Economics 435 Summer 1994 D. Maki FINAL EXAMINATION Instructions: Three 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. Part I. Answer one of the following two questions. 1. The following ten observations of annual time series data yield the same regression equation for all i if Yi (i = 1, 2, ... 6) is regressed on X: Y = 2.00 + 1.00 X, R2 = 0.87. Y1 Y2 Y3 Y4 Y5 Y6 X 5.90 3.63 4.14 0.99 4.69 3.98 2.00 1.25 2.53 3.82 4.17 4.69 3.98 1.00 3.67 4.83 4.46 5.83 4.69 3.48 3.00 7.11 5.93 4.78 6.66 5.19 6.98 4.00 7.93 7.73 5.10 7.49 5.89 5.48 5.00 9.22 8.93 10.56 9.98 9.69 10.98 8.00 8.35 10.03 10.24 9.15 7.89 7.98 7.00 8.54 8.53 9.92 8.32 6.69 9.48 6.00 10.54 13.03 10.88 10.81 11.69 9.98 9.00 12.38 9.83 11.20 11.64 13.79 12.70 10.00 For each of these six equations, note what econometric problems (if any) you suspect, how you would more formally test for the existence of each problem, and how you would correct if testing confirmed the existence of a problem. 2. Chow (1966) presents the following demand for money equation estimated using quarterly time-series data: Mt = .1365 + 1.069Ypt - .01321Yt - .7476Rt (7.22) (0.10) (13.84) R2=.9965 Where M is the money supply, Yp is permanent income, Y is current income and R is an interest rate (all variables are in logarithms). Taylor and Newhouse (1969), using the same data, present: Mt=.3067 + .06158Ypt + .3274Yt - .3325Rt + .5878Mt-1 (0.43) (3.48) (5.67) (8.79) R2=.9988 Chow claims his estimation shows permanent income is more important than current income in determining the demand for money; Taylor and Newhouse claim their estimation shows the opposite. a. Can the missing variable bias formula can be used to explain the large decrease in the coefficient of Yp when Mt-1 is added to the equation? Explain. b. Could a COMFAC test for AR(1) errors be used to aid in choosing between these two estimations? If so, explain how you would conduct such a test and interpret the results. c. What additional information about the two estimations presented would you find useful in choosing between them, and how would you use this information? Part II. Answer one of the following two questions. 3. Given the two equation model (Y1 and Y2 are endogenous): (1) Y1t = a1 + a2 Y2t + et (2) Y1t = b1 + b2 Y2t + b3 Xt + b4 Y2,t-1 + ut If neither et nor ut is autocorrelated: a. Which of the two equations are identified? If each of them are estimated OLS, which parameter estimators are unbiased, which are consistent, and which are efficient? b. If you were told to estimate the first equation using instrumental variables, what would you do? Be specific. c. If you were told to estimate the first equation using 2SLS, what would you do? Be specific. d. If you were told to estimate the first equation using Indirect Least Squares, what would you do? Be specific. e. How would your answers to parts a through d of this question change if it were known that et was AR(1)? 4. Answer all three (unrelated) parts of this question. A. The structure of a model with four endogenous and three exogenous variables is as follows (X represents presence and 0 absence of a variable in the equation): X 0 X X X 0 0 X X X 0 0 X X 0 0 X 0 X 0 0 X 0 X X 0 X 0 Which equations are underidentified, which are just identified, and which are overidentified? Explain. B. Given the pooled data model: Yit = a + b Xit + eit, where i = 1, 2, ... , N and t = 1, 2, ... , T. How would you estimate if the cross-section error component is known to be heteroscedastic? Would a similar method work when it is the time-series component which is known to be heteroscedastic? Explain. C. Assume that you are estimating a model with two explanatory variables, each of which has a geometric (Koyck) lag. Derive an equation to be estimated when both lags have identical weights. How would your answer change if the two weights were allowed to be different? Part III. Answer one of the following two questions. 5. Kochin and Parks (1984) investigated whether higher- priced houses get assessed (for property taxes) at a lower proportion of value than lower-priced houses, using a sample of 416 houses sold in King County, Washington in 1977-79. The results of five OLS estimations involving assessed value (A) and selling price (S) are reported below (t-values in parentheses). (1) A = 7505.40 + 0.3382 S, R2 = 0.597 (13.42) (24.79) (2) A/S = 0.7374 - 4.5714(10-6) S, R2 = 0.2917 (51.38) (-13.06) (3) ln(A) = 2.8312 + 0.6722 ln(S), R2 = 0.6547 (11.27) (28.01) (4) S = 2050.07 + 1.7669 A, R2 = 0.597 (1.34) (24.79) (5) A/S = 0.5556 + 3.8288(10-7) A, R2 = 0.0004 (27.26) (0.40) a. Interpret the results of each of these estimations in terms of whether there is inequity in real estate assessment. b. Which of the five estimations do you prefer? Explain. c. If there is reason to believe that sales price is a noisy measure (contains error) of the true value of a house, which of the five estimations do you prefer? Explain. d. If you suspect heteroscedasticity in the relationship between A and S, which of the five estimations do you prefer? Explain. e. Is there some estimation you would prefer to any of those given? Explain. 6. Consider two models. In the first, researcher A assumes that firms in a given industry relate their stocks of finished goods (Y) to their expected annual sales (XE) according to the linear relationship: Y = a + bXE, while actual sales (X) differ from expected sales by a random quantity (u) which is distributed with mean zero and constant variance: X = XE + u, and u is distributed independently of XE. In the second model, researcher B assumes that firms in the same industry relate their intended stocks of finished goods (Y*) to their expected annual sales (XE) according to the linear relationship: Y* = a + bXE, while actual sales (X) differ from expected sales by a random quantity (u) which is distributed with mean zero and constant variance: X = XE + u, and u is distributed independently of XE. Since unexpected sales lead to a reduction in stocks, actual stocks (Y) are given by: Y = Y* - u. If both researchers have access to cross- section data on X and Y (but not XE or Y*), what problems will they have in estimating the parameters a and b? How might they deal with these problems? Would it help either of these researchers if it were known that the amount of labour (L) employed by the firms in question is also a linear function of expected sales, and data are available on L? Explain.