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Chapter 7 Hypothesis Tests and Confidence Intervals in Multiple Regression Multiple Choice 1)

The confidence interval for a single coefficient in a multiple regression a. b. c. d.

makes little sense because the population parameter is unknown. should not be computed because there are other coefficients present in the model. contains information from a large number of hypothesis tests. should only be calculated if the regression R2 is identical to the adjusted R2.

Answer: c 2)

The following linear hypothesis can be tested using the F-test with the exception of a. b. c. d.

β 2 = 1 and β 3 = β 4 / β 5 . β2 = 0 . β1 + β 2 = 1 and β 3 = −2 β 4 . β0 = β1 and β1 = 0.

Answer: a 3)

The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables, a. b. c. d.

stays the same. changes, unless the second explanatory variable is a binary variable. changes. changes, unless you test for a null hypothesis that the addition regression coefficient is zero.

Answer: c

1

4)

All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of a. H 0 : β1 + β 2 = 1 .

β3 = β1 and β 4 = 0 . β2 c. H 0 : β 2 = 0 and β 3 = 0 . d. H 0 : β1 = − β 2 and β1 + β 2 = 1 .

b. H 0 :

Answer: a 5)

When testing joint hypothesis, you should a. use t-statistics for each hypothesis and reject the null hypothesis is all of the restrictions fail. b. use the F-statistic and reject all the hypothesis if the statistic exceeds the critical value. c. use t-statistics for each hypothesis and reject the null hypothesis once the statistic exceeds the critical value for a single hypothesis. d. use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.

Answer: d

6)

The overall regression F-statistic tests the null hypothesis that a. all slope coefficients are zero. b. all slope coefficients and the intercept are zero. c. the intercept in the regression and at least one, but not all, of the slope coefficients is zero. d. the slope coefficient of the variable of interest is zero, but that the other slope coefficients are not.

Answer: a

7)

For a single restriction (q = 1), the F-statistic a. b. c. d.

is the square root of the t-statistic. has a critical value of 1.96. will be negative. is the square of the t-statistic.

Answer: d 2

8)

The homoskedasticity-only F-statistic is given by the following formula ( SSRrestricted − SSRunrestricted ) / q .

SSRunrestricted /(n − kunrestricted − 1) ( SSRrestricted − SSRunrestricted ) / q b. F = .

SSRrestricted /(n − krestricted − 1) ( SSRunrestricted − SSRrestricted ) / q . c. F = SSRunrestricted /(n − kunrestricted − 1) ( SSRrestricted − SSRunrestricted ) /(q − 1) . d. F = SSRunrestricted /(n − kunrestricted )

a. F =

Answer: a 9)

All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of ( SSRrestricted − SSRunrestricted ) / q .

SSRunrestricted /(n − kunrestricted − 1) ( SSRunrestricted − SSRrestricted ) / q . b. F = SSRrestricted /(n − krestricted − 1) ( SSRrestricted − SSRunrestricted ) n − kunrestricted − 1 × . c. F = SSRunrestricted q

a. F =

⎛ SSRrestricted ⎞ (n − kunrestricted − 1) d. F = ⎜ . − 1⎟ × q ⎝ SSRunrestricted ⎠

Answer: b 10)

In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated a. by dividing the estimate by its standard error. b. from the square root of the F-statistic. c. by multiplying the p-value by 1.96. d. using the adjusted R2 and the confidence interval.

Answer: a

3

11)

To test joint linear hypotheses in the multiple regression model, you need to a. compare the sums of squared residuals from the restricted and unrestricted model. b. use the heteroskedasticity-robust F-statistic. c. use several t-statistics and perform tests using the standard normal distribution. d. compare the adjusted R2 for the model which imposes the restrictions, and the unrestricted model.

Answer: b

12)

The homoskedasticity-only F-statistic is given by the following formula a. F =

2 2 ( Runrestricted − Rrestricted /q .

2 1 − ( Runrestricted ) /(n − kunrestricted − 1)

b. F =

2 1 − Runrestricted /q .

2 Runrestricted /(n − kunrestricted − 1)

c. F =

2 2 ( Runrestricted − Runrestricted /q .

2 1 − ( Runrestricted ) /(n − krestricted − 1)

d. F =

2 2 ( Rrestricted − Runrestricted /q .

2 1 − ( Rrestricted ) /(n − krestricted − 1)

Answer: a 13)

2 2 Let Runrestricted and Rrestricted be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is

a. b. c. d.

4.61.

8.01.

10.34.

7.71.

Answer: b

4

14)

If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal to one, then you should a. subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the resulting ratio is larger than 1.96. b. add and subtract 1.96 from the slope and check if that interval includes 1. c. see if the slope coefficient is between 0.95 and 1.05. d. check if the adjusted R2 is close to 1.

Answer: a

15)

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution you can a. b. c. d.

safely assume that your regression results are significant. reject the null hypothesis. reject the assumption that the error terms are homoskedastic. conclude that most of the actual values are very close to the regression line.

Answer: b 16)

If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then a. b. c. d.

a series of t-tests may or may not give you the same conclusion. the regression is always significant. all of the hypotheses are always simultaneously rejected. the F-statistic must be negative.

Answer: a 17)

When your multiple regression function includes a single omitted variable regressor, then a. use a two-sided alternative hypothesis to check the influence of all included variables. b. the estimator for your included regressors will be biased if at least one of the included variables is correlated with the omitted variable. c. the estimator for your included regressors will always be biased. d. lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance of the coefficients of the included variables.

Answer: b

5

18)

A 95% confidence set for two or more coefficients is a set that contains a. b. c. d.

the sample values of these coefficients in 95% of randomly drawn samples. integer values only. the same values as the 95% confidence intervals constructed for the coefficients. the population values of these coefficients in 95% of randomly drawn samples.

Answer: d 19)

When there are two coefficients, the resulting confidence sets are a. b. c. d.

rectangles. ellipses. squares. trapezoids.

Answer: b 20)

When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot reject the null hypothesis at the 5% level, if the ellipse contains the point a. b. c. d.

(-1.96, 1.96). |(0, 1.96)|.

(0,0).

(1.962, 1.962).

Answer: c 21)

The OLS estimators of the coefficients in multiple regression will have omitted variable bias a. only if an omitted determinant of Yi is a continuous variable. b. if an omitted variable is correlated with at least one of the regressors, even though it is not a determinant of the dependent variable. c. only if the omitted variable is not normally distributed. d. if an omitted determinant of Yi is correlated with at least one of the regressors.

