Question 1

Consider the following Cobb-Douglas production function Y_i = AK $\begin{array} { c } \beta _ { 1 } \ i \end{array}$ L $\begin{array} { c } \beta _ { 2 } \ i \end{array}$ $e ^ { u _ { i } }$ (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: y_i = $\beta _ { 0 } ^ { * }$ + β₁k_i + β₂l_i + u_i 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.

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Question 2

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 (b)number of observations: 1,732; number of restrictions:7; F-statistic: 4.92 (c)number of observations: 63; number of restrictions: 1; F-statistic: 2.47 (d)number of observations: 4,000; number of restrictions: 5; F-statistic: 1.82 (e)Explain why you can use the F_q_,_∞ distribution to compute the critical values in (a)-(d).

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Question 3

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  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?
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?
Dependent Variable: TESTSCR
Method: Least Squares
Date: 07/30/06 Time: 17:55
Sample: 1 420
Included observations: 420

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Question 4

The homoskedasticity only F-statistic is given by the formula where SSR_restricted is the sum of squared residuals from the restricted regression, SSR_unrestricted is the sum of squared residuals from the unrestricted regression, q is the number of restrictions under the null hypothesis, and k_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 R² of the restricted and unrestricted regression:

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Question 5

Consider the following regression using the California School data set from your textbook. $\widehat{\text { TestScore }}$ = 681.44 - 0.61LchPct n=420, R²=0.75, SER=9.45 where TestScore is the test score and LchPct is the percent of students eligible for subsidized lunch (average = 44.7, max = 100, min = 0). a. What is the effect of a 20 percentage point increase in the student eligible for subsidized lunch? b. Your textbook started with the following regression in Chapter 4: $\widehat{\text { TestScore} }$ = 698.9 - 2.28STR n=420, R²=0.051, SER=18.58 where STR is the student teacher ratio. Your textbook tells you that in the multiple regression framework considered, the percentage of students eligible for subsidized lunch is a control variable, while the student teacher ratio is the variable of interest. Given that the regression R² is so much higher for the first equation than for the second equation, shouldn't the role of the two variables be reversed? That is, shouldn't the student teacher ratio be the control variable while the percent of students eligible for subsidized lunch be the variable of interest?

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Question 6

Consider the following multiple regression model Y_i = ?₀ + ?₁X₁_i + ?₂X₂_i + ?₃X₃_i + 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)?₁ = -?₂; ?₃= 0 (b)?₁ + ?₂+ ?₃= 1 (c)?₁ = ?₂; ?₃= 0

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Question 7

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.

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Question 8

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 (b)k = 3, test for the slope coefficient of X₁ to be unity, and the coefficients on the other explanatory variables to be zero (c)k = 10, test for the slope coefficient of X₁ to be zero, and for the slope coefficients of X₂and X₃ to be the same but of opposite sign. (d)k = 4, test for the slope coefficients to add up to unity

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Question 9

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?

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Question 10

To calculate the homoskedasticity-only overall regression F-statistic, you need to compare the SSR_restricted with the SSR_unrestricted. Consider the following output from a regression package, which reproduces the regression results of testscores on the student-teacher 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 Sum of squared resid corresponds to SSR_unrestricted. How are you going to find SSR_restricted?

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Question 11

Prove that

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Question 12

Using the 420 observations of the California School data set from your textbook, you estimate the following relationship: $\widehat{\text { TestScore }}$ = 681.44 - 0.61LchPct n=420, R²=0.75, SER=9.45 where TestScore is the test score and LchPct is the percent of students eligible for subsidized lunch (average = 44.7, max = 100, min = 0). a. Interpret the regression result. b. In your interpretation of the slope coefficient in (a)above, does it matter if you start your explanation with "for every x percent increase" rather than "for every x percentage point increase"? c. The "overall" regression F-statistic is 1149.57. What are the degrees of freedom for this statistic? d. Find the critical value of the F-statistic at the 1% significance level. Test the null hypothesis that the regression R²= 0. e. The above equation was estimated using heteroskedasticity robust standard errors. What is the standard error for the slope coefficient?

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Question 13

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: $\widehat { \text { Studentw } }$ = -229.21 - 6.36 × Female + 5.58 × Height, R²=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?

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Question 14

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

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Question 15

Consider the regression model Y_i = ?₀ + ?₁X₁_i + ?₂X₂_i+ ?₃X₃_i + u_i. Use "Approach #2" from Section 7.3 to transform the regression so that you can use a t-statistic to test: ?₁= $\frac { \beta _ { 2 } } { \beta _ { 3 } }$

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Question 16

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  Restricted model:
Dependent Variable: TESTSCR
Method: Least Squares
Date: 07/31/06 Time: 17:37
Sample: 1 420
Included observations: 420  Calculate the homoskedasticity only F-statistic and determine whether the null hypothesis can be rejected at the 5% significance level.

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Question 17

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

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Question 18

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.

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Question 19

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 H₀ : β₅ = 0, β₆= 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.

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Question 20

Give an intuitive explanation for  Name conditions under which the F-statistic is large and hence rejects the null hypothesis.

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Quiz 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.

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?

Prove that

Consider the regression model Yi = ?0 + ?1X1i + ?2X2i+ ?3X3i + ui. Use "Approach #2" from Section 7.3 to transform the regression so that you can use a t-statistic to test: ?1 = β2β3\frac { \beta _ { 2 } } { \beta _ { 3 } }β3​β2​​

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.

Give an intuitive explanation for Name conditions under which the F-statistic is large and hence rejects the null hypothesis.

Consider the regression model Y_i = ?₀ + ?₁X₁_i + ?₂X₂_i+ ?₃X₃_i + u_i. Use "Approach #2" from Section 7.3 to transform the regression so that you can use a t-statistic to test: ?₁= $\frac { \beta _ { 2 } } { \beta _ { 3 } }$