Exam 7: Hypothesis Tests and Confidence Intervals in Multiple Regression

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: =-229.92-6.52\times Female +0.51\times Sibs +5.58\times Height , (44.01)(5.52)(2.25)(0.62) =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?

At a mathematical level, if the two conditions for omitted variable bias are satisfied, then a. $\quad E \left( u _ { i } \mid X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i } \right) \neq 0$ b. there is perfect multicollinearity. c. large outliers are likely: $X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i }$ and $Y _ { i }$ have infinite fourth moments. d. $\left( X _ { 1 i } , X _ { 2 i } , \ldots , X _ { k i } , Y _ { i } \right) , i = 1 , \ldots , n$ are not i.i.d. draws from their joint distribution.

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: = -229.21-6.36\times Female +5.58\times Height ,=0.50, SER =20.99 (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?

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 =7,311.17+3,985.20\times Reputation -0.20\times Size (2,058.63)(664.58)(0.13) +8,406.79\times Dpriv -416.38\times Dlibart -2,376.51\times Dreligion (2,154.85)(1,121.92)(1,007.86) =0.72,SER=3,773.35 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.

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

In the multiple regression model with two explanatory variables $Y _ { i } = \beta _ { 0 } + \beta _ { 1 } X _ { 1 i } + \beta _ { 2 } X _ { 2 i } + u _ { i }$ the OLS estimators for the three parameters are as follows (small letters refer to deviations from means as in $z _ { i } = Z _ { i } - \bar { Z }$ ): =-- = = 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 $\left( X _ { 1 i } \right)$ and the saving rate $\left( X _ { 2 i } \right)$ (average investment share of GDP from 1980 to 1990 ) on GDP per worker (relative to the U.S.)in 1990.The various sums needed to calculate the OLS estimates are given below: =33.33;=2.025;=17.313 =8.3103;=.0122;=0.6422 =-0.2304;=1.5676;=-0.0520 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?

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: =15,005+201\times Temperat +465\times DodgNetWin +82\times OppNetWin (8,770)(121)(169)(26) +9647\times DFSaSu +1328\times Drain +1609\times D 150m+271\times DDiv -978\times D2001; (1505)(3355)(1819)(1,184)(1,143) =0.416,SER=6983 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?

For a single restriction (q=1) , the F -statistic

The general answer to the question of choosing the scale of the variables is

The homoskedasticity-only F-statistic is given by the following formula a. $\quad F = \frac { \left( R _ { \text {unnestricted } } ^ { 2 } - R _ { \text {restricted } } ^ { 2 } / q \right. } { 1 - \left( R _ { \text {unrestricted } } ^ { 2 } \right) / \left( n - k _ { \text {unrestricted } } - 1 \right) }$ . b. $\quad F = \frac { 1 - R _ { \text {unrestricted } } ^ { 2 } / q } { R _ { \text {unrestricted } } ^ { 2 } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$ . c. $\quad F = \frac { \left( R _ { \text {unrestricted } } ^ { 2 } - R _ { \text {unrestricted } } ^ { 2 } / q \right. } { 1 - \left( R _ { \text {unrestricted} } ^ { 2 } \right) / \left( n - k _ { \text {restricted } } - 1 \right) }$ . d. $\quad F = \frac { \left( R _ { \text {restricted } } ^ { 2 } - R _ { \text {uurestricted } } ^ { 2 } / q \right. } { 1 - \left( R _ { \text {restricted } } ^ { 2 } \right) / \left( n - k _ { \text {restricted } } - 1 \right) }$ .