Answer: d

6

22)

At a mathematical level, if the two conditions for omitted variable bias are satisfied, then a. b. c. d.

E (ui | X 1i , X 2i ,..., X ki ) ≠ 0 . there is perfect multicollinearity. large outliers are likely: X 1i , X 2i ,..., X ki and Yi have infinite fourth moments. ( X 1i , X 2i ,..., X ki , Yi ), i = 1,..., n are not i.i.d. draws from their joint distribution.

Answer: a 23)

All of the following are true, with the exception of one condition: 2

a. a high R 2 or R does not mean that the regressors are a true cause of the dependent variable.

2

b. a high R 2 or R does not mean that there is no omitted variable bias.

2

c. a high R 2 or R always means that an added variable is statistically significant.

2

d. a high R 2 or R does not necessarily mean that you have the most appropriate set of regressors.

Answer: c 24)

The general answer to the question of choosing the scale of the variables is a. b. c. d.

dependent on you whim. to make the regression results easy to read and to interpret. to ensure that the regression coefficients always lie between -1 and 1. irrelevant because regardless of the scale of the variable, the regression coefficient is unaffected.

Answer: b 25)

If the estimates of the coefficients of interest change substantially across specifications, a. then this can be expected from sample variation. b. then you should change the scale of the variables to make the changes appear to be smaller. c. then this often provides evidence that the original specification had omitted variable bias. d. then choose the specification for which your coefficient of interest is most significant.

Answer: c

7

Essays and Longer Questions

1)

The F-statistic with q = 2 restrictions when testing for the restrictions β1 = 0 and β 2 = 0 is given by the following formula: ⎛ 2 2 l tt ⎞ 1 ⎜ t1 + t2 − 2 ρ t1 ,t2 1 ⎟ F= 2 ⎟ l 2⎜ 1− ρ t1 ,t2 ⎝ ⎠ Discuss how this formula can be understood intuitively.

Answer: For the case when there is no correlation between the two explanatory variables, the formula reduces to a simple average of the squared t-statistics, i.e., 1 F = ( t12 + t22 ) . The F2,∞ distribution is the distribution of a random variable 2 with a chi-squared distribution with 2 degrees of freedom, divided by 2.

Equivalently, the F2,∞ distribution is the distribution of the average of 2 squared standard normal random variables. Because the t-statistics are uncorrelated by assumption, they are independent standard normal random variables under the null hypothesis. If either β1 or β 2 are nonzero (or both), then either t12 or t22 or both will be large. This leads to a large F-statistic, and hence a rejection of the null hypothesis.

2)

The cost of attending your college has once again gone up. Although you have been told that education is investment in human capital, which carries a return of roughly 10% a year, you (and your parents) are not pleased. One of the administrators at your university/college does not make the situation better by telling you that you pay more because the reputation of your institution is better than that of others. To investigate this hypothesis, you collect data randomly for 100 national universities and liberal arts colleges from the 2000-2001 U.S. News and World Report annual rankings. Next you perform the following regression n = 7,311.17 + 3,985.20 × Reputation – 0.20 × Size Cost (2,058.63) (664.58) (0.13)

+ 8,406.79 × Dpriv – 416.38 × Dlibart – 2,376.51 × Dreligion (2,154.85) (1,121.92) (1,007.86)

R2=0.72, SER = 3,773.35

8

where Cost is Tuition, Fees, Room and Board in dollars, Reputation is the index used in U.S. News and World Report (based on a survey of university presidents and chief academic officers), which ranges from 1 (“marginal”) to 5 (“distinguished”), Size is the number of undergraduate students, and Dpriv, Dlibart, and Dreligion are binary variables indicating whether the institution is private, a liberal arts college, and has a religious affiliation. The numbers in parentheses are heteroskedasticity-robust standard errors. (a)

Indicate whether or not the coefficients are significantly different from zero.

Answer: The coefficient on liberal arts colleges, is not significantly different from zero.

All other coefficients are statistically significant at conventional levels, with the exception of the size coefficient, which carries a t-statistic of 1.54, and hence is not statistically significant at the 5% level (using a one-sided alternative hypothesis).

(b)

What is the p-value for the null hypothesis that the coefficient on Size is equal to zero?

Based on this, should you eliminate the variable from the regression? Why or why not?

Answer: Using a one-sided alternative hypothesis, the p-value is 6.2 percent. Variables should not be eliminated simply on grounds of a statistical test. The sign of the coefficient is as expected, and its magnitude makes it important. It is best to leave the variable in the regression and let the reader decide whether or not this is convincing evidence that the size of the university matters.

(c)

You want to test simultaneously the hypotheses that β size = 0 and β Dlibart = 0 . Your regression package returns the F-statistic of 1.23. Can you reject the null hypothesis?

Answer: The critical value for F2,∞ is 3.00 (5% level) and 4.61 (1% level). Hence you cannot reject the null hypothesis in this case.

(d)

Eliminating the Size and Dlibart variables from your regression, the estimation regression becomes n = 5,450.35 + 3,538.84 × Reputation + 10,935.70 × Dpriv – 2,783.31 × Dreligion; Cost (1,772.35) (590.49) (875.51) (1,180.57)

R2=0.72, SER = 3,792.68 Why do you think that the effect of attending a private institution has increased now?

Answer: Private institutions are smaller, on average, and some of these are liberal arts colleges. Both of these variables had negative coefficients.

9

(e)

You give a final attempt to bring the effect of Size back into the equation by forcing the assumption of homoskedasticity onto your estimation. The results are as follows: n = 7,311.17 + 3,985.20 × Reputation – 0.20 × Size Cost (1,985.17) (593.65) (0.07)

+ 8,406.79 × Dpriv – 416.38 × Dlibart – 2,376.51 × Dreligion (1,423.59) (1,096.49) (989.23)

R2=0.72, SER = 3,682.02 Calculate the t-statistic on the Size coefficient and perform the hypothesis test that its coefficient is zero. Is this test reliable? Explain.

Answer: Although the coefficient would be statistically significant in this case, the test is unreliable and should not be used for statistical inference. There is no theoretical suggestion here that the errors might be homoskedastic. Since the standard errors are quite different here, you should use the more reliable ones, i.e., the heteroskedasticity-robust.

