科目：计量经济学 教授：刘锦添 试别：期中考 时间：2005.11.25 by ShowEye ---------------------------------------------------------------------- 试题 : 1. The following regression equation is estimated as a production function. logＱ = 1.37 + 0.632 logＫ + 0.452 logＬ Ｒ^2 = 0.98 se = (0.257) (0.219) cov(ｂK,ｂL) = -0.044. The sample size is 40. Test the following hypothesis at the 5% level of significance. (a) bK = bL (b) There are constant returns to scale. 2. The model ｙt = βo + β1ｘ1t + β2ｘ2t + β3ｘ3t + ｕt was estimated by ordinary least squares from 26 observations. The results were ^ ｙt = 2 + 3.5ｘ1t - 0.7ｘ2t + 2.0 ｘ3t (1.9) (2.2) (1.5) t-ratios are in parentheses and Ｒ^2 = 0.982. The same model was estimated with the restriction β1 = β2. Estimates were: ^ ｙt = 1.5 + 3(ｘ1t + ｘ2t) - 0.6 ｘ3t Ｒ^2 = 0.876 (2.7) (2.4) (a) Test the significance of the restriction β1 = β2. State the assumptions under which the test is valid. _ (b) Suppose that ｘ2t is dropped from the equation: would the Ｒ^2 rise or fall? (c) would the Ｒ^2 rise or fall if ｘ2t is dropped? 3. The demand for Ceylonese tea in the United States is given by the equation logＱ = βo + β1 logＰc + β2 logＰI + β3 logＰB + β4 logＹ + ｕ where Ｑ = imports of Ceylon tea in the United States 　　　　 Ｐc = price of Ceylon tea ＰI = price of Indian tea ＰB = price of Brazilian coffee Ｙ = disposable income The following results were obtained from T = 22 observations. logＱ = 2.837 - 1.481 logＰc + 1.181 logＰI + 0.186 logＰB + 0.257 logＹ (2.0) (0.987) (0.690) (0.134) (0.370) RSS = 0.4277 logＱ + logＰc = -0.738 + 0.199 logＰB + 0.261 logＹ RSS = 0.6788 (0.820) (0.155) (0.165) Figures in parentheses are standard errors. (a) Test the hypothesis β1 = -1, β2 = 0, and β3,β4≠0 against βi≠0 for i = 1,2,3,4. (b) Discuss the economic implications for these results. 4. A staff member for a political campaign estimated the model Ｖt = α + βＰt + ｕt, for t = 1,2,...,22, where Ｖt is voter turnout in precinct t, and Ｐt is the precinct's population. When the results were being printed out, the printer malfunctioned, smudging some of the results. With the information already provided, fill in the blanks. Coefficient Estimate Standard Error t-ratio ︿ α 26.034 ______________ 14.955 ︿ β 0.137 0.028 _______ _ ESS=305.96 P=54.478 Ｓv^2=31.954 Ｓp^2=925.91 ︿ _ R^2=____ σ^2=_______ V=_____ 5. Assume that the midel is Ｙt = α + βＸt + ｕt. Given the n observations (X1,Y1), (X2,Y2),..., (Xn,Yn), construct an estimate of β as follows: First connect the first and second points of the scatter diagram and compute the slope of that line. Then connect the first and third points and compute its slope. Proceed similarly and finally connect the first and last points and compute the slope of that line. finally, average all these ︿ slopes and call that β , the estimate of β. ︿ 　 Draw a scatter diagram, give a geometric representation of β, and derive ︿ an algebraic expression for it. Next compute the expected value of β. Be sure to state any assumptions you made in computing the expectation. Is the estimate biased or unbiased? Explain. Finally, prove without any derivations why this estimate is inferior to the one we obtained earlier using the OLS procedure. Explain what you mean by "inferior."　 ∮完卷∮ --

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