Practice exam - chs 4-7 material

Practice exam questions – Chapters 4-7 (the final is cumulative, so please also review the midterm practice questions) 1. Assume that you have collected a sample of observations from over 400 households on their consumption spending. If you regress household consumption on a constant, A. Your coefficient estimate will always be significant. B. The coefficient estimate will be equal to the average consumption in the sample. C. You most likely have an omitted variable bias. D. Your R 2 will be 1. Answer: B 2. Using wage data from 240 public and 204 private firms, you run a regression model with ‘wage’ as the dependent variable and ‘public’ as an explanatory variable which takes a value of 1 if the firm is a public firm and 0 otherwise. A. In this regression, the estimated slope equals the average wage of private firms in the sample. B. In this regression, the estimated intercept equals the average wage of private firms in the sample. C. In this regression, the estimated slope equals the average wage of public firms in the sample. D. In this regression, the estimated intercept equals the average wage of public firms in the sample. Answer: B 3. If an independent variable in a multiple linear regression model is an exact linear combination of other independent variables, the model suffers from the problem of _____. A. perfect collinearity B. homoskedasticity C. heteroskedasticty D. omitted variable bias Answer: A 4. Suppose the correlation between regressor X and another variable that affects the dependent variable but omitted from the model is positive. The slope coefficient estimate for X A. is positively biased. B. is negatively biased. C. is unbiased. D. may be positively or negatively biased depending on the relation between the omitted variable and the dependent variable. Answer: D

For the next 3 questions (5-7): A regression between foot length (response variable in cm) and height (explanatory variable in inches) for 33 students resulted in the following regression equation: 𝒀𝒀 � = 10.9 + 0.23 x. 5. One student in the sample was 73 inches tall with a foot length of 29 cm. What is the predicted foot length for this student? A. 17.57 cm B. 27.69 cm C. 29 cm D. 33 cm Answer: B 6. One student in the sample was 73 inches tall with a foot length of 29 cm. What is the residual for this student? A. 29 cm B. 1.31 cm C. 0.00 cm D. -1.31 cm Answer: B 7. What is the estimated average foot length for students who are 70 inches tall? A. 27 cm B. 28 cm C. 29 cm D. 30 cm Answer: A 8. Assume that you have collected a sample of observations from over 100 households and their consumption and income patterns. Using these observations, you estimate the following regression C i = β 0 +β 1 Y i + u i where C is consumption and Y is disposable income. The estimate of β 1 will tell you: A) B) The amount you need to consume to survive C) D) Answer: D

11. The R 2 of model 4 is: A) 0.623 B) 0.626 C) 0.773 D) 0.629 Answer: D 12. The homoskedasticity-only F-statistic for testing β 3 = β 4 =0 in the regression shown in model 5 is: A) 8.01 B) 0.0019 C) 320.93 D) 319.02 Answer: C 13. A regression between foot length (response variable in cm) and height (explanatory variable in inches) for 33 students resulted in the following regression equation: 𝒀𝒀 � = 10.9 + 0.23 x. Karl is 7 inches taller than his younger brother Mark. What is the expected difference in foot length: A) 12.51 cm B) 1.61 cm C) 11.13 cm D) Not enough information to determine the difference. Answer: B 14. The following relationship is estimated between ice cream sales and outside temperature. 𝒚𝒚 � = 4.75 + 0.23X, R 2 = 0.86, SER = 0.78 (0.12) (0.04) What is the t-statistic for the null hypothesis that the slope coefficient is zero? A) 5.75 B) -5.75 C) 39.58 D) -0.29 Answer: A 15. The slope estimator, β 1 , has a smaller standard error, other things equal, if: A) there is more variation in the explanatory variable, X . B) there is a large variance of the error term, u . C) the sample size is smaller. D) the intercept, β 0 , is small. Answer: A