3)

In the multiple regression model with two explanatory variables

Yi = β 0 + β 1 X 1i + β 2 X 2i + u i the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in zi = Z i − Z ): βˆ = Y − βˆ X − βˆ X 0

n

1

n

1

2

2

n

n

∑ y x ∑x −∑ y x ∑x x

βˆ1 = i =1

i 1i

2 2i

i =1

i 2i

i =1

n

n

n

i =1

i =1

i =1

i =1

1i 2 i

∑ x12i ∑ x22i − (∑ x1i x2i )2

n

n

n

i =1 n 2 1i i =1

i =1

n

∑ yi x2i ∑ x12i − ∑ yi x1i ∑ x1i x2i

βˆ2 = i =1

n

i =1

n

∑ x ∑ x − (∑ x x ) i =1

2 2i

i =1

2

1i 2 i

You have collected data for 104 countries of the world from the Penn World Tables and want to estimate the effect of the population growth rate (X1i) and the saving rate (X2i) (average investment share of GDP from 1980 to 1990) on GDP per worker (relative to 10

the U.S.) in 1990. The various sums needed to calculate the OLS estimates are given below: n

n

n

i =1

i =1

i =1

∑ Yi = 33.33; ∑ X1i = 2.025; ∑ X 2i = 17.313 n

n

n

∑ y = 8.3103; ∑ x = .0122; ∑ x = 0.6422 i =1

n

2 i

i =1

2 1i

2 2i

i =1

n

n

∑ y x = −0.2304; ∑ y x = 1.5676; ∑ x x = −0.0520 i =1

i 1i

i =1

i 2i

i =1

1i 2 i

The heteroskedasticity-robust standard errors of the two slope coefficients are 1.99 (for population growth) and 0.23 (for the saving rate). Calculate the 95% confidence interval for both coefficients. How many standard deviations are the coefficients away from zero?

Answer: The 95% confidence interval for the population growth is (–16.85, -9.05), and the 95% confidence interval for the saving rate is (0.94, 1.84). The population growth coefficient has a t-statistic of -6.51, and the saving rate coefficient of 6.04. These represent standard deviations away from zero.

4)

A subsample from the Current Population Survey is taken, on weekly earnings of individuals, their age, and their gender. You have read in the news that women make 70 cents to the $1 that men earn. To test this hypothesis, you first regress earnings on a constant and a binary variable, which takes on a value of 1 for females and is 0 otherwise.

The results were: n = 570.70 - 170.72 × Female, R2=0.084, SER = 282.12.

Earn (9.44) (13.52)

(a)

Perform a difference in means test and indicate whether or not the difference in the mean salaries is significantly different. Justify your choice of a one-sided or two-sided alternative test. Are these results evidence enough to argue that there is discrimination against females? Why or why not? Is it likely that the errors are normally distributed in this case? If not, does that present a problem to your test?

Answer: The t-statistic is -12.63, while the critical value is –1.64. The difference is therefore statistically significant. A one-sided alternative was chosen since the claim is that females make less than males. This represents little evidence of discrimination, since attributes of males and females have not been included.

Given that earnings distributions are not normally distributed, the errors will also not be distributed normally, and assuming that they are, results in problematic inference.

11

(b)

Test for the significance of the age and gender coefficients. Why do you think that age plays a role in earnings determination?

Answer: The t-statistics are 9.36 for the age coefficient, and -13.00 for the gender coefficient. Both of these values are greater than the (absolute) critical value from the standard normal distribution (1.64). Hence you can reject the null hypothesis that these coefficients are zero. Age proxies “on the job training.” A better proxy that has been used frequently in the past is the Mincer experience variable (Age-Education-6). Obviously this is a better proxy for some subsample of individuals than for others.

5) You have collected data from Major League Baseball (MLB) to find the determinants of winning. You have a general idea that both good pitching and strong hitting are needed to do well. However, you do not know how much each of these contributes separately. To investigate this problem, you collect data for all MLB during 1999 season. Your strategy is to first regress the winning percentage on pitching quality (“Team ERA”), second to regress the same variable on some measure of hitting (“OPS – On-base Plus Slugging percentage”), and third to regress the winning percentage on both.

Summary of the Distribution of Winning Percentage, On Base plus Slugging percentage, and Team Earned Run Average for MLB in 1999 Average Standard deviation

Percentile 10%

25%

40%

Team ERA OPS

4.71

0.53

3.84

4.35

4.72

50% (median) 4.78

0.778

0.034

0.720

0.754

0.769

0.780

0.790 0.798 0.820

Winning Percentage

0.50

0.08

0.40

0.43

0.46

0.48

0.49

The results are as follows: n = 0.94 – 0.100 × teamera , R2 = 0.49, SER = 0.06.

Winpct (0.08) (0.017) n = –0.68 + 1.513 × ops , R2=0.45, SER = 0.06.

Winpct (0.17) (0.221)

12

60%

75%

90%

4.91

5.06

5.25

0.59

0.60

n = –0.19 – 0.099 × teamera + 1.490 × ops , R2=0.92, SER = 0.02.

Winpct (0.08) (0.008) (0.126)

(a)

Use the t-statistic to test for the statistical significance of the coefficient.

Answer: The t-statistics for team ERA and OPS are -12.38 and 11.83. Both of these are highly significant.

(b)

There are 30 teams in MLB. Does the small sample size worry you here when testing for significance?

Answer: The t-statistic is only normally distributed in large samples. As a result, inference is problematic here.

6)

In the process of collecting weight and height data from 29 female and 81 male students at your university, you also asked the students for the number of siblings they have.

Although it was not quite clear to you initially what you would use that variable for, you construct a new theory that suggests that children who have more siblings come from poorer families and will have to share the food on the table. Although a friend tells you that this theory does not pass the “straight-face” test, you decide to hypothesize that peers with many siblings will weigh less, on average, for a given height. In addition, you believe that the muscle/fat tissue composition of male bodies suggests that females will weigh less, on average, for a given height. To test these theories, you perform the following regression: n = –229.92 – 6.52 × Female + 0.51 × Sibs+ 5.58 × Height, Studentw (44.01) (5.52) (2.25) (0.62)

R2=0.50, SER = 21.08 where Studentw is in pounds, Height is in inches, Female takes a value of 1 for females and is 0 otherwise, Sibs is the number of siblings (heteroskedasticity-robust standard errors in parentheses). (a)

Carrying out hypotheses tests using the relevant t-statistics to test your two claims separately, is there strong evidence in favor of your hypotheses? Is it appropriate to use two separate tests in this situation?

Answer: The t-statistics for gender and number of siblings are -1.18 and 0.23 respectively. Neither coefficient is statistically significant at conventional levels. If you wanted to test the two hypothesis simultaneously, then you should use an F-test.

13

(b)

You also perform an F-test on the joint hypothesis that the two coefficients for females and siblings are zero. The calculated F-statistic is 0.84. Find the critical value from the Ftable. Can you reject the null hypothesis? Is it possible that one of the two parameters is zero in the population, but not the other?

