. Suppose that the United States Golf Associate (USGA) wants to compare the mean distances traveled by four brands of golf balls when struck by a driver. A completely randomized design is employed with Tigger Wedge, the USGA s robot golfer, using a driver to hit a random sample of golf balls ( balls for each brand) in a random sequence. The distance is recorded for each hit, and the resulting means are organized by brand in the following table: Mean Brand A 5.8 Brand B 6.6 Brand C 69.95 Brand D 9. Judy Bates was hired by the USGA to determine whether the mean distances traveled by the four brands of golf balls were the same. To do this she estimated the following linear regression model: Distancei = + BrandAi + BrandBi + BrandCi + i Where: Distancei = Distance traveled for the ith golf ball BrandAi = Dummy variable equal to for the distance traveled by BrandA golf balls; otherwise. BrandBi = Dummy variable equal to for the distance traveled by BrandB golf balls; otherwise. BrandCi = Dummy variable equal to for the distance traveled by BrandC golf balls; otherwise. Judy produced the following results for her USGA employers: Distance R i ˆ ˆ BrandA (57.6) (.8).7 ESS 79. i ˆ BrandB ˆ BrandC i i (5.8) (9.) (t statistics in parentheses) RSS 97.86 TSS 67.6 a. Test, using Judy s results and a 5% level of significance, whether the brand of golf ball matters in explaining the mean distance traveled. b. Judy forgot to include the values of her parameter estimates. Help her out by calculating the numerical values for each of the estimated coefficients above. Page
. Consider the following regression output using data from the Colby Student Lifestyle Survey for the Fall of. Dependent Variable: GPA Method: Least Squares Included observations: 8 Variable Coefficient Std. Error Intercept.5.68 SAT SKIP FEMALE WHITE.7.66 -.6975.68.5579.676.77675.66 R-squared.65 Mean dependent var.68 Adjusted R-squared.8989 S.D. dependent var.556 S.E. of regression.998 F-statistic.578 Sum squared resid 5.885 Prob(F-statistic). Where: GPA is the student s self-reported Grade Point Average. SAT is the student s self-reported SAT score SKIP is the number of classes skipped each semester FEMALE is a dummy variable equal to if the student is female; otherwise WHITE is a dummy variable equal to if the student is white, otherwise. a. Provide an interpretation for each of the estimated coefficients in the model above. b. Develop and complete hypothesis tests for each of the coefficients of this model, stating the null and alternative hypotheses. Include a sentence that outlines the intuition of your alternative hypothesis. Test your hypotheses at the 5% level of significance. c. Use the regression output and correlation matrix below, and apply the specification criteria we studied this semester to determine whether or not WHITE is an irrelevant variable in this equation. Dependent Variable: GPA Method: Least Squares Sample: 86 Included observations: 9 Variable Coefficient Std. Error t-statistic Prob. Intercept.6799.8 7.595. SAT.57.6 7.7587. SKIP -.6.6 -.578. FEMALE.556.66.866.689 R-squared.575 Mean dependent var.695 Adjusted R-squared. S.D. dependent var.5 S.E. of regression.5 F-statistic 5.579 Sum squared resid 5.8 Prob(F-statistic). Page
GPA SAT SKIP FEMALE WHITE GPA. SAT.8. SKIP -.79.8665. FEMALE.65 -.6 -.665. WHITE.65.55 -.8.69. 5. Consider the following estimated model of demand for housing (standard errors in parentheses): log Qi =.7.7 log Pi +.96 log Yi (.7) (.6) N =, families R =.7 Where Q = measure of quantity of housing in square feet consumed by each family P = price of a unit of housing in family s locality Y = measure of family income a. Calculate the price and income elasticities of demand for this model. b. Test, at the 5% level of significance, if the income elasticity of demand for housing is equal to in this model. Page
a. : ANSWERS H 8 points F F,6,5%,,5% ESS F K RSS n K 79. 97.86 5.8.87.8 (nearest) (interpolated) 9.67 5.8 Reject H and conclude that this model is useful for predicting distance. This implies that the brand of the golf ball must have some association with mean distance since Brand is the only factor in the regression model. Therefore, mean distance will vary with Brand which implies that at least two of the brands have different mean distances. b. 8 points ˆ 9., i.e., the mean distance of Brand D golf balls (the omitted dummy variable). ˆ 5.8 9..86, i.e., the difference in the mean distance of Brand A and Brand D golf balls. ˆ 6.6 9..6, i.e., the difference in the mean distance of Brand B and Brand D golf balls. ˆ 69.95 9..5, i.e., the difference in the mean distance of Brand C and Brand D golf balls.. Dependent Variable: GPA Sample: 86 Included observations: 8 Variable