16. In the simple linear regression model Y i = β 0 + β 1 X i + u i : A) the intercept is typically small and unimportant. B) β 0 + β 1 X i represents the population regression function. C) the absolute value of the slope is typically between 0 and 1. D) β 0 + β 1 X i represents the sample regression function. Answer: B 17. To obtain the slope estimator using the least squares principle, you divide the: A) sample variance of X by the sample variance of Y . B) sample covariance of X and Y by the sample variance of Y . C) sample covariance of X and Y by the sample variance of X . D) sample variance of X by the sample covariance of X and Y . Answer: C 18. The following model is estimated to predict house prices: 𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷𝑷 � = 𝟏𝟏𝟏𝟏𝟏𝟏 . 𝟐𝟐 + 𝟎𝟎 . 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 + 𝟐𝟐𝟐𝟐 . 𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒𝟒 + 𝟎𝟎 . 𝟏𝟏𝟒𝟒𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝑷𝑷 + 𝟎𝟎 . 𝟎𝟎𝟎𝟎𝟐𝟐𝟎𝟎𝟏𝟏𝟏𝟏𝟏𝟏𝑷𝑷 + 𝟎𝟎 . 𝟎𝟎𝟏𝟏𝟎𝟎𝟎𝟎𝟎𝟎𝑷𝑷 − 𝟒𝟒𝟒𝟒 . 𝟒𝟒𝑷𝑷𝟖𝟖𝟖𝟖𝑷𝑷 (23.9) (2.61) (8.94) (0.011) (0.00048) (0.311) (10.5) Where HSize is the size of the house, LSize is the size of the lot, both measured in sqft. Home prices are measured in $1,000. A homeowner purchases 200 sqft from an adjacent lot. The 95% confidence interval for the change in value of her home is: A) [$1.06, $2.94] B) [$211.84, $588.16] C) [$2118.4, $5881.6] D) [$4531.9, $6423.3] Answer: B 19. Using the example of a simple linear regression (population regression function) of test scores and student-teacher ratio, a negative error means: A) the regression is useless for predicting test scores B) the district did worse than predicted C) the student-teacher ratio must be incorrect D) the district did better than predicted Answer: B

25. You have to worry about perfect multicollinearity in the multiple regression model because: A) many economic variables are perfectly correlated. B) the OLS estimator is no longer unbiased. C) the OLS estimator cannot be computed in this situation. D) in real life, economic variables change together all the time. Answer: C 26. The sample regression line estimated by OLS: A) has an intercept that is equal to zero. B) is the same as the population regression line. C) cannot have negative and positive slopes. D) is the line that minimizes the sum of squared prediction mistakes. Answer: D 27. The OLS residuals in the multiple regression model: A) cannot be calculated because there is more than one explanatory variable. B) can be calculated by subtracting the fitted values from the actual values. C) are zero because the predicted values are another name for forecasted values. D) are typically the same as the population regression function errors. Answer: B 28. Omitted variable bias: A) will always be present as long as the regression R 2 < 1. B) is always there but is negligible in almost all economic examples. C) exists if the omitted variable is correlated with the included regressor but is not a determinant of the dependent variable. D) exists if the omitted variable is correlated with the included regressor and is a determinant of the dependent variable. Answer: D 29. The following OLS assumption is most likely violated by omitted variables bias: A) E ( u i X i ) = 0 B) ( X i , Y i ) i =1,..., n are i.i.d draws from their joint distribution C) there are no outliers for X i , u i D) there is heteroskedasticity Answer: A 30. In a regression study, a 95% confidence interval for β1 was given as: ( -5.65, 2.61). What would a test for H 0 : β 1 =0 vs H a : β 1 ≠ 0 conclude? A) reject the null hypothesis at α= 0.05 and all smaller α B) fail to reject the null hypothesis at α= 0.05 and all smaller α C) reject the null hypothesis at α= 0.05 and all larger α D) fail to reject the null hypothesis at α= 0.05 and all larger α Answer: B