Answer: The critical value is 3.00 at the 5% level, and 4.61 at the 1% level. Hence you cannot reject the null hypothesis. The hypothesis is that both coefficients are zero, and this cannot be rejected. Had you rejected the null hypothesis, then the alternative hypothesis states that one or both of the restrictions do not hold.

(c)

You are now a bit worried that the entire regression does not make sense and therefore also test for the height coefficient to be zero. The resulting F-statistic is 57.25. Does that prove that there is a relationship between weight and height?

Answer: Although you cannot prove anything in this context with certainty, there is a very high probability that there is a relationship between height and weight in the population, given the sample result. The critical value from the F-table is 3.78 at the 1% level.

7)

You have collected data for 104 countries to address the difficult questions of the determinants for differences in the standard of living among the countries of the world.

You recall from your macroeconomics lectures that the neoclassical growth model suggests that output per worker (per capita income) levels are determined by, among others, the saving rate and population growth rate. To test the predictions of this growth model, you run the following regression: n = 0.339 – 12.894 × n + 1.397 × sK , R2=0.621, SER = 0.177 RelPersInc (0.068) (3.177) (0.229)

where RelPersInc is GDP per worker relative to the United States, n is the average population growth rate, 1980-1990, and sK is the average investment share of GDP from 1960 to1990 (remember investment equals saving). Numbers in parentheses are for heteroskedasticity-robust standard errors. (a)

Calculate the t-statistics and test whether or not each of the population parameters are significantly different from zero.

Answer: The t-statistics for population growth and the saving rate are –4.06 and 6.10, making both coefficients significantly different from zero at conventional levels of significance.

14

(b)

The overall F-statistic for the regression is 79.11. What is the critical value at the 5% and 1% level? What is your decision on the null hypothesis?

Answer: The critical value is 3.00 and 4.61 respectively, allowing you to reject the null hypothesis that all slope coefficients are zero.

(c)

You remember that human capital in addition to physical capital also plays a role in determining the standard of living of a country. You therefore collect additional data on the average educational attainment in years for 1985, and add this variable (Educ) to the above regression. This results in the modified regression output: n = 0.046 – 5.869 × n + 0.738 × sK + 0.055 × Educ, R2=0.775, SER = 0.1377 RelPersInc (0.079) (2.238) (0.294) (0.010)

How has the inclusion of Educ affected your previous results?

Answer: The coefficient on the population growth rate is roughly half of what it was originally, while the coefficient on the saving rate has approximately doubled.

The regression R2 has increased significantly. (d)

Upon checking the regression output, you realize that there are only 86 observations, since data for Educ is not available for all 104 countries in your sample. Do you have to modify some of your statements in (d)?

Answer: When comparing results, you should ensure that the sample is identical, since comparisons are not valid otherwise. In addition, there are now less than 100 observations, making inference based on the standard normal distribution problematic.

8)

Attendance at sports events depends on various factors. Teams typically do not change ticket prices from game to game to attract more spectators to less attractive games.

However, there are other marketing tools used, such as fireworks, free hats, etc., for this purpose. You work as a consultant for a sports team, the Los Angeles Dodgers, to help them forecast attendance, so that they can potentially devise strategies for price discrimination. After collecting data over two years for every one of the 162 home games of the 2000 and 2001 season, you run the following regression: n Attend = 15,005 + 201 × Temperat + 465 × DodgNetWin + 82 × OppNetWin (8,770) (121) (169) (26)

+ 9647 × DFSaSu + 1328 × Drain + 1609 × D150m + 271 × DDiv – 978 × D2001; (1505) (3355) (1819) (1,184) (1,143)

R2=0.416, SER = 6983 15

where Attend is announced stadium attendance, Temperat it the average temperature on game day, DodgNetWin are the net wins of the Dodgers before the game (wins-losses), OppNetWin is the opposing team’s net wins at the end of the previous season, and DFSaSu, Drain, D150m, Ddiv, and D2001 are binary variables, taking a value of 1 if the game was played on a weekend, it rained during that day, the opposing team was within a 150 mile radius, the opposing team plays in the same division as the Dodgers, and the game was played during 2001, respectively. Numbers in parentheses are heteroskedasticity- robust standard errors. (a)

Are the slope coefficients statistically significant?

Answer: The t-statistics for Temperat, DodgNewWin, OppNetWin, and DFSaSu are all statistically significant at the 5% level, using a one-sided test. The constant is insignificant using a two-sided test. All the other coefficients are not statistically significant at the 5% level.

(b)

To test whether the effect of the last four binary variables is significant, you have your regression program calculate the relevant F-statistic, which is 0.295. What is the critical value? What is your decision about excluding these variables?

Answer: The critical value at the 5% level is 2.37. Hence you cannot reject the null hypothesis that all four coefficients are simultaneously zero.

9)

The administration of your university/college is thinking about implementing a policy of coed floors only in dormitories. Currently there are only single gender floors. One reason behind such a policy might be to generate an atmosphere of better “understanding” between the sexes. The Dean of Students (DoS) has decided to investigate if such a behavior results in more “togetherness” by attempting to find the determinants of the gender composition at the dinner table in your main dining hall, and in that of a neighboring university, which only allows for coed floors in their dorms. The survey includes 176 students, 63 from your university/college, and 113 from a neighboring institution.

The Dean’s first problem is how to define gender composition. To begin with, the survey excludes single persons’ tables, since the study is to focus on group behavior. The Dean also eliminates sports teams from the analysis, since a large number of single-gender students will sit at the same table. Finally, the Dean decides to only analyze tables with three or more students, since she worries about “couples” distorting the results. The Dean finally settles for the following specification of the dependent variable:

GenderComp=|(50%-% of Male Students at Table)| Where “|Z|” stands for absolute value of Z. The variable can take on values from zero to fifty.

16

After considering various explanatory variables, the Dean settles for an initial list of eight, and estimates the following relationship, using heteroskedasticity-robust standard errors (this Dean obviously has taken an econometrics course earlier in her career and/or has an able research assistant): n = 30.90 – 3.78 × Size – 8.81 × DCoed + 2.28 × DFemme +2.06 × DRoommate GenderComp (7.73) (0.63) (2.66) (2.42) (2.39)

- 0.17 × DAthlete + 1.49 × DCons – 0.81 SAT + 1.74 × SibOther, R2=0.24, SER = 15.50 (3.23) (1.10) (1.20) (1.43) where Size is the number of persons at the table minus 3, DCoed is a binary variable, which takes on the value of 1 if you live on a coed floor, DFemme is a binary variable, which is 1 for females and zero otherwise, DRoommate is a binary variable which equals 1 if the person at the table has a roommate and is zero otherwise, DAthlete is a binary variable which is 1 if the person at the table is a member of an athletic varsity team, DCons is a variable which measures the political tendency of the person at the table on a seven-point scale, ranging from 1 being “liberal” to 7 being “conservative,” SAT is the SAT score of the person at the table measured on a seven-point scale, ranging from 1 for the category “900-1000” to 7 for the category “1510 and above,” and increasing by one for 100 point increases, and SibOther is the number of siblings from the opposite gender in the family the person at the table grew up with. (a)

Indicate which of the coefficients are statistically significant.

