2) For a normal distribution, the skewness and kurtosis measures are as follows: A) 1.96 and 4 B) 1 and 2 C) 0 and 3 D) 0 and 0

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1 Introduction to Econometrics Midterm April 26, 2011 Name Student ID MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. (5,000 credit for each correct answer.) 1) Ideal randomized controlled experiments in economics are 1) A) often performed in practice. B) often used by the Federal Reserve to study the effects of monetary policy. C) useful because they give a definition of a causal effect. D) sometimes used by universities to determine who graduates in four years rather than five. 2) For a normal distribution, the skewness and kurtosis measures are as follows: 2) A) 1.96 and 4 B) 1 and 2 C) 0 and 3 D) 0 and 0 3) Which of the following statements is correct? 3) A) ESS > TSS B) ESS = SSR + TSS C) TSS = ESS + SSR D) R2 = 1 - (ESS/TSS) 4) An implication of ^ ^ n ( 1 1) d var(vi) N(0, ) is that 4) [var(xi)]2 A) OLS is BLUE. B) ^ 1 is unbiased. C) ^ 1 is consistent. D) there is heteroskedasticity in the errors. 5) The slope estimator, 1, has a smaller standard error, other things equal, if 5) A) there is more variation in the explanatory variable, X. B) there is a large variance of the error term, u. C) the intercept, 0, is small. D) the sample size is smaller. 6) The power of the test 6) A) is the probability that the test correctly rejects the null when the alternative is true. B) depends on whether you use Y or Y2 for the t-statistic. C) is the probability that the test actually incorrectly rejects the null hypothesis when the null is true. D) is one minus the size of the test. 1

2 ^ 7) The normal approximation to the sampling distribution of 1 is powerful because 7) A) many explanatory variables in real life are normally distributed. B) is implies that OLS is the BLUE estimator for 1. C) it allows econometricians to develop methods for statistical inference. D) many other distributions are not symmetric. 8) In multiple regression, the R2 increases whenever a regressor is 8) A) added. B) added unless there is heterosckedasticity. C) added unless the coefficient on the added regressor is exactly zero. D) greater than 1.96 in absolute value. 9) The adjusted R2, or R 2, is given by 9) A) ESS n-1 SSR B) 1- TSS n - k -1 TSS C) 1- n-2 n - k -1 SSR TSS D) 1- n-2 n - k -1 ESS TSS 10) Consider the following multiple regression models (a) to (d) below. DFemme = 1 if the individual is 10) 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 unless: A) Earni = 1 DFemme + 2 Dmale + 3 DMarried + 4 DSingle + 5 X3i B) Earni = DFemme + 3 X3i C) Earni = DFemme + 2 Dmale + 3 X3i D) Earni = DMarried + 2 DSingle + 3 X3i 11) In the multiple regression model, the SER is given by 11) n 1 ^ A) n- k-2 ui 2 n 1 ^ B) n- k-1 u 2 i i=1 i=1 n n 1 ^ 1 C) n-k-1 ui D) n - k -2 ui 2 i=1 i=1 2

3 12) Imagine you regressed earnings of individuals on a constant, a binary variable ( Male ) which 12) takes on the value 1 for males and is 0 otherwise, and another binary variable ( Female ) which takes on the value 1 for females and is 0 otherwise. Because females typically earn less than males, you would expect A) the coefficient for Male to have a positive sign, and for Female a negative sign. B) this to yield a difference in means statistic. C) none of the OLS estimators to exist because there is perfect multicollinearity. D) both coefficients to be the same distance from the constant, one above and the other below. 13) The following OLS assumption is most likely violated by omitted variables bias: 13) A) E(ui Xi) = 0 B) (Xi, Yi) i=1,..., n are i.i.d draws from their joint distribution C) there is heteroskedasticity D) there are no outliers for Xi, ui 14) Panel data estimation can sometimes be used 14) A) to avoid the problems associated with misspecified functional forms. B) to counter sample selection bias. C) in the case of omitted variable bias when data on the omitted variable is not available. D) in case the sum of residuals is not zero. 15) If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then 15) A) all of the hypotheses are always simultaneously rejected. B) a series of t-tests may or may not give you the same conclusion. C) the F-statistic must be negative. D) the regression is always significant. 16) Let R2 unrestricted and R2 restricted be and respectively. The difference between the 16) unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is A) 8.01 B) C) 4.61 D) ) All of the following are true, with the exception of one condition: 17) A) a high R2 or R 2 does not mean that there is no omitted variable bias. B) a high R2 or R 2 always means that an added variable is statistically significant. C) a high R2 or R 2 does not mean that the regressors are a true cause of the dependent variable. D) a high R2 or R 2 does not necessarily mean that you have the most appropriate set of regressors. 3

4 18) The following is not one of the Gauss-Markov conditions: 18) A) the errors are normally distributed. B) E(uiuj X1,, Xn) = 0, i = 1,, n, j = 1,..., n, i j C) E(ui X1,, Xn) = 0 D) var(ui X1,, Xn) = 2 u, 0 < 2 u < for i = 1,, n, 19) In the case of regression with interactions, the coefficient of a binary variable should be interpreted 19) as follows: A) first set all explanatory variables to one, with the exception of the binary variables. Then allow for each of the binary variables to take on the value of one sequentially. The resulting predicted value indicates the effect of the binary variable. B) first compute the expected values of Y for each possible case described by the set of binary variables. Next compare these expected values. Each coefficient can then be expressed either as an expected value or as the difference between two or more expected values. C) there are really problems in interpreting these, since the ln(0) is not defined. D) for the case of interacted regressors, the binary variable coefficient represents the various intercepts for the case when the binary variable equals one. 20) Sample selection bias 20) A) is more important for nonlinear least squares estimation than for OLS. B) results in the OLS estimator being biased, although it is still consistent. C) occurs when a selection process influences the availability of data and that process is related to the dependent variable. D) is only important for finite sample results. 4

5 PROBLEMS. Write your answer in the space provided or on a separate sheet of paper. 21) To investigate possible gender discrimination in a firm, a sample of 100 men and 64 women with similar job descriptions are selected at random. A summary of the resulting monthly salaries follows: _ Average Salary( Y ) Standard Deviation (s Y ) n Men $3100 $ Women $2900 $ a) What do these data suggest about wage differences in the firm? Do they represent staistically significant evidence that wages of men and women are different?(to answer this question, you need to use statistical inference approach..) (20,000 RMB) b) Do these data suggest that the firm is guilty of gender discrimination in its compensation policies? Explain.(5,000 RMB) 22) You recall from one of your earlier lectures in macroeconomics that the per capita income depends on the savings rate of the country: those who save more end up with a higher standard of living. To test this theory, you collect data from the Penn World Tables on GDP per worker relative to the United States (RelProd) in 1990 and the average investment share of GDP from (SK ), remembering that investment equals saving. The regression results in the following output: RelProd = SK, R2=0.46, SER = 0.21 (0.04) (0.38) (a) Interpret the regression results carefully. ((15,000RMB) 5

6 (b) Calculate the t-statistics to determine whether the two coefficients are significantly different from zero. Justify the use of a one-sided or two-sided test.((20,000rmb) (c) You accidentally forget to use the heteroskedasticity-robust standard errors option in your regression package and estimate the equation using homoskedasticity-only standard errors. This changes the results as follows: RelProd = SK, R2=0.46, SER = 0.21 (0.04) (0.26) You are delighted to find that the coefficients have not changed at all and that your results have become even more significant. Why haven t the coefficients changed? Are the results really more significant? Explain.(15,000RMB) (d) Upon reflection you think about the advantages of OLS with and without homoskedasticity-only standard errors. What are these advantages? Is it likely that the error terms would be heteroskedastic in this situation? (20,000RMB) 6

7 23) Earnings functions attempt to predict the log of earnings from a set of explanatory variables, both binary and continuous. You have allowed for an interaction between two continuous variables: education and tenure with the current employer. Your estimated equation is of the following type: ^ ^ ^ ^ ^ ln(earn) = Femme + 2 Educ + 3 Tenure + 4 x (Educ Tenure) + where Femme is a binary variable taking on the value of one for females and is zero otherwise, Educ is the number of years of education, and tenure is continuous years of work with the current employer. i) What is the effect of an additional year of education on earnings ( returns to education ) for men? For women?(5,000rmb) ii) If you allowed for the returns to education to differ for males and females, how would you respecify the above equation? What is the effect of an additional year of tenure with a current employer on earnings?(15,000rmb) 24) (30,000RMB)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: Included observations: 420 7