31. In multiple regression, the R 2 increases whenever a regressor is: A) added unless the coefficient on the added regressor is exactly zero. B) excluded from the model. C) added unless there is heteroskedasticity. D) greater than 1.96 in absolute value. Answer: A 32. The adjusted R 2 , or , is given by: A) 1- B) 1- C) 1- D) Answer: C 33. Consider the following multiple regression models (a) to (d) below. DFemme = 1 if the individual is a female, and is zero otherwise; DMale is a binary variable which takes on the value one if the individual is male, and is zero otherwise; DMarried is a binary variable which is unity for married individuals and is zero otherwise, and DSingle is (1-DMarried). Regressing weekly earnings ( Earn ) on a set of explanatory variables, you will experience perfect multicollinearity in the following cases EXCEPT for: A) i = + DFemme + Dmale + X 3 i B) i = + DMarried + DSingle + X 3 i C) i = + DFemme + X 3 i D) i = DFemme + Dmale + DMarried + DSingle + X 3 i Answer: C 34. Compute the simple linear regression equation if: A) 𝑦𝑦 � = 1297.5 − 2.6 𝑥𝑥 B) 𝑦𝑦 � = 874.1 − 0.774 𝑥𝑥 Answer: A

40. Let R 2 unrestricted and R 2 restricted 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 unrestricted model includes 3 regressors. The homoskedasticity-only F -statistic in this case is: A) 4.61 B) 8.01 C) 10.34 D) 7.71 Answer: B 41. The OLS estimators of the coefficients in multiple regression will have omitted variable bias: A) only if an omitted determinant of Y i 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 Y i is correlated with at least one of the regressors. Answer: D 42. If R 2 ur = 0.6873, R 2 r = 0.5377, number of restrictions = 3, and n – k – 1 = 229, F-statistic equals: A. 21.2 B. 28.6 C. 36.5 D. 42.1 Answer: C 43. Consider the following regression output where the dependent variable is test scores and the two explanatory variables are the student-teacher ratio and the percent of English learners: = 698.9 - 1.10× STR - 0.650× PctEL . You are told that the t -statistic on the student-teacher ratio coefficient is 2.56. The standard error therefore is approximately: A) 0.25. B) 1.96. C) 0.650. D) 0.43. Answer: D Short Answer Questions 1. Sir Francis Galton, a cousin of James Darwin, examined the relationship between the height of children and their parents towards the end of the 19th century. It is from this study that the name "regression" originated. You decide to update his findings by collecting data from 110 college students, and estimate the following relationship:

= 19.6 + 0.73 × Midparh , R 2 = 0.45, SER = 2.0 (7.2) (0.10) where Studenth is the height of students in inches, and Midparh is the average of the parental heights. Values in parentheses are heteroskedasticity robust standard errors. (Following Galton's methodology, both variables were adjusted so that the average female height was equal to the average male height.) (a) Test for the statistical significance of the slope coefficient. (b) If children, on average, were expected to be of the same height as their parents, then this would imply two hypotheses, one for the slope and one for the intercept. i. What should the null hypothesis be for the intercept? Calculate the relevant t -statistic and carry out the hypothesis test at the 1% level. ii. What should the null hypothesis be for the slope? Calculate the relevant t -statistic and carry out the hypothesis test at the 5% level. (c) Can you reject the null hypothesis that the regression R 2 is zero? (d) Construct a 95% confidence interval for a one inch increase in the average of parental height. Answer: (a) S ince t=0.73/0.1=7.3 is larger than all of our critical values, we reject H 0 : β 1 = 0 , that is, β 1 is statistically significant(ly different from zero). (b) H 0 : β 0 = 0, t =19.6/7.2=2.72, for H 1 : β 0 ≠ 0, the critical value for a two -sided alternative is 2.58. Hence we reject the null hypothesis in (i). For the slope we have H 0 : β 1 = 1, t =(0.73 - 1)/0.1= - 2.70, for H 1 : β 1 ≠ 1, the critical value for a two -sided alternative is 1.96. Hence we reject the null hypothesis in (ii). (c) For the simple linear regression model, H 0 : β 1 = 0 implies that R 2 = 0. So to test the null hypothesis that R 2 = 0, we simply test H 0 : β 1 = 0 , which is the same test as in (a). (d) (0.73 – 1.96 × 0.10, 0.73 + 1.96 × 0.10) = (0.53, 0.93). 2. You have obtained measurements of height in inches of 29 junior and 81 senior high-school students ( Studenth ). A regression of the height on a constant and a binary variable ( BJunior ), which takes a value of one for juniors and is zero otherwise, yields the following result: = 71.0 – 4.84× BJunior , R 2 = 0.40, SER = 2.0 (0.3) (0.57) (a) What is the interpretation of the intercept? What is the interpretation of the slope? How tall are juniors, on average? (b) Test the hypothesis that juniors, on average, are shorter than seniors, at the 1% level.

4. A subsample from the Current Population Survey is taken, on weekly earnings of individuals, their age, and their occupation (service industry (‘SI’) vs. non-service industry (‘non-SI’)). To test the hypothesis that people in the service industry earn less, you first regress earnings on a constant and a binary variable, which takes on a value of 1 for workers in the service industry and is 0 otherwise. The results were: = 570.70 – 170.72 × SI , = 0.084, SER = 282.12. (a) There are 850 workers from the service industry in your sample and 894 from other industries. What are the mean earnings of SI and non-SI workers in this sample? What is the percentage of average SI income to non-SI income? (b) You decide to control for age (in years) in your regression results because older people, up to a point, earn more on average than younger people. This regression output is as follows: = 323.70 – 169.78 × SI + 5.15 × Age , = 0.135, SER = 274.45. Interpret these results carefully. How much, on average, does a 40-year-old SI worker make per year in your sample? What about a 20-year-old non-SI worker? Does this represent stronger evidence for the difference in earnings? Answer: (a) Non-SI workers earn $570.70, SI workers 570.70 - 170.72= $399.98. Percentage of average SI income to non-SI income is 399.98/570.70 = 70.1% in the sample. (b) As individuals become one year older, they earn $5.15 more, on average. SI workers earn significantly less money on average and for a given age. 13.5 percent of the earnings variation is explained by the regression. A 40-year-old SI worker earns $359.92 (=323.70 - 169.78*1+5.15*40) , while a 20-year-old non-SI worker makes $426.70 (=323.70+5.15*20) . There is somewhat more evidence here, since age has been added as a regressor. However, many attributes, which could potentially explain this difference, are still omitted. 5. You collect data randomly for 100 national universities and liberal arts colleges from the U.S. News and World Report annual rankings. Next you perform the following regression = 7,311.17 + 3,985.20 × Reputation – 0.20 × Size (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) R 2 = 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. (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? (c) You want to test simultaneously the hypotheses that β size = 0 and β Dilbert = 0. Your regression package returns the F -statistic of 1.23. Can you reject the null hypothesis? (d) Eliminating the Size and Dlibart variables from your regression, the estimation regression becomes = 5,450.35 + 3,538.84 × Reputation + 10,935.70 × Dpriv – 2,783.31 × Dreligion ; (1,772.35 (590.49) (875.51) (1,180.57) R 2 = 0.72, SER = 3,792.68 Why do you think that the effect of attending a private institution has increased now? Answer: (a) The coefficient on liberal arts colleges, is not significantly different from zero (as usual, divide the coefficient estimate by the corresponding standard error, so for Dlibart, it is - 416.38/1,221.92) . 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) Using a one-sided alternative hypothesis, the p -value is 6.2 percent (t= - 0.2/0.13= -1.54 & P(Z<- 1.54)=0.062). 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) The critical value for F 2,∞ is 3.00 (5% level) and 4.61 (1% level). Hence you cannot reject the null hypothesis in this case. (d) Private institutions are smaller, on average, and some of these are liberal arts colleges. Both of these variables had negative coefficients.