## Temple University Department of Economics

### Hypothesis Testing Homework

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 b1 , b2 , and b3 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 t-Statistic 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 R-squared 0.991784 Mean dependent var 33.35294 Adjusted R-squared 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 Hannan-Quinn criter. 3.568145 F-statistic 845.0196 Durbin-Watson stat 1.387993 Prob(F-statistic) 0.000000

(b) An estimate for the error variance (sigma hat squared).

sigma hat squared = SSR/df = 24.43275/(17-3) = 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 b2 .

Var(b2) = .2589892 = .06707

(d) R2, SSE, SST, and SSR.

R2 = .991

SSR = 24.43275

R2 = 1-SSR/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 t-Statistic 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 R-squared 0.098652 Mean dependent var 3.265625 Adjusted R-squared 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 Hannan-Quinn criter. 4.191548 F-statistic 1.587016 Durbin-Watson stat 1.652507 Prob(F-statistic) 0.221790

B. How much of the variation in rdintens is explained by the two independent variables?

You are being asked to report the R-sq = .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 beta1 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 t-test 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 p-value 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 F-statistic and p-value 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 Champaign-Urbana, 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 t-Statistic 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 R-squared 0.794053 Mean dependent var 5.749841 Adjusted R-squared 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 Hannan-Quinn criter. -0.859347 F-statistic 158.0806 Durbin-Watson stat 1.795774 Prob(F-statistic) 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.