Set #2 Economics 141 Fall 2016 Due September 29 Review Exercises (not to turn in): Exercises 2.2, 2.4, 2.5, and 2.7 from Wooldridge's text. 1. Suppose you are given the following (semi-fabricated) data on a typical automobile's fuel consumption (F ) and automobile speed (S): Fi (miles/gallon) Si (miles/hour) 10 18 25 29 30 28 25 22 18 15 11 8 10 20 30 40 50 60 70 80 90 100 110 120 with P i Fi P P 2 i Fi i Fi P i Si = 239 P 2 i Si = 5417 Si = 14350: = 780 = 65000 It is conjectured that there is a linear relationship between Fi and Si : Fi = 0 + 1 S i + Ui ; where Ui is a random error term assumed to have zero mean and constant variance (a) What are the least squares estimates of 0 (b) What are the residual sum of squares and and R2 2: 1? for this regression? (c) Plot Fi as a function of Si and discuss the appropriateness of the speci...cation used above. On the same graph, plot the ...tted values F^i ^ 0 + ^ 1 S_ i against Si : Are the estimated residuals \"nicely behaved,\" i.e., do they look independent and centered at zero? 1 (d) Re-estimate 0 and R2 of this regression. 1 using only the last eight observations (with Si 50) and calculate the (e) On the basis of this exercise what might you conclude about the relationship between speed and fuel economy? What might be a better model for this relationship than the one considered so far? 2. Short Answer: Give a brief answer, explanation, and/or mathematical derivation to the questions below. 1. (a) Suppose it is observed that, for a set of data points f(Xi ; Yi )g which are assumed to satisfy a simple linear regression model, the sample mean of Yi is greater than the absolute value of the sample mean of Xi ; but the sample variance of the dependent variable Yi is less than the sample variance of the independent variable Xi - that is, Y > jXj and s2Y < s2X : What, if anything, does this imply about the absolute value of the least-squares slope coe cient estimate ^ ? How about the sign of the intercept term ^ ? (b) A fellow researcher ...ts a simple linear regression model Yi = 0 + 1 Xi + Ui by least squares, but has doubts that the standard assumption E(Ui ) = 0 is correct for her data set. She decides to try to test for this assumption by calculating the residuals from her regression, e^i = Yi Y^i ; and then performing the usual t-test of the null hypothesis that their (population) mean is zero, assuming the errors are normally distributed. Evaluate this proposal. (c) Suppose that, instead of ...tting the regression model Yi = 0 + 1 Xi + Ui by least squares, you instead ...t the model Yi = 0 + 1 Zi + Ui ; where Zi 3 + 10Xi : How are the least squares estimators ^ 0 and ^ 1 for this second model related to the corresponding LS estimators ^ 0 and ^ of the ...rst model? [Hint: ...gure out how Z; 2 ; and ^ Y Z are related to X; ^ X ; and ^ Y X :] 1 Z (d) In trying to model the demand for money as a function of interest rates (using a simple regression model), would you rather observe economic data during a period in which interest rates were relatively stable, or a period in which rates were volatile? Why? (e) Two research workers, working independently of each other, estimated the coe cients of a simple linear model Yi = + Xi + Ui by the method of least squares, each drawing samples of the same size from data with the same population parameters (but dierent nonrandom regressors). When they found out about each other's work, they decided to pool their results to obtain one joint estimator of : Two possible ways of doing this were considered: (i) Taking a simple arithmetic mean ( ^ 1 + ^ 2 )=2 of the two estimators; or (ii) Combining the two samples and obtaining a new estimator of by least squares. Do these two methods dier? If so, which is preferred under the Gauss-Markov assumptions for the simple linear model? (f) A research paper reports, \"...in a simple linear regression with 225 observations, the estimated slope coe cient ^ = 0:6 is statistically signi...cantly dierent from zero at a 5% level, since its t-statistic is 3.0.\"Suppose you want to test the null hypothesis that the true slope coe cient is equal to one, not zero. Would you reject this null hypothesis at a 5% signi...cance level, assuming the error terms in the model are normally distributed and the other standard assumptions of the simple linear model hold? 2 (g) "In the simple linear regression model under the assumptions of the Gauss-Markov Theorem, a necessary and su cient condition for the least-squares estimators of the intercept and slope coe cients to be uncorrelated is that the sample average of the dependent variable is zero.\" True or False? Explain. 3. Here are some summary statistics for a sample of N = 20 observations on pairs f(Xi ; Yi )g: P i Xi P 2 i Xi P i Yi P = 40 i Yi P 2 i Yi = 160 Xi = 100 = 30 = 105 Assume these data were generated from a simple linear model Yi = 0 + 1 Xi + Ui ; where the regressors fXi g are nonrandom and the error terms Ui are independently, identically, and normally distributed, i.e., Ui i:i:d: N (0; 2 ): (a) What are the least squares estimates of 0 and 1 ? (b) Calculate the sum of squared residuals (SSR) and the R2 for this regression. (c) Test the null hypothesis H0 : 1 = 0 against the alternative HA : 1 6= 0 at a 5% signi...cance level. Be sure to explicitly state the critical value you use for your test. [Hint: there are dierent ways to perform this test; not all are equally cumbersome.] (d) Give an expression for a 95% con...dence interval for the parameter = 0 + 1 : Your expression may be a bit messy and involve square roots, but simplify as much as possible, and show your intermediate calculations. Also, be sure to explicitly state any \"critical value\"you use. Is 0 = 0 in the con...dence interval? 4. In a study of the eects of job training (T RAIN IN G) on earnings (EARN IN GS), the following least squares regression results were reported: d EARN IN GS = 14:8 + 0:5 T RAIN IN G (1:9) (0:3) Sample Size N = 400 Standard Error of the Regression SER = 6:4 (Heteroskedasticity-Consistent Standard Errors in Parentheses) 1. (a) Test the null hypothesis that the true eect of job training on earnings is non-positive (i.e., H0 : 0, where is the true slope coe cient) at an approximate 5% signi...cance level. (b) Suppose you believe that the assumption of homoskedastic errors is appropriate, and that the sample variance of job training, ^ 2T RAIN IN G , equals 1:28: Construct an approximate 95% con...dence interval for the true slope coe cient under the homoskedasticity assumption (that is, using a "non-robust" standard error), and compare it to the corresponding 95% con...dence interval using the heteroskedasticity-robust standard errors. 3