Economics 835 Spring 1998 D. Maki Midterm Examination Instructions: There are three parts to this examination, each with two questions. Answer one question from each part (a total of three questions) Questions are equally weighted. Total time allowed = three hours. This is a closed book examination. Throughout the examination, I am not looking for SHAZAM commands as answers. Part I: Answer one of the following two questions. 1. a. A two-factor Translog production function, constrained for Hick's Neutral Technical Change is: lnQ=á0 + á1T + á2lnL + á3lnK + á4(lnL)2 + á5(lnL)(lnK) + á6(lnK)2, where Q is output, k is capital input, L is labour input and T is a time trend. Constant Returns To Scale (output doubles when both inputs double) implies that á2+á3=1, and á4=á5=á6. Explain in detail how you would test for CRTS given HNTC. b. If HNTC did not hold, three terms would be added to the above equation: + á7T(lnL) + á8T(lnK) + á9T2. Explain in detail how you would test for HNTC, if CTRS is not assumed. c. A Cobb-Douglas production function is the same as the Translog constrained for HNTC, except all the second order terms are eliminated. Explain in detail how you would test whether the Cobb-Douglas or the following linear production function is appropriate, using Box-Cox estimation: Q=á0 + á1eT + á2L + á3K. 2. Answer all three unrelated parts of this question. a. Assume a theoretical production function: lnQ=á1 + á2lnL + á3lnK + á4lnM + u, while the sample equation estimated is lnQ=B1 + B2lnL+ B3lnK + v. Here Q is output, L is labour input, K is capital input and M is managerial input, which is unobservable. If the estimate of the scale effect is given by S=B2+B3, is S biased? Always or only under some conditions? Explain. If it is biased, derive an expression for the bias, and suggest a sign for the bias under reasonable assumptions (state your assumptions). b. The exponential distribution has the probability density function: f(X) = (1/é)exp(-X/é) for X>0. Derive the Maximum Likelihood Estimator for é. c. For the general linear model, derive the formulas for confidence intervals for forecasts (both interpretations). Part II. Answer one of the following two questions. 3. What would the following functions look like in Y-X space, and how would you estimate the parameters? (Assume all values of the variables and parameters are positive). a. Y=á1 + á2X + á3(1/X) + u b. lnY=á1 + á2X + á3X2 + u c. Y=á1 + á2X + á3lnX + u d. Y=á1 + exp(á2+á3X2+u) e. Y=1/[1+exp(á1+á2X+u) 4. For the three variable model, if you mean deviate all three variables, a. Show what the XtX and XtY matrices look like. b. Find the elements of (XtX)-1 in notation of your choice. If you went one step further in standardizing the variables by transforming each variable by subtracting the sample mean and dividing the result by the sample standard deviation, the resulting coefficients are the standardized partial regression coefficients (Beta coefficients). c. Show the relationship between these standardized partial regression coefficients and the usual partial regression coefficients. d. Now what would the XtX, XtY and (XtX)-1 matrices look like? Part III. Answer one of the following two questions. 5. Using census data for the 50 U.S. states, the following model was estimated (standard errors in parentheses): PC = - 0.365 - 0.0009 D + 0.0094 NW + 0.0003 Y - 0.099 U + (0.978) (0.0006) (0.0104) (0.0001) (0.084) 1.519 COP - 0.0068 AGE1 + 0.0077 AGE2, R2=0.484, RSS=33.41 (0.276) (0.000034) (0.0038) where PC=property crime index, D=population density, NW=percent of population non-white, Y=per capita income, U=unemployment rate, COP=police force per thousand population, AGE1=population (000 persons) in the age group 15-24, AGE2=population (000 persons) in the age group 25-34. a. Are any of the coefficient signs (with t-values > 1) surprising? Explain. Another regression was run, dropping variables with coefficients having t-values < 2: PC = - 0.037 + 0.0002 Y + 1.428 COP - 0.0061 AGE1 + ((0.84) (0.00008) (0.266) (0.839) 0.0068 AGE 2, R2=0.468, RSS=36.85 (0.839) b. In the second estimation, are any of the coefficient signs (with t-values > 1) surprising? Explain. c. Calculate the appropriate test statistic to see whether dropping the coefficients with t-values < 2 produced a significantly worse "fit". d. In the first equation presented, explain in detail how would you test whether the two age distribution variables, considered together, matter. e. For the first equation presented, calculate as much of the ANOVA table as you can. 6. A few years ago, someone took data on the top 25 business schools in the U.S., and regressed the average gain in salary for persons graduating from the MBA program (SG) on annual tuition (TUT) and external and internal ratings of reputation (Z1 - Z5), taking on the value 1 (highest rating) through 5 (lowest rating). Z1=rating of MBA skills at being analysts, Z2=MBA skills at being team players, and Z3 is MBA skills in having a global view, all three of these as rated by recruiters(external). Z4 is teaching evaluations by MBA's, and Z5 is curriculum evaluation by MBA's (internal). Some estimated equations follow (t-values in parentheses, intercepts estimated but not reported). (1) SG=0.031*TUT-1.25*Z1+0.56*Z2-1.76*Z3-0.23*Z4-0.57*Z5 R2=.51,(0.42) (-1.46) (0.83) (-2.09) (-0.24) (-0.46) (2) SG = -1.78*Z1+0.51*Z2-1.85*Z3-0.25*Z4-0.74*Z5 R2=.50, (-1.78) (0.78) (-2.33) (-0.27) (-0.65) (3) SG = -2.32*Z3-1.06*Z5 R2=0.40, (-3.72) (-1.24) (4) TUT = -4.38*Z1-1.85*Z2-2.98*Z3-0.69*Z4-5.56*Z5 R2=0.46, (-1.76) (-0.89 (-1.17) (-0.23) (-1.52) (5) TUT = -6.40*Z1-5.52*Z5 R2=.37, (-3.04) (-2.06) a. Which of the estimated coefficients have theoretically expected signs? Explain. b. Do external ratings affect salary gain? Explain. c. Do internal ratings affect salary gain? Explain. d. What is the relationship between tuition and salary gain? Explain. e. What additional regressions, involving the same data noted above, would you need to more rigorously test (b) and (c) above? What other estimations might you find useful? Explain.