Take-home examination for Econ 140A, Summer 2020, Week 2 • Your answers should be submitted in a single pdf document on GauchoSpace. • You may either type or handwrite your answers, or some combination if you like. You can take pictures of your handwritten responses, then include them in the single pdf document that you submit. Regardless of what you decide, it is important that your answers are as clear as possible. Your answers should appear in the order in which the questions are asked. Please review your answers before submitting them to confirm that they are easily readable. • Be sure that you explicitly answer each question and explain each step, as if you were writing solutions so that another student in the class would be able to follow your thoughts. Part of your grade will depend on explaining each step of your answers. 1. (10 points) Table 1 presents data on number of interviews (X) and number of job offers (Y ) for 15 job seekers. Table 1: Responses by sampled job seekers Job Seeker (k) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Interviews (Xi ) 3 5 1 4 6 10 0 11 4 6 7 7 6 3 4 Job Offers (Yi ) 0 2 1 2 2 8 1 1 4 5 3 8 4 5 2 (a) (1 point) You want to make a prediction of the number of job offers as a function of the number of interviews, Yi = f (Xi ). In particular, you want to make a linear prediction of the number of job offers, i.e., f (Xi ) = aXi . If a = 0.8, what is the P average squared prediction error, i.e., n1 ni=1 (Yi − aXi )2 ? P (b) (1 point) If a = 1, what is the average squared prediction error, i.e., n1 ni=1 (Yi − aXi )2 ? Using your answer to questions (a) and (b), which value of a is a better adjustment in terms of the average squared prediction error, a = 0.8 or a = 1? (c) (1 point) One of your friends suggests that you should compute the regression estimator of a, namely α̂ = Pn Xi Yi Pi=1 n 2 , i=1 Xi to find the best linear predictor of the number of job offers. Calculate α̂. (d) (1 point) What is the average squared prediction error of the regression estimator P α̂, i.e., n1 ni=1 (Yi − α̂Xi )2 . Compare your answer to the answers to questions (a) and (b). P (e) (1 point) Calculate the average prediction error n1 ni=1 (Yi − α̂Xi ). Calculate 1 Pn i=1 Xi (Yi − α̂Xi ). n P (f) (2 points) Prove that n1 ni=1 Xi (Yi − α̂Xi ) = 0 in general. (g) (1 point) Make a scatter plot using the data from Table 1 where the number of interviews is in the x axis and the number of job offers is in the y axis. Add a line representing the best linear predictor, i.e., α̂Xi . (h) (1 point) If you would like to make a prediction for a job seeker that had 8 interviews, how many job offers would you expect the job seeker to have? (i) (1 point) Calculate the R2 of your prediction, i.e., R2 = 1 − P Y = n1 ni=1 Yi . Pn 2 i=1 (Yi −α̂Xi ) P , n 2 i=1 (Yi −Y ) where 2. Use the information on Table 1 to solve the following questions. (a) (1 point) You want to make a prediction of the number of job offers as a function of the number of interviews, Yi = f (Xi ). In particular, you want to make an affine prediction of the number of job offers, i.e., f (Xi ) = b0 +b1 Xi . If b0 = 1 and b1 = 0.2, P what is the average squared prediction error, i.e., n1 ni=1 (Yi − b0 − b1 Xi )2 ? (b) (1 point) If b0 = 0.5 and b1 = 0.8, what is the average squared prediction error, P i.e., n1 ni=1 (Yi − b0 − b1 Xi )2 ? Using your answer to questions 2(a) and 2(b), which values of b0 and b1 are a better adjustment in terms of the average squared prediction error, (b0 , b1 ) = (1, 0.2) or (b0 , b1 ) = (0.5, 0.8)? (c) (1 point) One of your friends suggests that you should estimate the OLS regression Pn (X −X)(Y −Y ) i Pn i function coefficients of b0 and b1 , namely β̂0 = Y − β̂1 X and β̂1 = i=1 (Xi −X)2 i=1 P P where X = n1 ni=1 Xi and Y = n1 ni=1 Yi , to find the best linear predictor of the number of job offers. Calculate β̂0 and β̂1 . (d) (1 point) What is the average squared prediction error of the OLS regression funcP tion, i.e., n1 ni=1 (Yi −β̂0 −β̂1 Xi )2 . Compare your answer to the answers to questions 2(a) and 2(b). (e) (1 point) Calculate the average prediction error 1 n Pn i=1 (Yi − β̂0 − β̂1 Xi ). Do you find this result intuitive? Why? P (f) (1 point) Prove that n1 ni=1 (Yi − β̂0 − β̂1 Xi ) = 0 in general. P (g) (2 points) Prove that n1 ni=1 Xi (Yi − βˆ0 − βˆ1 Xi ) = 0 in general. Show that that equality approximately holds using the data from Table 1. (h) (2 points) Make a scatter plot using the data from Table 1 where the number of interviews is in the x axis and the number of job offers is in the y axis. Add a line representing the OLS linear predictor, i.e., β̂0 + β̂1 Xi . (i) (2 points) Calculate the R2 of the regression, i.e., R2 = 1 − P where Y = n1 ni=1 Yi . Pn (Y −β̂ −β̂ X ) i=1 Pn i 0 1 2 i i=1 (Yi −Y ) 2 , (j) (2 points) Compare your answers to questions 1(i) and 2(i). What method is better? Discuss in no more than five lines. Page 2 (k) (2 points) Cathy who works at the Job Opportunities Office says "everyone of our job seekers receives no job offer if the job seeker has no interviews, but for each additional interview the job seeker receives 0.8 job offers on average." Comment on her statement based on your answer to question 2(c). (l) (2 points) Your friend Paul, who took Econ 140A last year, says that you should run a hypothesis test to evaluate Cathy’s statement. If the null hypothesis is that β1 = 0.8 versus the alternative that β1 6= 0.8, can you reject a 5% level test? Clearly state the standard error of β̂1 , the test statistic, and the critical value. (Hint: the standard error is s.e.(β̂1 ) = √ σ̂ 2 = SX 2 nSX Pn 2 i=1 (Xi −X) n where σ̂ 2 = Pn 2 i=1 (Yi −β̂0 −β̂1 Xi ) n−1 and .) (m) (2 points) Construct a 95% confidence interval for β1 . Interpret this confidence interval in no more than five lines. Would you argue in favor or against Cathy’s statement? 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