1 Economics 835 Spring 1994 D. Maki FINAL EXAMINATION Instructions: There are three parts to this examination, each consisting of two questions. Answer one question from each part. All parts count equally. Please use a separate examination booklet for each question answered. This is a closed book examination - no books or notes are allowed. Total time allowed = 4 hours (must stop promptly at 4:00). I. Part I. Answer one of the following two questions. 1. The following equation was estimated to explain a short- term interest rate (standard errors in parentheses): Yt = 5.5 + 0.93 xt - 0.38 xt-1 - 5.2 (Pt/Pt-4) + 0.50 Yt-1 (1.3) (0.04) (0.09) (1.3) (0.07) - 0.05 (D1-D4) + 0.08 (D2-D4) + 0.06 (D3-D4) (0.04) (0.04) (0.04) _ R2 = 0.90 R2 = 0.89 SEE = 0.19 DW = 1.3 N = 92 where Y = interest rate on 4-6 month commercial paper (percent), x = interest rate on 90 day T-bills (percent), P = an aggregate price index, and Di = a dummy variable = unity for quarter i and zero otherwise. a. What is the estimated seasonal pattern? If you had access to the original data, how would you test whether there is significant seasonality in the data? b. Would the corrected R2 increase or decrease if (Pt/Pt-4) were dropped? Explain. c. What are the long-run (steady state) values of the partial derivatives of Y with respect to x and (Pt/Pt-4)? Explain how you would obtain confidence intervals for these partial derivatives, if you had access to the requisite output. d. Instead of using (Pt/Pt-4), suppose we use the percentage rate of inflation, dP, defined by: dP = 100((Pt - Pt-4)/Pt-4)) What will be the new coefficients and their standard errors? 2. Five equations are reported regressing Y (birth weight of babies, in grams) on D1 (unity for mothers who have given birth before, zero otherwise), D2 (unity for unsupported mothers, zero otherwise) and X (number of cigarettes smoked per day by the mother during pregnancy). Standard errors are in parentheses, and N = 964. 2 1. Y = 3,418 - 7.2X, R2 = 0.012 (2.1) 2. Y = 3,412 - 169D2, R2 = 0.014 (47) 3. Y = 3,386 + 109D1 - 132D2 - 7.2X, R2 = 0.040 (27) (47) (2.1) 4. Y = 3,363 + 143D1 - 4.0X - 8.1(D1*X), R2 = 0.036 (29) (2.8 (4.1) 5. Y = 3,385 + 113D1 - 117D2 - 72(D1*D2) - 7.3X, R2= 0.041 (28) (52) (115) (2.1) A. For each of the following hypotheses, either calculate the test statistic or explain what additional information you would need in order to perform the test. Assume D1 measures "parity", D2 "support", and Y "weight". (a) The effect of smoking is unaffected by parity or support. (b) Given parity and support, one more cigarette per day reduces weight by 4 grams. (c) The effect of parity and support, given smoking, is zero. (d) H0: Eqn. 5 is correct, versus H1: Eqn. 4 is correct. (e) The number of previous children matters. B. In equation 3, provide an expression for the bias in the coefficient of X if W = mother's weight is a missing variable. C. How would you estimate, if the effect of X is different for < one pack a day versus > one pack a day? Part II. Answer one of the following two questions. 3. Given the following ten observations of annual time series data on seven variables: Y1 Y2 Y3 Y4 Y5 Y6 X 3.98 4.69 0.99 4.14 3.63 5.90 2.00 4.48 5.19 4.67 4.32 3.03 1.75 1.00 2.98 4.19 5.33 3.96 4.33 3.17 3.00 5.98 4.19 5.66 3.78 4.93 6.11 4.00 3.98 4.39 5.99 3.60 6.23 6.43 5.00 7.98 6.69 6.98 7.56 5.93 6.22 8.00 5.48 5.39 6.65 7.74 7.53 5.85 7.00 7.48 4.69 6.32 7.92 6.53 6.54 6.00 6.48 8.19 7.31 7.38 9.53 7.04 9.00 8.70 9.79 7.64 7.20 5.83 8.38 10.00 If you regress any of the Y's on X the same equation results: Yi = 3.00 + 0.50 X R2 = 0.63 (3.6) (3.7) (t-values in parentheses) 3 For each of these six equations, note whether you suspect any econometric problems, what problems (if any) you suspect, how you would more formally test for the existence of this problem, and how you would correct the problem if testing indicated it exists. 4. This question consists of three unrelated parts. Answer all parts. a. Suppose income is the dependent variable in a regression, and contains errors of measurement caused by (a) people not knowing their exact income, and always guessing within 5% of the true value, or (b) people rounding their reported income to the nearest $1000. What is the effect of each of these situations on the properties of the OLS estimators, and how would you estimate the coefficients of the model? b. Assume the CLR model applies to Y = a + b1X1 + b2X2 + e. Most samples are such that X1 and X2 are correlated, but by extreme good fortune you obtain a sample where they are not. You regress Y on X1 and an intercept, producing b1*. Are b1* and its estimated variance unbiased? Explain. c. An equation is estimated using time series data: Yt = 2.7 + 0.4 Xt + 0.9 Yt-1 R2 = 0.98, DW = 1.97 (0.4) (0.06) Standard errors in parentheses. Which of the following statments are correct? Explain. (i) Since the DW is so close to two, the estimators are unbiased. (ii) Since the R2 is so high, this is a useful equation for explaining the data generating process for Y. (iii) One should estimate this equation using Xt-1 as an instrument for Yt-1. Part III. Answer one of the following two questions. 5. Consider the four equation supply-demand system for two commodities, A and B: 1. QA = a0 + a1 PA + a2 X1 + a3 X2 + u1 2. PA = b0 + b1 QA + b2 X1 + u2 3. QB = c0 + c1 PB + c2 QA + c3 X2 + c4 X3 + u3 4. PB = d0 + d1 QB + d2 PA + d3 X1 + d4 X4 + u4 where Q and P are price and quantity respectively, the subscripted X's are predetermined variables, and the subscripted a, b, c and d's are coefficients to be 4 estimated. If it helps, you can consider equations 1 and 3 to be demand functions, and 2 and 4 to be supply functions; with commodity B used as an input in producing commodity A (e.g. steel and automobiles). The X2 variable could represent income, X1 something that affects both supply and demand in the auto industry, and X3 and X4 factors that affect the demand and supply of steel, respectively. a. Is this system identified, using the rank and order conditions? b. How would you estimate the coefficients of the identified equations? (Be specific). c. If any equations are not identified, how might you change the specification to obtain identification using zero restrictions, assuming you cannot get more data, drop intercepts or resort to nonlinearities? 6. This question consists of two unrelated parts. Answer both parts. a. Econometrician A states: "There is no real need to worry about the identification properties of a system of equations before estimation, since attempting to estimate an underidentified equation using 2SLS in most computer programs, including SHAZAM, will result in an error message". Econometrician B rejoins: "You are only half right. There is no error message, but if the 2SLS results are the same as the OLS results, this means the equation was underidentified". Do you agree with either of these people? Explain. b. Consider the simultaneous equation model (all variables mean deviated: y1 = a1 y2 + u1 y2 = a2 y1 + b1 x1 + b2 x2 + u with XtX = 1 0 and XtY = 2 3 0 1 3 4 What are the 2SLS estimates of the identified parameters? END OF EXAMINATION _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Question 2 is based on an example in Dougherty, 4 and 6 are from Kennedy (slightly revised), and the rest are based on examples and questions in Maddala.