Economics 435 Spring 1999 D. Maki FINAL 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. The following model for the income inequality coefficient, called the Gini coefficient (Y), which ranges from 0 to 1 (0 being perfect equality) is estimated on cross-section data for 40 countries. Y=b1 + b2GDP + b3POP + b4URB + b5LIT + b6EDU + b7AGR + u where GDP is the growth rate of gross domestic product, POP is the growth rate of population, URB is the percentage of people living in urban areas, LIT is the percentage of people who can read and write, EDU is the secondary school enrollment as a percentage of the people of secondary school age and AGR is the share of agriculture in GDP. a. Derive another model that will enable you to use the Lagrange Multiplier (LM) test to test whether the above structure is different for socialist versus non-socialist countries (assume it is known which countries fall in what category). Define any new notation, and detail how you would conduct the test. b. Explain how you would do the same thing using a Wald test. c. If you suspected heteroscedasticity was a problem, how would you (i) test for it, (ii) correct for it, if it is found to exist, and (iii) test to see if your correction "worked" if you made one? 2. An index of air quality (AIRQ) for 30 California cities and towns was regressed on: POP=population, VA=value added by industrial manufactureres, RAIN=rainfall, COAST=unity for towns on the coast, DEN=population density, INC=per capita income, POV=poverty rate, ELEC=electricity consumed by industrial manufacturers, FUEL=fuel oil consumed in industrial manufacturing, and EST=number of industrial establishments with 20 or more employees. The initial estimation produced an R2=0.6213 with positive coefficients on POP, RAIN, and DEN, with only POP, COAST and INC having t-values greater than 1.2. The least significant variables were dropped, one at a time, until all t-values were greater than 1.2. At that point the R2=0.5983, with POP, VA, COAST, INC and POV remaining in the equation. No sign changes occurred. a. Test whether the variables dropped from the estimation were jointly significant. b. What were the expected signs for the coefficients (possibly not all expectations were unambiguous)? Explain. c. If there were any unexpected signs in the the initial estimation, why do you think this occurred? Explain. d. If there were any unexpected signs in the the final estimation, why do you think this occurred? Explain. e. If you suspected heteroscedasticity was a problem, how would you (i) test for it, (ii) correct for it, if it is found to exist, and (iii) test to see if your correction "worked" if you made one? PART II. Answer one of the following two questions. 3. Using annual data for the U.S. for the 90 year period covering 1900-1989, a number of equations were run relating the logarithm of income (LY) to the logarithm of the money supply (LM) and an interest rate (IR). The sample statements were adjusted so that all estimations used the same 80 observations (1910-1989). The SHAZAM commands and the resulting R2 values are given below. Eqn. 1: OLS LY LM IR; R2=0.9627 Eqn. 2: OLS LY LM(0.5) IR(0.5); R2=0.9816 Eqn. 3: OLS LY LM(0.10) IR(0.10); R2=0.9879 Eqn. 4: OLS LY LM(0.10,2) IR(0.10,2); R2=0.9865 a. Test the following (use a critical value of 2.00 for all tests): (i). Whether the second degree constraint on 10 lags is binding. (The degrees of freedom is tricky). (ii). Whether 10 lags is better than 5 lags. (iii). Whether 5 lags is better than no lags. b. The coefficient estimates from Eqn. 4 are (t-values in parentheses): Lag LM IR 0 0.53 (9.45) 0.020 (6.08) 1 0.31 (11.80) 0.015 (7.60) 2 0.14 (10.84) 0.011 (7.98) 3 0.02 (0.71) 0.008 (5.45) 4 -0.07 (-2.23) 0.006 (3.47) 5 -0.11 (-3.31) 0.005 (2.65) 6 -0.12 (-3.73) 0.004 (2.69) 7 -0.08 (-3.48) 0.005 (3.59) 8 0.00 (0.20) 0.006 (4.15) 9 0.12 (4.74) 0.008 (3.43) 10 0.29 (5.22) 0.012 (2.88) Is there anything here that you find bothersome? Explain. What would you do about it? 4. The following general version of the capital asset pricing model (CAPM) is used in the finance literature: (A) SRt = B1 + B2MRt + B3RFRt + ut where SR is the rate of return of a company's stock, MR is the rate of return of a market average protfolio (such as the Standard & Poor's stock average), and RFR is the return of a risk-free asset (such as, for example, the three-month Treasury bill). You have data on SR, MR and RFR for a company for a number of periods. a. A more commonly used version of CAPM is the following: (B) SRt - RFRt = b0(MRt - RFRt) + vt Write the null hypothesis on the B's (should not involve b0) which will make model (B) the restricted model and (A) the unrestricted model. Explain how you would perform the Wald test. b. Suppose you have calculated the Durbin-Watson statistic and it indicates the presence of autocorrelation. How would you proceed to deal with the problem, and test whether your procedure corrected this problem? (Remember, SHAZAM commands are not allowed as answers). PART III. Answer one of the following two questions. 5. Consider the three-equation model: A = X - M Endogenous: A, M, X M = a1 + a2Y + a3P + a4U + u Exogenous: Y, P, U X = B1 + B2P + B3A + v Error terms: u, v a. Is the order condition satisfied for this model? Explain. b. Derive the reduced form equations. c. If we estimated the second equation using OLS, what properties would the estimators have? Explain. How about the third equation? d. Would TSLS be useful in estimating the coefficients of this model? Explain. If so, note specifically how this would be done. 6. State whether each of the following statements is true (T), false (F) or uncertain (U), and give a short explanation. a. If the R2 values for the reduced form equations are nearly unity, the OLS and TSLS estimates of the parameters will be very close to each other. b. If any equation in a system is underidentified, the entire system is underidentified, and none of the parameters in any equations can be consistently estimated. c. If the number of parameters in the reduced form model is unequal to the number of parameters in the structural form model the model is underidentified. d. Which variables should be treated as endogenous and which as exogenous cannot be determined from the data. e. If a model is underidentified, the only solution is to find more exogenous variables to be entered into other equations. f. Estimating the structural model with TSLS and then solving for the implied reduced form coefficients is more efficient than estimating the reduced form coefficients directly.