1 Economics 435 Spring 1997 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. If you have data on two variables, Y and X, and you wish to regress Y on X with a second-degree Almon lag applied to the current term and three lags on X, in SHAZAM you would give the command: OLS Y X(0.3,2). a. Explain in detail how you would estimate the lag weights (beta coefficients) if you had to use some program which did multiple regression, but had no special features for estimating Almon lags (or if you didn't know about the "(0.3,2)" operation in SHAZAM). b. How would you test whether the second-degree constraint is binding? c. How would you test whether the lag length should have been 6 instead of 3? d. How would you go about estimating the standard errors of the lag weights (beta coefficients)? 2. Suppose M = demand for real cash balances, Y* = expected real income, and R = an interest rate. Assume expectations are formulated as follows: Y*t = a1Yt + (1 - a1)Y*t-1 If the equation of interest is: Mt = B1 + B2Y*t + B3R + ut a. What would the estimating equation look like? b. What estimation problems would you forsee? c. If the equation of interest was changed to: Mt = B1 + B2Y*t + B3R*t + ut, where R*t is defined as the expected rate of interest, expectations being formulated as follows: R*t = a2Rt + (1 - a2)R*t-1, what would the estimating equation look like? d. For the revised problem of part c, what estimation problems would you forsee? Part II. Answer one of the following two questions. 3. Given 100 time series observations on a variable (Y), a number of persons attempted to estimate the trend in that variable. They all worked with a time variable (X), defined as a sequence of consecutive integers starting with unity. Person A estimated (t-values in parentheses throughout): Y = 106.40 - 0.004 X R-square = 0.0002 DW = 0.12 (-0.13) and concluded there was no trend in Y. 2 Person B estimated:Y=106.40+0.89 X-0.009 X2 R-square = 0.52 (9.92) (-10.28) DW = 0.26 and concluded there was a nonlinear trend in Y. Person C corrected for AR(1) errors using EGLS: Y=114.77+0.08X R-square=0.88, DW=2.34, Rho=0.94 (0.67) and agreed with Person A that there was no trend. Person D noted that since C's rho value was so close to unity, one might as well estimate in first differences (sample 2-100): dY = 0.77 - 0.006 dX2 R-square=0.01 (-1.01) DW=2.55 and agreed with C and A that there was no trend. (Note: dY=Yt-Yt-1 and similarly for dX2). Person E didn't do any estimations, but noted that Person D was wrong for forgetting dX and for estimating an intercept when there shouldn't be one in a first difference equation. a. Critique each of the five arguments presented above. b. With which person do you agree? Explain. c. If you had access to the data, are there any additional estimations you would perform? If so, explain why. 4. A Cobb-Douglas production function was estimated by Dougherty using 24 observations of annual time series data. The OLS estimation was (t-values in parentheses): (i) lnQ = 2.81 - 0.53 lnK + 0.91 lnL + 0.05 X R-square=0.97 (-1.54) (6.48) (2.26) DW = 1.63 where Q = output, K = capital input, L = labour input and X = a time trend. He also estimated the equation: (ii) ln(Q/L)= -0.005 + 0.11 ln(K/L) + 0.006 X R-square=0.65 (0.73) (1.01) DW = 1.44 a. Do you suspect multicollinearity in equation (i)? Why? b. Why do you think the time trend is included in these equations? Could inclusion of this cause any problems? c. What do you suppose was the logic of estimating equation (ii)? Explain. d. Are the R2's of the two equations comparable? If they are, discuss how you could test for constant returns to scale. If they are not, how would you test for CRTS? e. The values of dL and dU are 1.188 and 1.546 at the .05 level for equation (ii). Is there an autocorrelation problem here? How would you investigate further? If there is a problem, would EGLS be likely to cure the "problem" of non-significant coefficients? Explain. Part III. Answer one of the following two questions. 5. In Table 20.2, Gujarati provides annual time series data on five series covering the years 1970-1991, inclusive. The series are: Y = GDP, M = money supply (M2 definition), I = private investment, G = federal government expenditure and R = an interest rate (6 mo. t-bills). a. Consider the three equation model: 3 (i) R = a0 + a1M + a2Y + u1 (ii) Y = b0 + b1R + b2I + u2 (iii) I = c0 + c1R + u3 Which equations are identified? How would you obtain consistent estimators of the coefficients of the identified equations? If the model is not identified, how might you change it to make it identified, and still retain consistency with (some) economic theory? How would you obtain consistent estimators of the coefficients of your (revised) model? b. OLS estimates of the above equations are (t-values in parentheses): (i) R = 7.72 - 0.02 M + 0.01 Y R-square=0.26 (-2.49) (2.53) (ii) Y = 312.74 - 73.80 R + 6.58 I R-squared=0.95 (-2.14) (18.63) (iii) I = 306.52 + 24.97 R R-squared=0.07 (1.18) Given that it is customary to conduct specification searches for simultaneous equation models using OLS, is there anything in the above estimations which is bothersome? Explain. c. How would you test whether investment should be considered endogenous in the above model? 6. This question has three unrelated parts. Answer all parts. a. From the data referred to in problem 5, someone generated consumption (C), as Y-(I+G), and estimated two equations (t-values in parentheses): (i) C = -53.37 + 0.85 Y - 0.98 G R-square=0.996 (7.20) (-2.02) (ii) (C/Y) = 0.65 + 31.9(1/Y) - 0.25(G/Y) R-square=0.34 (6.64) (1.42) (-0.63) Why do you suppose equation (ii) was estimated? What specific assumption must have been made? How would you test whether this assumption appears to be correct? b. If you wanted a two-piece linear spline function (a continuous function, piecewise linear) to describe the relationship between two variables, Y and X, such that dY/dX is different for X<=X1 and for X>=X1, how would you perform the estimation? How would you test whether the restriction of continuity is binding? c. If you have a simple regression problem: Y=B1+B2X+u, and you suspect X contains measurement error, will OLS yield unbiased and consistent estimators? Explain. If there is a problem with OLS, how would you estimate the parameters?