Answer: Only the constant, Size, and DCoed are statistically significant at the 5% level.

(b)

Based on the above results, the Dean decides to specify a more parsimonious form by eliminating the least significant variables. Using the F-statistic for the null hypothesis that there is no relationship between the gender composition at the table and DFemme, DRoommate, DAthlete, and SAT, the regression package returns a value of 1.10. What are the degrees of freedom for the statistic? Look up the 1% and 5% critical values from the F- table and make a decision about the exclusion of these variables based on the critical values.

Answer: The F4, ∞ is 2.37 at the 5% level, and 3.32 at the 1% level. Hence you cannot reject the null hypothesis that all four coefficients are zero.

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(c)

The Dean decides to estimate the following specification next: n = 29.07 – 3.80 × Size – 9.75 × DCoed + 1.50 × DCons + 1.97 × SibOther, GenderComp (3.75) (0.62) (1.04) (1.04) (1.44)

R2=0.22 SER = 15.44 Calculate the t-statistics for the coefficients and discuss whether or not the Dean should attempt to simplify the specification further. Based on the results, what might some of the comments be that she will write up for the other senior administrators of your college?

What are some of the potential flaws in her analysis? What other variables do you think she should have considered as explanatory factors?

Answer: The t-statistics for the five coefficients are as follows: 7.75, -6.13, -9.38, 1.44 and 1.37. The Dean should leave the specification as is and allow readers to decide if they want to place much weight on the insignificant coefficients. The variable of interest is DCoed and she will most likely focus on that, concluding that having coed floors in dormitories will increase the gender balance at dining hall tables. She will most likely go further in her report and suggest that communication between the sexes will improve as a result of coed floors.

One of the major flaws in the analysis is that students from one college do not have coed floors in dormitories while students from the other college do not have single gender floors. Ideally you would like to survey students from the same college where some of the students lived on single gender floors while others did not. Answers on omitted variables will obviously vary. Ideally some survey question should be included which would indicate the student’s attitude towards the other sex.

10)

The Solow growth model suggests that countries with identical saving rates and population growth rates should converge to the same per capita income level. This result has been extended to include investment in human capital (education) as well as investment in physical capital. This hypothesis is referred to as the “conditional convergence hypothesis,” since the convergence is dependent on countries obtaining the same values in the driving variables. To test the hypothesis, you collect data from the Penn World Tables on the average annual growth rate of GDP per worker (g6090) for the 1960-1990 sample period, and regress it on the (i) initial starting level of GDP per worker relative to the United States in 1960 (RelProd60), (ii) average population growth rate of the country (n), (iii) average investment share of GDP from 1960 to1990 (sK - remember investment equals savings), and (iv) educational attainment in years for 1985 (Educ). The results for close to 100 countries is as follows (numbers in parentheses are for heteroskedasticity-robust standard errors):

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gn 6090 = 0.004 - 0.172 × n + 0.133 × sK + 0.002 × Educ – 0.044×RelProd60, (0.007) (0.209) (0.015) (0.001) (0.008) 2 R =0.537, SER = 0.011

(a)

Is the coefficient on this variable significantly different from zero at the 5% level? At the 1% level?

Answer: The coefficient has a t-statistic of 5.50 and is therefore statistically significant at both the 5% and the 1% level.

(b)

Test for the significance of the other slope coefficients. Should you use a one-sided alternative hypothesis or a two-sided test? Will the decision for one or the other influence the decision about the significance of the parameters? Should you always eliminate variables which carry insignificant coefficients?

Answer: The t-statistics are –0.82. 8.87, and 2.00. Hence the coefficient on population growth is not statistically significant. You should use a one-sided alternative hypothesis test since economic theory gives you information about the expected sign on these variables. In the above case, the decision will not be influenced by the choice of a one-sided or two-sided test, since the (absolute value of the) critical value is 1.64 or 1.96 at the 5% significance level. If there is a strong prior on the sign of the coefficient, then the variable should not be eliminated based on the significance test. Instead it should be left in the equation, but the low p-value should be flagged to the reader, and the reader should decide herself how convincing the evidence is in favor of the theory.

Mathematical and Graphical Problems 1)

Explain carefully why testing joint hypotheses simultaneously, using the F-statistic, does not necessarily yield the same conclusion as testing them sequentially (“one at a time” method), using a series of t-statistics.

Answer: Testing a joint hypothesis sequentially does not result in the desired significance level. Even if this were not a problem, then the shape of the confidence set of the textbook suggests another reason for this strategy to be problematic. Drawing a confidence interval for both parameters and extending the lines up and to the right, results in a rectangle, indicating the area where the joint hypothesis would be rejected using the t-statistic. Obviously the confidence set does not coincide with the rectangle, and there are therefore various outcomes possible under which both strategies would come to the same conclusion or different conclusions. Since the proper testing strategy involves using the F-statistic, the t-statistic could result in improper inference under circumstances where the two areas do not coincide.

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2)

Set up the null hypothesis and alternative hypothesis carefully for the following cases:

(a)

k = 4, test for all coefficients other than the intercept to be zero Answer: H 0 : β1 = 0, β 2 = 0, β3 = 0, β 4 = 0

(b)

k = 3, test for the slope coefficient of X 1 to be unity, and the coefficients on the other explanatory variables to be zero Answer: H 0 : β1 = 1, β 2 = 0, β 3 = 0

(c)

k = 10, test for the slope coefficient of X 1 to be zero, and for the slope coefficients of X 2 and X 3 to be the same but of opposite sign.

Answer: H 0 : β1 = 0, β 2 + β 3 = 0

(d)

k = 4, test for the slope coefficients to add up to unity Answer: H 0 : β1 + β 2 + β 3 + β 4 = 1

3)

Consider a situation where economic theory suggests that you impose certain restrictions on your estimated multiple regression function. These may involve the equality of parameters, such as the returns to education and on the job training in earnings functions, or the sum of coefficients, such as constant returns to scale in a production function. To test the validity of your restrictions, you have your statistical package calculate the corresponding F-statistic. Find the critical value from the F-distribution at the 5% and 1% level, and comment whether or not you will reject the null hypothesis in each of the following cases.

