Purpose To learn how to conduct subset F-tests in multiple linear by heatherrhunt


									ECO 5350                                                                           T. Fomby
Intro. Econometrics                                                                Fall 2008

                                         EXERCISE 4

Purpose: To learn how to conduct “subset” F-tests in multiple linear regression.
You are to hand in this homework on Monday, October 20.

Go to the website of the Hill, et. al. textbook that we are using in our course. It is at
http://www.wiley.com/college/hill/. Download the two files beer.def and beer.dat. This
data set represents the consumption of beer by a single household as a function of the
price of beer, the price of other liquor, the price of remaining goods and services and
income. A convenient iso-elastic demand curve for beer can be estimated using the
following demand specification:

        log(Q) = β 1 + β 2 log( PB) + β 3 log( PL) + β 4 log( PR) + β 5 log( I ) + e .     (1)

The coefficients β 2 , β 3 , β 4 , and β 5 represent, respectively, the own-price elasticity of the
demand for beer, the cross-elasticity of demand due to the price of liquor, the cross-
elasticity of demand due to the price of other goods, and the income elasticity of demand
for beer. Cut and paste the data from the beer.dat file into a new SAS program file and
name it Beer.sas, let’s say. Then I want you to create a data step that stores this data in a
data set call it “beer”. Then using this data I want you to define the variables lq = log(q),
lpb = log(pb), lpl = log(pl), lpr = log(pr), and li = log(i). Then in a Proc Reg step
estimate model (1) and get the various elasticities of demand. Then use your estimated
model to answer the following questions.

        a) The own-price elasticity of demand for beer is _______________. The
demand for beer for this household is (inelastic / elastic / unitary elastic). Circle the
correct alternative.

         b) If we wanted to test if the demand for beer of this household was unitary
elastic, how would we do it? What would be the form of the t-statistic? What would be
the null hypothesis that you would use? Calculate the t-statistic for this test and report it.
Does it seem that the demand for beer by this household is unitary elastic? Explain your
         c) