科目：計量經濟學一
教授：劉錦添
試別：94上期中考
時間：2005.11.25
提供：ShowEye
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試題 :

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."　


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