(a)

number of observations: 152; number of restrictions: 3; F-statistic: 3.21 Answer: F3,∞ = 2.60 (5% level), F3,∞ = 3.78 (1% level). Reject the null hypothesis at the 5% level, but not at the 1% level.

(b)

number of observations: 1,732; number of restrictions:7; F-statistic: 4.92 Answer: F7,∞ = 2.01 (5% level), F7,∞ = 2.64 (1% level). Reject the null hypothesis at the 5% level and at the 1% level.

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(c)

number of observations: 63; number of restrictions: 1; F-statistic: 2.47 Answer: F1,∞ = 3.84 (5% level), F1,∞ = 6.63 (1% level). Cannot reject the null hypothesis at the 5% level or at the 1% level.

(d)

number of observations: 4,000; number of restrictions: 5; F-statistic: 1.82 Answer: F5,∞ = 2.21 (5% level), F5,∞ = 3.02 (1% level). Cannot reject the null hypothesis at the 5% level or at the 1% level.

(e)

Explain why you can use the Fq,∞ distribution to compute the critical values in (a)-(d).

Answer: The F-statistic is distributed Fq,∞ in large samples. Although strictly speaking this only holds for the limiting case of n = ∞ , for practical purposes the approximation is close for n > 100. This is therefore problematic for (c) above, where n = 63.

4)

Females, on average, are shorter and weigh less than males. One of your friends, who is a pre-med student, tells you that in addition, females will weigh less for a given height. To test this hypothesis, you collect height and weight of 29 female and 81 male students at your university. A regression of the weight on a constant, height, and a binary variable, which takes a value of one for females and is zero otherwise, yields the following result: n = –229.21 – 6.36 × Female + 5.58 × Height , R2=0.50, SER = 20.99 Studentw (43.39) (5.74) (0.62)

where Studentw is weight measured in pounds and Height is measured in inches (heteroskedasticity-robust standard errors in parentheses).

Calculate t-statistics and carry out the hypothesis test that females weigh the same as males, on average, for a given height, using a 10% significance level. What is the alternative hypothesis? What is the p-value? What critical value did you use?

Answer: The t-statistics for the intercept, the gender binary variable, and the height variable are -5.28, -1.11, and 9.00, respectively. For a one-sided alternative hypothesis, β Female < 0 , the critical value from the standard normal table is –1.28. Hence you cannot reject the null hypothesis at the 10% level. The pvalue is 13.4%.

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5)

You are presented with the following output from a regression package, which reproduces the regression results of testscores on the student-teacher ratio from your textbook Dependent Variable: TESTSCR Method: Least Squares Date: 07/30/06 Time: 17:44 Sample: 1 420 Included observations: 420 Variable C STR R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient

Std. Error

t-Statistic

Prob.

698.93 -2.28

9.47 0.48

73.82 -4.75

0.00 0.00

0.05 0.05 18.58 144315.48 -1822.25 0.13

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 8.69 8.71 22.58 0.00

Std. Error are homoskedasticity only standard errors. a)

What is the relationship between the t-statistic on the student-teacher ratio coefficient and the F-statistic?

Answer: The F-statistic tests the null hypothesis that all slope coefficients are zero. In the case of a single explanatory variable, this is the same as testing for the significance of the explanatory variable coefficient. In that case, the F-statistic is the same as the square of the t-statistic in the case of a single restriction (q = 1).

b)

Next, two explanatory variables, the percent of English learners (EL_PCT) and expenditures per student (EXPN_STU) are added. The output is listed as below. What is the relationship between the three t-statistics for the slopes and the homoskedasticity-only F-statistic now?

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Dependent Variable: TESTSCR Method: Least Squares Date: 07/30/06 Time: 17:55 Sample: 1 420 Included observations: 420 Variable C STR EL_PCT EXPN_STU R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient

Std. Error

t-Statistic

Prob.

649.58 -0.29 -0.66 0.00

15.21 0.48 0.04 0.00

42.72 -0.60 -16.78 2.74

0.00 0.55 0.00 0.01

0.44 0.43 14.35 85699.71 -1712.81 0.74

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 8.18 8.21 107.45 0.00

Answer: There is no simple relationship between the F-statistic and the three t-statistics now. The F-statistic tests the null hypothesis that H 0 : β STR = β EL _ PCT = β EXPN _ STU = 0 simultaneously. The t-statistics test the

significance of each slope coefficient separately.

6)

Consider the following multiple regression model

Yi = β 0 + β 1 X 1i + β 2 X 2i + β 3 X 3i + u i You want to consider certain hypotheses involving more than one parameter, and you know that the regression error is homoskedastic. You decide to test the joint hypotheses using the homoskedasticity-only F-statistics. For each of the cases below specify a restricted model and indicate how you would compute the F-statistic to test for the validity of the restrictions. (a)

β1 = − β 2 ; β 3 = 0 Answer: The restricted model is Yi = β 0 + β 2 ( X 2i − X 1i ) + ui and the rule-of-thumb F( SSRrestricted − SSRunrestricted ) / 2 statistic would be F = .

SSRunrestricted / n − 3 − 1)

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(b)

β1 + β 2 + β 3 = 1

(c)

Answer: (Yi − X 3i ) = β 0 + β1 ( X 1i − X 3i ) + β 2 ( X 2i − X 3i ) + ui and the rule-of-thumb F( SSRrestricted − SSRunrestricted ) /1 statistic would be F = SSRunrestricted / n − 3 − 1) β1 = β 2 ; β 3 = 0 Answer: Yi = β 0 + ( β1 X 1i + X 2i ) + ui and the homoskedasticity-only F-statistic would be

F=

7)

( SSRrestricted − SSRunrestricted ) / 2 SSRunrestricted /(n − 3 − 1)

( SSRrestricted − SSRunrestricted ) / q . Name conditions SSRunrestricted /(n − kunrestricted − 1) under which the F-statistic is large and hence rejects the null hypothesis.