Coefficient Std. Error t-statistic Prob. Intercept.5.68 6.959. SAT.7.66 7.67968. SKIP -.6975.68 -.878. FEMALE.5579.676.8875.696 WHITE.77675.66.7.87 R-squared.65 Mean dependent var.68 Adjusted R-squared.8989 S.D. dependent var.556 S.E. of regression.998 F-statistic.578 Sum squared resid 5.885 Prob(F-statistic). a. Algebraically, the OLS estimate of the intercept term is the value of the GPA when all of the explanatory variables and the error term are equal to zero. However, because the intercept term in an OLS regression equation also equals the sum of the constant effect of omitted explanatory variables and the nonzero mean of the sample error observations, it has no interpretive value. For this reason there should be no hypothesis test in part b for. pts Ceteris paribus, a point increase in a student s SAT score will, on average, increase that students GPA by. points. points Page
Ceteris paribus, the more classes a student skips the lower their GPA on average. Note that the size of this coefficient seems large a student who skips just classes will earn a GPA that is. lower on average according to this result. The units of the SKIP variable are not defined. In fact, the SKIP variable is an ordinal variable, meaning that SKIP equals for to classes skipped in a semester, for - classes skipped, 5 for 5-7 classes skipped, 8 for 8- classes skipped, and for or more classes skipped. Therefore, the interpretation of can only be made in relative terms as in the first sentence of this paragraph. points Ceteris paribus, females have a mean GPA that is.56 points greater than males in this sample. points Ceteris paribus, white students in this sample have a mean GPA that is.78 points higher than students of color. However, this result is not statistically significant, i.e., there is no statistically significant difference in GPAs among white and non-white students in this sample, controlling for their SAT score, the number of classes skipped, and their sex. points b. points Coefficient: Hypothesized Sign Ceteris paribus, students with higher SAT scores should have higher GPAs. Ceteris paribus, the more classes a student skips the lower their mean GPA should be. There is no convincing evidence that GPAs at Colby should differ between men and women. There is no convincing evidence that GPAs at Colby should be any different for student of color. t-statistic: t c =.65 (one-sided) t c =.96 (two-sided) Degrees of freedom = 8--=78 (Normal) Decision Rule: H : H A: > Reject H because t-stat >.65 and the estimate has the correct sign. H : H A: < Reject H because t-stat >.65 and the estimate has the correct sign. H : = H A: Do not reject H because t-stat <.96. H : = H A: Do not reject H because t-stat <.96. Page 5
c. Theory: There s no a priori evidence that GPAs at Colby should differ due to race, so theory is no help here in determining whether or not WHITE is an irrelevant variable. points t-test: The estimated coefficient on WHITE is not statistically significant at any reasonable level of significance, thus we can suspect that it is an irrelevant variable. points R : The regression omitting the WHITE variable has a slightly lower however, given that the value of the t-statistic for was greater than.. R. This comes as no surprise, points Expected Bias: Without an expected sign for we must examine both alternatives in conjunction with the reported sign of the correlation coefficient and compare this with the regression results. points For the regression that omits the WHITE dummy variable, the coefficient estimates for and are essentially unchanged indicating a possible [very small] negative bias. Similarly, the coefficient for is essentially unchanged but indicates a possible [very small] positive bias. The correlation coefficient for WHITE and SAT is positive as is the correlation coefficient for WHITE and FEMALE. The correlation coefficient for WHITE and SKIP is negative. To be consistent with the observed changes in coefficient estimates between the two regressions, we d therefore have to expect to be negative. Given the lack of statistical significance for and the fact that there is no significant change in the parameter estimates when WHITE is omitted, meaning that there does not appear to be any bias, we can conclude that WHITE is an irrelevant variable. 5a. For a double-log specification no calculation is necessary. The parameter estimates are elasticities. Therefore the price elasticity of demand is.7 and the income elasticity is.96.. 5 points b. 5 points H H : : logincome logincome...96. t.58.6 t.96 c A Since -.58 <.96 we cannot reject H and therefore cannot conclude that the income elasticity of demand is significantly Page 6
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