Temple University
Department of Economics

Introduction to Econometrics

Hypothesis Testing Homework

 

1. Consider a multiple regression;

eqn01, 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 eqn02 (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

 

R&D

 

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

lev-log

 

beta1/x

e-2

 

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

 

rent

 

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

hypo01

The test statistic is

t-stat

 

with

 

Cov01

So the observed t is

obs_t

This is such a small t that we cannot reject the null.