98 
Based on computer estimation (attach a printout), find:
(a) The least squares estimates b_{1} , b_{2} , and b_{3} for the model coefficients.
From EVIEWS
Dependent Variable: Y 



Method: Least Squares 



Date: 03/26/11 Time: 08:26 



Sample: 1 17 




Included observations: 17 













Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 










C 
44.09476 
14.11416 
3.124150 
0.0075 
X2 
0.481075 
0.258989 
1.857511 
0.0844 
X3 
0.090392 
0.032315 
2.797211 
0.0143 










Rsquared 
0.991784 
Mean dependent var 
33.35294 

Adjusted Rsquared 
0.990611 
S.D. dependent var 
13.63333 

S.E. of regression 
1.321059 
Akaike info criterion 
3.553529 

Sum squared resid 
24.43275 
Schwarz criterion 
3.700567 

Log likelihood 
27.20500 
HannanQuinn criter. 
3.568145 

Fstatistic 
845.0196 
DurbinWatson stat 
1.387993 

Prob(Fstatistic) 
0.000000 













(b) An estimate for the error variance (sigma hat squared).
sigma hat squared = SSR/df = 24.43275/(173) = 1.74519 Note that this is also the square of the " S.E. of regression" in the above table.
(c) An estimate for the variance for b_{2} .
Var(b_{2}) = .258989^{2} = .06707
(d) R^{2}, SSE, SST, and SSR.
R^{2} = .991
SSR = 24.43275
R^{2} = 1SSR/SST ==> .991784 = 1  (24.43275/SST) ==> SST = 2973.795
SST = SSR + SSE ==> 2973.79503 = 24.43275 + SSE ==> SSE = 2949.362
2. For this problem you will need the data in rdchem.wf1 for 32 firms in the chemical industry. This is an EVIEWS workfile. You will need the EVIEWS software to do the homework. The variable rdintens is is expenditures on research and development as a percent of company sales. Sales and R&D expenditures are both measured in millions of dollars. The variable profmarg is profits as a percent of sales; both in millions of dollars. From the data use EVIEWS to construct estimates of the model parameters of
A. Report your estimates of the coefficients and their standard errors.
Dependent Variable: RDINTENS 



Method: Least Squares 



Date: 03/26/11 Time: 08:53 



Sample: 1 32 




Included observations: 32 













Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 










C 
0.469548 
1.676242 
0.280120 
0.7814 
LSALES 
0.321472 
0.215592 
1.491111 
0.1467 
PROFMARG 
0.050167 
0.045780 
1.095830 
0.2822 










Rsquared 
0.098652 
Mean dependent var 
3.265625 

Adjusted Rsquared 
0.036490 
S.D. dependent var 
1.874079 

S.E. of regression 
1.839569 
Akaike info criterion 
4.146000 

Sum squared resid 
98.13642 
Schwarz criterion 
4.283412 

Log likelihood 
63.33599 
HannanQuinn criter. 
4.191548 

Fstatistic 
1.587016 
DurbinWatson stat 
1.652507 

Prob(Fstatistic) 
0.221790 













B. How much of the variation in rdintens is explained by the two independent variables?
You are being asked to report the Rsq = .098.
C. Interpret your estimate of the coefficient on the log of sales. In particular, if sales increases by 10%, what is the estimated percentage increase in rdintens? Is this and economically large effect?
From Chapter 2 of the text (Wooldridge, Introduction to Econometrics) we know that the marginal effect a change in sales is in the third column of this table and the elasticity of rdintens with respect to sales is in the fourth column.
Level  Log 
Our estimate of beta_{1} is .32. The variable rdintens is the ratio of R&D spending to sales. Its mean is 3.26. Therefore the elasticity of RDINTENS with respect to sales is .32*3.26 = 1.04. So, if sales rise 10% then we expect RDINTENS to rise by 10%. If the denominator of RDINTENS rises by 10% and RDINTENS has risen by 10% then R&D spending must have risen 100%, i.e. it doubled! This is an economically meaningful magnitude.
D. At the 5% level, test the hypothesis that sales has no impact on rdintens.
This is just a ttest that we can read out of the table of estimation results. The observed t is 1.49 which is smaller than the critical t of 2.045. Also, the pvalue is much greater than the stipulated 5% in the two tails.
Do not reject the null that sales don't matter.
E. Are sales and profmarg jointly significant in explaining rdintens?
The Fstatistic and pvalue reported in the table are 1.587 and .221 respectively. The critical F(29,.05) = 3.328. We do not reject the null that sales and profmarg do not matter.
3. Some cities are economically dominated by the universities that they host. Examples include University of Illinois in ChampaignUrbana, University of Wisconsin in Madison, University of MIchigan in Ann Arbor, or Penn State in Happy Valley. A perennial complaint in these towns is that the student population drives the monthly rent fo apartments. The data for 127 college towns is in the EVIEWS file rental .wf1. Let rent be average monthly rent paid on apartments in the town, pop denote the total city population, avginc the average city income, and pctstu the student population as a percent of the total population. A model of rental costs in such towns is
A. What do you expect for the sign of the coefficient on log(avginc)?
As income in the community rises we expect rents to rise as well.
B. Use the data to fit the stated model.
Dependent Variable: LRENT 



Method: Least Squares 



Date: 03/26/11 Time: 09:44 



Sample (adjusted): 1 127 



Included observations: 127 after adjustments 












Variable 
Coefficient 
Std. Error 
tStatistic 
Prob. 










C 
3.355510 
0.468364 
7.164325 
0.0000 
LPOP 
0.031448 
0.027184 
1.156853 
0.2496 
LAVGINC 
0.875790 
0.041813 
20.94546 
0.0000 
PCTSTU 
0.006569 
0.001209 
5.434882 
0.0000 










Rsquared 
0.794053 
Mean dependent var 
5.749841 

Adjusted Rsquared 
0.789030 
S.D. dependent var 
0.331448 

S.E. of regression 
0.152239 
Akaike info criterion 
0.895742 

Sum squared resid 
2.850731 
Schwarz criterion 
0.806162 

Log likelihood 
60.87964 
HannanQuinn criter. 
0.859347 

Fstatistic 
158.0806 
DurbinWatson stat 
1.795774 

Prob(Fstatistic) 
0.000000 













C. Was your supposition in part A. confirmed? (Note that your supposition is a testable hypothesis.)
Yes, the supposition is confirmed and the coefficient on LAVGINC is statistically significant (t=20.9 and p = 0.0).
D. Test the hypothesis, at the 1% level, that the effect of log(pop) is five times as great as the effect of pctstu on log(rent).
This is a test of significance of a linear combination of random variables.
At the 1% level the crtical t(123,.005) = +/ 2.616.
The statement of hypothesis is
The test statistic is
with
So the observed t is
This is such a small t that we cannot reject the null.