Exam 7: Hypothesis Tests and Confidence Intervals in Multiple Regression

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.

When testing joint hypothesis, you should

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 $H _ { 0 } : \beta _ { 5 } = 0 , \beta _ { 6 } = 0$ . 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 F- statistic.

When there are two coefficients, the resulting confidence sets are

The confidence interval for a single coefficient in a multiple regression a. makes little sense because the population parameter is unknown. b. should not be computed because there are other coefficients present in the model. c. contains information from a large number of hypothesis tests. d. should only be calculated if the regression $R ^ { 2 }$ is identical to the adjusted $R ^ { 2 }$ .

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

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: = 570.70-170.72\times Female ,=0.084, SER =282.12. (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?

The following linear hypothesis can be tested using the F-test with the exception of a. $\quad \beta _ { 2 } = 1$ and $\beta _ { 3 } = \beta _ { 4 } / \beta _ { 5 }$ . b. $\beta _ { 2 } = 0$ . c. $\beta _ { 1 } + \beta _ { 2 } = 1$ and $\beta _ { 3 } = - 2 \beta _ { 4 }$ . d. $\quad \beta _ { 0 } = \beta _ { 1 }$ and $\beta _ { 1 } = 0$ .

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 $= \mid ( 50 \% - \%$ of Male Students at Table $) \mid$ Where " $| Z |$ " stands for absolute value of $Z$ . The variable can take on values from zero to fifty. 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): =30.90-3.78\times Size -8.81\times DCoed +2.28\times DFemme +2.06\times DRoommate (7.73)(0.63)(2.66)(2.42)(2.39) -0.17\times DAthlete +1.49\times DCons -0.81 SAT +1.74\times SibOther, =0.24,=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.

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: 1420 Included observations: 420 Restricted model: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:37 Sample: 1420 Included observations: 420 Calculate the homoskedasticity only F-statistic and determine whether the null hypothesis can be rejected at the 5% significance level.

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

You have estimated the following regression to explain hourly wages, using a sample of 250 individuals: = -2.44-1.57\times DFemme +0.27\times DMarried +0.59\times Educ +0.04\times Exper -0.60\times DNonwhite (1.29)(0.33)(0.36)(0.09)(0.01)(0.49) +0.13\times NCentral -0.11\times South (0.59)(0.58) =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.

You have estimated the following regression to explain hourly wages, using a sample of 250 individuals: = -2.44-1.57\times DFemme +0.27\times DMarried +0.59\times Educ +0.04\times Exper -0.60\times DNonwhite (1.29)(0.33)(0.36)(0.09)(0.01)(0.49) +0.13\times NCentral -0.11\times South (0.59)(0.58) =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 $\beta _ { \text {maxriad } } = \beta _ { \text {momwhite } } = \beta _ { \text {noentral } } = \beta _ { \text {saurh } } = 0$ is $0.61$ . Do you reject the null hypothesis?

Let $R _ { \text {unrestricted } } ^ { 2 } \text { and } R _ { \text {restricted} } ^ { 2 }$ 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

All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of a. $H _ { 0 } : \beta _ { 1 } + \beta _ { 2 } = 1$ . b. $H _ { 0 } : \frac { \beta _ { 3 } } { \beta _ { 2 } } = \beta _ { 1 }$ and $\beta _ { 4 } = 0$ . c. $H _ { 0 } : \beta _ { 2 } = 0$ and $\beta _ { 3 } = 0$ . d. $H _ { 0 } : \beta _ { 1 } = - \beta _ { 2 }$ and $\beta _ { 1 } + \beta _ { 2 } = 1$ .

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: = 0.339-12.894\timesn+1.397\times,=0.621,SER=0.177 (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.

Prove that $F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestriced } } \right) / q } { S S R _ { \text {urrestricted } } / \left( n - k _ { \text {unrestriced } } - 1 \right) } = \frac { \left( R _ { \text {unrestricted } } ^ { 2 } - R _ { \text {restricted } } ^ { 2 } / q \right. } { 1 - R _ { \text {unrestrocted } } ^ { 2 } / \left( n - k _ { \text {unrestriceed } } - 1 \right) }$

To calculate the homoskedasticity-only overall regression $F$ -statistic, you need to compare the $S S R _ { \text {restricted } }$ with the $S S R _ { \text {urrestricted } }$ . 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: 1420 Included observations: 420 Sum of squared resid corresponds to $S S R _ { \text {unrestricted } }$ . How are you going to find $S S R _ { \text {restricted } }$ ? ii

Consider the following two models to explain testscores. Model 1: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:52 Sample: 1420 Included observations: 420 : Model 2: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:56 Sample: 1420 Included observations: 420 Explain why you cannot use the F-test in this situation to discriminate between Model 1 and Model 2.

The homoskedasticity only F-statistic is given by the formula $F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestricted } } \right) / q } { S S R _ { \text {unrestricted } } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$ where $S S R _ { \text {restricted } }$ is the sum of squared residuals from the restricted regression, $S S R _ { \text {unrestricted } }$ is the sum of squared residuals from the unrestricted regression, $q$ is the number of restrictions under the null hypothesis, and $k _ { \text {unrestricted } }$ is the number of regressors in the unrestricted regression. Prove that this formula is the same as the following formula based on the regression $R ^ { 2 }$ of the restricted and unrestricted regression: $F = \frac { \left( E S S _ { \text {wrratricted } } - E S S _ { \text {rastricted } } \right) / q } { 1 - E S S _ { \text {wurrestricted } } / \left( n - k _ { \text {umrestricted } } - 1 \right) }$