Give an intuitive explanation for F =

Answer: First rewrite

F=

( SSRrestricted − SSRunrestricted ) / q ( SSRrestricted − SSRunrestricted ) (n − kunrestricted − 1) = × SSRunrestricted /(n − kunrestricted − 1) SSRunrestricted q

The numerator for the first expression is the difference between the sum of squared residuals between the restricted and the unrestricted model. Anytime you place restrictions on the model, the SSR will increase (or, strictly speaking, at least no decrease). Hence if the explanatory power (SSR) of your regression decreases (increase) by much as a result of the restrictions you have placed on the model, then the numerator will be large. However, the SSR depend on units of measurement. To make the first expression independent of the units of measurement, the difference is divided by the unrestricted residual sums of squares. The first fraction now represents the percentage increase in the SSR that result from the imposition of the restrictions. The second fraction has the degrees of freedom of the denominator in its numerator, and the degrees of freedom of the numerator in its denominator. The degrees of freedom of the numerator is the difference of the degrees of freedom of the restricted and the unrestricted regression respectively, i.e., (n − krestricted − 1) − (n − kunrestricted − 1) = kunrestricted − krestricted = q .

As the degrees of freedom (number of observations) increase, we are closer to observing the population rather than the sample. Since the null hypothesis is a statement about the population, even small differences in parameters should become statistically significant eventually.

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8)

Prove that F =

2 2 ( SSRrestricted − SSRunrestricted ) / q ( Runrestricted − Rrestricted /q = 2 SSRunrestricted /(n − kunrestricted − 1) 1 − Runrestricted /(n − kunrestricted − 1)

Answer: Note that SSR = TSS − ESS . Hence we get (TSS − ESS restricted − (TSS − ESSunrestricted )) / q . Next, dividing numerator and F= (TSS − ESSunrestricted ) /(n − kunrestricted − 1) ESSunrestricted ESSrestricted − /q TSS TSS denominator by TSS, gives us F = . Since TSS − ESSunrestricted /(n − kunrestricted − 1) TSS ESS , this gives us the expression we were looking for.

R2 = TSS 9)

To calculate the homoskedasticity-only overall regression F-statistic, you need to compare the SSRrestricted with the SSRurrestricted . Consider the following output from a regression package, which reproduces the regression results of testscores on the studentteacher ratio, the percent of English learners, and the expenditures per student from your textbook: Dependent Variable: TESTSCR Method: Least Squares Date: 07/30/06 Time: 17:55 Sample: 1 420 Included observations: 420 Variable C STR EL_PCT EXPN_STU R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

Coefficient

Std. Error

t-Statistic

Prob.

649.58 -0.29 -0.66 0.00

15.21 0.48 0.04 0.00

42.72 -0.60 -16.78 2.74

0.00 0.55 0.00 0.01

0.44 0.43 14.35 85699.71 -1712.81 0.74

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 8.18 8.21 107.45 0.00

Sum of squared resid corresponds to SSRunrestricted . How are you going to find SSRrestricted ?

25

Answer: You could simply run a regression of Testscr on a constant. However, for the case n = βm + β l × STR + β l l l Testscore STR EL _ PCT × EL _ PCTi + β EXPN _ STU × EXPN _ STU i + ui i 0 i the restricted residuals are Yl = βm + ul , and for the restricted sum of square i

0

i

residuals, you get simply the variation in n

testscores SSRrestricted = ∑ (Testscorei − Testscore) 2 . i =1

10)

Adding the Percent of English Speakers (PctEL) to the Student Teacher Ratio (STR) in your textbook reduced the coefficient for STR from 2.28 to 1.10 with a standard error of 0.43. Construct a 90% and 99% confidence interval to test the hypothesis that the coefficient of STR is 2.28.

Answer: The 90% confidence interval is (1.10 ± 1.64× 0.43) = (0.39, 1.81). The 99% confidence interval is (-0.01, 2.21). Hence you can reject the null hypothesis at both the 90% and 99% confidence level.

11)

The homoskedasticity only F-statistic is given by the formula F=

( SSRrestricted − SSRunrestricted ) / q SSRunrestricted /( n − k unrestricted − 1)

where SSRrestricted is the sum of squared residuals from the restricted regression, SSRunrestricted is the sum of squared residuals from the unrestricted regression, q is the number of restrictions under the null hypothesis, and kunrestricted is the number of regressors in the unrestricted regression. Prove that this formula is the same as the following formula based on the regression R2 of the restricted and unrestricted regression: F=

( ESSunrestricted − ESSrestricted ) / q 1 − ESSunrestricted /(n − kunrestricted − 1)

Answer: Note that SSR = TSS − ESS . Hence we get (TSS − ESS restricted − (TSS − ESSunrestricted )) / q F= , which gives the above (TSS − ESSunrestricted ) /(n − kunrestricted − 1) expression once the TSS in the numerator are cancelled.

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12)

Trying to remember the formula for the homoskedasticity-only F-statistic, you forgot whether you subtract the restricted SSR from the unrestricted SSR or the other way around. Your professor has provided you with a table containing critical values for the F distribution. How can this be of help?

Answer: All the values in the F table are positive. Hence the correct answer must produce a positive value in the numerator and denominator (or negative expressions in ( SSR? − SSR? ) / q and hence the denominator is both). But F = SSRunrestricted /(n − kunrestricted − 1) positive. Hence for the numerator to be also positive, you must have SSRrestricted − SSRunrestricted .

13)

Consider the following regression output for an unrestricted and a restricted model.

Unrestricted model: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:35 Sample: 1 420 Included observations: 420 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C STR EL_PCT LOG(AVGINC) MEAL_PCT CALW_PCT

658.47 -0.76 -0.19 11.69 -0.37 -0.07

7.68 0.23 0.03 1.74 0.04 0.06

85.73 -3.27 -5.62 6.71 -9.53 -1.21

0.00 0.00 0.00 0.00 0.00 0.23

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.80 0.79 8.64 30888.64 -1498.51 1.51

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

27

654.16 19.05 7.16 7.22 324.94 0.00

Restricted model: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:37 Sample: 1 420 Included observations: 420 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C STR EL_PCT LOG(AVGINC)

593.48 -0.39 -0.43 28.36

6.96 0.27 0.03 1.40

85.32 -1.42 -14.34 20.32

0.00 0.16 0.00 0.00

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.71 0.71 10.26 43792.42 -1571.82 1.30

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 7.50 7.54 342.98 0.00

Calculate the homoskedasticity only F-statistic and determine whether the null hypothesis can be rejected at the 5% significance level.

Answer: There are two restrictions, namely H 0 : β meal _ pct = 0, β calw _ pct = 0 . The F-statistic ⎛ 43792.42 ⎞ 420 − 5 − 1 − 1⎟ × = 86.47 . The 5% critical value from the is F = ⎜ 2 ⎝ 30888.64 ⎠ F2,∞ distribution is 3.00. Hence we easily reject the two restrictions at the 5% level of significance.

