科目：計量經濟學一
教授：劉錦添
試別：93上期中考
時間：2004.11.19
提供：ShowEye
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1. Consider the linear model ｙt = ｘ_t1β1 + ｘ_t2β2 + ｅt, where the ｅt
   are independent normal random variables with mean zero and variance σ^2.
   Suppose that we have only the following 4 observations:

                         ｙt     ｘ_t1     ｘ_t2
                          2        1        0
                         -1        0        1
                          4        1        1
                          0        1       -1

   Note that this is a special model without an intercept or constant term.

   (a) Define ｙ, Ｘ, and β such that the model can be written in matrix
       algebra form.

   (b) Find Ｘ'Ｘ, Ｘ'ｙ and ｙ'ｙ.

   (c) Find ｂ1 and ｂ2.
　　　　　　　 ︿　　　　　　　　 ︿                       ︿
   (d) Compute ｅ = ｙ-Ｘｂ = ｙ- ｙ and use the estimator ｅ'ｅ/(T-2) to
       compute an estimate of the variance σ^2.

   (e) Find estimated variances for ｂ1 and ｂ2.

　 (f) Find an estimate of the covariance between ｂ1 and ｂ2.



2. Are rent rates influenced by the student population in a college town?
   Let rent be the average monthly rent paid on rental units in a college
   town in the United States. Let pop denote the total city population,
   avginc the average city income, and pctstu the student population as a
   percent of the total population.

   One model to test for a relationship is

   log(rent) = βo + β1 log(pop) + β2 log(avginc) + β3 pctstu + ｕ

   (a) State the null hypothesis that size of the student body relative to
       the population has no ceteris paribus effect on monthly rents.
       State the alternative that there is an effect.

   (b) What signs do you expect for β1 and β2 ?

   (c) The equation estimated using 1990 data for 64 college towns is

       log(rent) = .043 + .066 log(pop) + .507 log(avginc) + .0056 pctstu
                  (.844) (.039)          (.081)             (.0017)

                                n = 64 , R^2 = .458

       What is wrong with the statement: "A 10% increase in population is
       associated with about a 6.6% increase in rent?

   (d) Test the hypothesis stated in part (a) at the 1% level.



3. We test the rationality of assessments og housing prices. Here, we use
   a level-level formulation.

   (a) The simple regression model
                    ︿
                  price = -14.47 + .976 assess
                          (16.27) (.049)

                  n = 88 , RSS = 165644.51 , R^2 = .820

       First, test the hypothesis that Ho:βo = 0 against the two-sided
       alternative. Then, test Ho:β1 = 1 against the two-sided alternative.

       What do you conclude?

   (b) To test the joint hypothesis that βo = 0 and β1 = 1, we need the
       SSR in the restricted model. This amounts to computing

        n
       Σ (pricei－assessi)^2, where n = 88, since the residuals in the　  　
       i=1

       restricted model are just pricei－assessi (No estimation is needed for
       the restricted model because both parameters are specified under Ho)
       This turns out to yield RSS = 209448.99. Carry out the F test for the
       joint hypothesis.

   (c) Now, test Ho:β2=0, β3=0, and β4=0 in the model

       price = β0 + β1 asess + β2 lotsize + β3 sqrft + β4 bdrms + ｕ
　　　 The R-squared from estimating this model using he same 88 houses is .829

   (d) If the variance of price changes with assess, lotsize, aqrft, or bdrms,
       what can you say about the F test from part (c)?



4. Suppose you have the observations 0, 0, 4, 4 and 0, 4, 0, 4 on x and y,
   respectively, from the classical linear model y = α + βｘ　+ ε

   (a) Graph these observations and draw in the OLS estimatinglines.

   (b) Draw in the OLS estimating line that incorporates the constraint
       that α=0.

   (c) Calculate the R^2s associated with both these estimating lines,
       using R^2 = 1－RSS / TSS.

   (d) Calculate these R^2s using R^2 = ESS / TSS.

   (e) What is the lesson here?


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