8 Variable Coefficient Std. Error t-statistic Prob. C STR EL_PCT LOG(AVGINC) MEAL_PCT CALW_PCT R-squared 0.80 Mean dependent var Adjusted R-squared 0.79 S.D. dependent var S.E. of regression 8.64 Akaike info criterion 7.16 Sum squared resid Schwarz criterion 7.22 Log likelihood F-statistic Durbin-Watson stat 1.51 Prob(F-statistic) 0.00 Restricted model: Dependent Variable: TESTSCR Method: Least Squares Date: 07/31/06 Time: 17:37 Sample: Included observations: 420 Variable Coefficient Std. Error t-statistic Prob. C STR EL_PCT LOG(AVGINC) R-squared 0.71 Mean dependent var Adjusted R-squared 0.71 S.D. dependent var S.E. of regression Akaike info criterion 7.50 Sum squared resid Schwarz criterion 7.54 Log likelihood F-statistic Durbin-Watson stat 1.30 Prob(F-statistic) 0.00 Calculate the homoskedasticity only F-statistic and determine whether the null hypothesis can be rejected at the 5% significance level. 8

9 25) You have been asked by your younger sister to help her with a science fair project. During the previous years she already studied why objects float and there also was the inevitable volcano project. Having learned regression techniques recently, you suggest that she investigate the weight-height relationship of 4th to 6th graders. Her presentation topic will be to explain how people at carnivals predict weight. You collect data for roughly 100 boys and girls between the ages of nine and twelve and estimate for her the following relationship: Weight = Height4, R2 = 0.55, SER = (3.81) (0.46) where Weight is in pounds, and Height4 is inches above 4 feet. (a) Interpret the results.((15,000rmb) (b) You remember from the medical literature that females in the adult population are, on average, shorter than males and weigh less. You also seem to have heard that females, controlling for height, are supposed to weigh less than males. To see if this relationship holds for children, you add a binary variable (DFY) that takes on the value one for girls and is zero otherwise. You estimate the following regression function: Weight = DFY Height (DFY Height4), (5.99) (7.36) (0.80) (0.90) R2 = 0.58, SER = What do you learn from this regression? For example, i) Are the signs on the new coefficients as expected? Explain.(5,000RMB) ii)are the new coefficients individually statistically significant? You need to give specific steps.calculation in deriving your answer.(10,000rmb) iii) Write down and sketch the regression function for boys and girls separately.(15,000rmb) 9

10 (c) The medical literature provides you with the following information for median height and weight of nine- to twelve-year-olds: Median Height and Weight for Children, Age 9-12 Boys' Weight Boys' Height Girls' Weight Girls' Height 9-year-old year-old year-old year-old Insert two height measures each for boys (9-year-old, and 11-year-old) and girls (10-year-old, and 12-year-old) and see how accurate your predictions are compared to the table above.(10,000rmb) (d) To test if the intercept and slope for boys and girls are identical, how to set up your hypothesis? (5,000RMB) If the test statistic is Use the given table to find the critical values at the 5% and 1% level, and make a decision. (10,000RMB) 10

11 Allowing for a different intercept with an identical slope results in a t-statistic for DFY of ( 0.35). Having identical intercepts but different slopes gives a t-statistic on (DFYHeight4) of ( 0.35) also. Does this affect your previous conclusion? (e) Assume that you also wanted to test if the relationship changes by age. Briefly outline how you would specify the regression including the gender binary variable and an age binary variable (Older) that takes on a v alue of one for eleven to twelve year olds and is zero otherwise. Indicate in a table of two rows and two columns how the estimated relationship would vary between younger girls, older girls, younger boys, and older boys. 11

Contest Quiz 3. Question Sheet. In this quiz we will review concepts of linear regression covered in lecture 2.

Contest Quiz 3. Question Sheet. In this quiz we will review concepts of linear regression covered in lecture 2. Updated: November 17, 2011 Lecturer: Thilo Klein Contact: tk375@cam.ac.uk Contest Quiz 3 Question Sheet In this quiz we will review concepts of linear regression covered in lecture 2. NOTE: Please round