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14)

Consider the regression output from the following unrestricted model: Unrestricted model: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:35 Sample: 1 420 Included observations: 420 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C STR EL_PCT LOG(AVGINC) MEAL_PCT CALW_PCT

658.47 -0.76 -0.19 11.69 -0.37 -0.07

7.68 0.23 0.03 1.74 0.04 0.06

85.73 -3.27 -5.62 6.71 -9.53 -1.21

0.00 0.00 0.00 0.00 0.00 0.23

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.80 0.79 8.64 30888.64 -1498.51 1.51

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 7.16 7.22 324.94 0.00

To test for the null hypothesis that neither coefficient on the percent eligible for subsidized lunch nor the coefficient on the percent on public income assistance is statistically significant, you have your statistical package plot the confidence set.

Interpret the following graph and explain what it tells you about the null hypothesis.

29

.10 .05

C(6)

.00 -.05 -.10 -.15 -.20 -.25 -.48

-.44

-.40

-.36

-.32

-.28

-.24

C(5) Answer: The dot in the center of the ellipse is the point estimate for the two coefficients (-0.37,-0.07). Since the (0,0) point is not inside the ellipse, you reject the null hypothesis.

15)

Consider the regression model Yi = β 0 + β 1 X 1i + β 2 X 2i + β 3 X 3i + u i . Use “Approach #2” from Section 7.3 to transform the regression so that you can use a t-statistic to test:

β1 =

β2 β3

Answer: This is not a linear restriction. Hence you cannot use the F-test to test for its validity.

16)

Consider the following Cobb-Douglas production function Yi = AK iβ1 Lβi 2 eui (where Y is output, A is the level of technology, K is the capital stock, and L is the labor force), which has been linearized here (by using logarithms) to look as follows: yi = β 0* + β1ki + β 2li + ui Assuming that the errors are heteroskedastic, you want to test for constant returns to scale. Using a t-statistic and “Approach #2,” how would you proceed.

30

Answer: Under constant returns to scale, β1 + β 2 = 1 . Hence you need to transform the unrestricted model above by subtracting l from both sides, and by adding and subtracting β1li . This results in ( yi − li ) = β 0* + β1 (ki − li ) + ( β1 + β 2 − 1)li + ui .

The left hand side variable is now the (log of the) output-labor ratio, and the first explanatory variable on the right hand side is the (log of the) capital-labor ratio. If the null hypothesis of constant returns to scale holds, then the coefficient on l should be zero. This can be directly tested using a t-statistic.

17) Consider the following two models to explain testscores.

Model 1: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:52 Sample: 1 420 Included observations: 420 Variable

:

Coefficient

Std. Error

t-Statistic

Prob.

C STR EL_PCT LOG(AVGINC) MEAL_PCT

658.55 -0.73 -0.18 11.57 -0.40

7.68 0.23 0.03 1.74 0.03

85.70 -3.18 -5.52 6.65 -13.09

0.00 0.00 0.00 0.00 0.00

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.80 0.79 8.64 30998.01 -1499.25 1.52

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

31

654.16 19.05 7.16 7.21 405.36 0.00

Model 2: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:56 Sample: 1 420 Included observations: 420 Variable

Coefficient

Std. Error

t-Statistic

Prob.

C STR EL_PCT LOG(AVGINC) CALW_PCT

620.92 -0.66 -0.39 21.87 -0.41

7.27 0.25 0.03 1.52 0.05

85.41 -2.58 -14.05 14.41 -8.22

0.00 0.01 0.00 0.00 0.00

R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat

0.75 0.75 9.53 37659.29 -1540.13 1.41

Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion F-statistic Prob(F-statistic)

654.16 19.05 7.36 7.41 315.31 0.00

Explain why you cannot use the F-test in this situation to discriminate between Model 1 and Model 2.

Answer: Neither model is contained (“nested”) in the other, in the sense that you cannot place restrictions on Model 1 to obtain Model 2 (and vice versa). Hence there is no unrestricted and restricted model in this case.

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18)

Your textbook has emphasized that testing two hypothesis sequentially is not the same as testing them simultaneously. Consider the following confidence set below, where you are testing the hypothesis that H 0 : β5 = 0, β 6 = 0 .

Your statistical package has also generated a dotted area, which corresponds to drawing two confidence intervals for the respective coefficients. For each case where the ellipse does not coincide in area with the corresponding rectangle, indicate what your decision would be if you relied on the two confidence intervals vs. the ellipse generated by the Fstatistic.

Answer: The following possible outcomes can be seen in the figure above: (i) both Fstatistic and the two confidence intervals generate the same result; (ii) you do not reject the null hypothesis using the F-statistic, but you do so by using the confidence intervals (these are the points in the area at the “tip” of the ellipse); (iii) you reject the null hypothesis using the confidence intervals but not the Fstatistic.

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19)

You have estimated the following regression to explain hourly wages, using a sample of 250 individuals:

AHEi = −2.44 − 1.57 × DFemme + 0.27 × DMarried + 0.59 × Educ + 0.04 × Exper − 0.60 × DNonwhite (1.29) (0.33) (0.36) (0.09) (0.01) (0.49) +0.13 × NCentral − 0.11× South (0.59) (0.58)

R 2 = 0.36, SER = 2.74, n = 250 Numbers in parenthesis are heteroskedasticity robust standard errors. Add “*”(5%) and “**” (1%) to indicate statistical significance of the coefficients.

Answer: AHEi = −2.44 − 1.57** × DFemme + 0.27 × DMarried + 0.59** × Educ (1.29) (0.33) (0.36) (0.09) ** +0.04 × Exper − 0.60 × DNonwhite + 0.13 × NCentral − 0.11× South (0.01) (0.49) (0.59) (0.57) 20)

You have estimated the following regression to explain hourly wages, using a sample of 250 individuals:

AHEi = −2.44 − 1.57 × DFemme + 0.27 × DMarried + 0.59 × Educ + 0.04 × Exper − 0.60 × DNonwhite (1.29) (0.33) (0.36) (0.09) (0.01) (0.49) +0.13 × NCentral − 0.11× South (0.59) (0.58)

R 2 = 0.36, SER = 2.74, n = 250 Test the null hypothesis that the coefficients on DMarried, DNonwhite, and the two regional variables, NCentral and South are zero. The F-statistic for the null hypothesis β married = β nonwhite = β ncentral = β south = 0 is 0.61. Do you reject the null hypothesis?

Answer: The critical value for F4,∞ =3.32 at the 1% significance level. Hence you cannot reject the null hypothesis.

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