Final Exam - section 1. Thursday, December hours, 30 minutes

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1 Econometrics, ECON312 San Francisco State University Michael Bar Fall 2013 Final Exam - section 1 Thursday, December 19 1 hours, 30 minutes Name: Instructions 1. This is closed book, closed notes exam. 2. You can use one double-sided sheet of paper, letter size (8½ 11 in or mm), with any content you want. 3. No calculators of any kind are allowed. 4. Show all the calculations. 5. If you need more space, use the back of the page. 6. Fully label all graphs. Good Luck

2 1. (50 points). Professor Stephen Curry is a chair of economics department in a large state university. In his department, students are not required to take econometrics course, but can take it as an elective. In order to study the effects of econometrics course on starting salary of economics majors, prof. Curry collected a sample of 50 recent economics graduates (20 female, and 30 male), with the following variables: salary gpa metrics female starting annual salary in dollars grade point average, on a 4.0 scale dummy variable (= 1 if a student took econometrics course, 0 otherwise) dummy variable (= 1 if the student is female, 0 if male) Curry s Stata command and regression output are presented below: regress salary gpa metrics female Source SS df MS Number of obs = 50 F( 3, 46) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = salary Coef. Std. Err. t P> t [95% Conf. Interval] gpa metrics female _cons salary a. What is the dependent variable in the above regression? b. Write the fitted (predicted) equation for the regression model estimated by Curry. salary b1 b2 gpa b3metrics b4 female 1

3 c. Based on the fitted equation from the last section, how would you predict the starting salary of a male student, with 2.0 gpa, who did not take econometrics course. You do not need to calculate the final answer. Just write the formula, and substitute the numbers. salary b gpa b 2 b metrics 0 b female 0 d. Interpret the estimated coefficient on gpa. b means that an increase in student s gpa by 1 point (say from 2 to 3), all else being equal, is predicted to increase the starting salary by $ per year. e. Interpret the estimated coefficient on metrics b means that the starting salary of economics graduates who took econometrics, is predicted to be $ higher than the salary of students who did not take econometrics, all else being equal. 2

4 f. Interpret the estimated coefficient on female. b means that the starting salary of female economics graduates, all else being equal, is $ lower than that of male economics graduates. g. Suppose that prof. Curry wants to test whether starting salaries of female economists are different from those of male. Write the null and alternative hypotheses for this test. H H 0 1 : : h. Based on the reported p-values, what is your conclusion about the test in the last section, assuming significance level of 0. 05? Explain your answer. The reported p-value in Stata output is the smallest significance level at which we can reject the null hypothesis 0 against 0. The relevant p-vale for the above test is > 0.05, and therefore we fail to reject the null hypothesis at significance level of We conclude that there is not enough evidence that starting salaries of female economists differ from those of male. 3

5 i. Suppose that prof. Curry wants to test whether econometrics adds $9000 to the starting salary of economics graduates. Write the null and alternative hypotheses for this test. H H 0 1 : : j. Based on the reported confidence intervals, what is your conclusion about the test in the last section? Explain your answer. Stata output reports the 95% confidence intervals, which contain all the null values of unknown coefficients, that cannot be rejected at significance level of 0. 05, based on the sample at hand. The relevant confidence interval for the above test is [ , ], which contains Therefore, we fail to reject the null hypothesis at significance level of 0. 05, and conclude that the value of econometrics to the starting salaries of economists, could be $

6 2. (20 points). Cassidy is a banker, who wants to study the factors that affect the chances of mortgage delinquency. He collected data on 1000 single family homes, with the following information about their mortgages: delinquent dummy variable (= 1 if payment late by 90+ days, 0 otherwise) arm dummy variable (= 1 if adjustable rate mortgage, 0 if fixed) ref dummy variable (= 1 if for refinance, 0 if for purchase) insur dummy variable (= 1 if has mortgage insurance, 0 otherwise) lvr loan to value ratio, in percent rate initial interest rate, in percent amount loan amount in $100,000 units credit credit score, in 100s points (ranges ) term loan term in years Cassidy s Stata commands and output are presented below. probit delinquent lvr ref insur rate amount credit term arm Probit regression Number of obs = 1000 LR chi2(8) = Prob > chi2 = Log likelihood = Pseudo R2 = delinquent Coef. Std. Err. z P> z [95% Conf. Interval] mfx lvr ref insur rate amount credit term arm _cons Marginal effects after probit y = Pr(delinquent) (predict) = variable dy/dx Std. Err. z P> z [ 95% C.I. ] X lvr ref* insur* rate amount credit term arm* (*) dy/dx is for discrete change of dummy variable from 0 to 1 5

7 a. Interpret the estimated marginal effect of lvr. A 1% increase in loan to value ratio, increases the probability of delinquency by , or 0.15%, holding all other regressors at their sample mean values. b. Interpret the estimated marginal effect of arm. The probability of delinquency on adjustable rate mortgage is 13.5% higher than that of a fixed rate mortgage, holding all other regressors at their sample mean values. c. Interpret the estimated marginal effect of credit. An increase in credit score by 100 points, reduces the probability of delinquency by 3.9%, holding all other regressors at their sample mean values. d. Before estimating the probit model, Cassidy estimated the linear probability model. His Stata commands and output are as follows: regress delinquent lvr ref insur rate amount credit term arm predict P summarize P Variable Obs Mean Std. Dev. Min Max P Based on the above Stata commands and output, briefly explain why Cassidy prefers the probit model over the linear probability model. The linear probability model predicts probabilities that are not necessarily within the [0, 1] interval. In the above table we see that the linear probability model predicts negative probability of delinquency for some mortgages, which does not make sense. 6

8 3. (10 points). The following graph shows the predicted probability, based on the logit model, of graduating from high school, as a function of ASVABC scores (a test of knowledge in English and Math). Predicted Probabilities, Logit Model Pr(GRAD) ASVABC Based on the above figure, which statement about the marginal effect of ASVABC on the probability of graduating from high school is true? Circle the correct answer. a. The marginal effect of ASVABC is constant for all levels of ASVABC b. The marginal effect of ASVABC is increasing in ASVABC c. The marginal effect of ASVABC is decreasing in ASVABC 7

9 4. (10 points). Suppose that theory and common sense suggest that housing prices depend on living area, number or rooms, age of the house, crime rate in the neighborhood, and quality of schools in the area. a. Suppose that a researcher estimates housing prices, but forgot to include the quality of schools as one of the regressors. What are the likely consequences for the estimated coefficients on the included variables? Circle the correct answer. i. The OLS estimators are biased and inconsistent. ii. The OLS estimators are biased but consistent. iii. The OLS estimators unbiased but inconsistent. iv. The OLS estimators are unbiased and consistent, but inefficient. v. The OLS estimators are biased and inconsistent, but efficient. b. Suppose that the researcher realizes that quality of schools is a relevant variable affecting housing prices, but unfortunately there is no data on quality of schools. Propose a solution to this problem. Be specific. The researcher can use proxy variables, such as teacher/students ratio, or students achievements on national tests in English, Math, etc. Another proxy could be teacher evaluations or credentials. Proxy variables are strongly correlated with the variable of interest (here quality of schools), and therefore can replace the missing variable. 8

10 5. (10 points). Define the following concepts: a. Estimator An estimator is a function of a random sample. b. Unbiased estimator An estimator ˆ of the true population parameter is unbiased, if E ( ˆ). In words, an estimator is unbiased if its mean is equal to the true population parameter it tries to estimate. 9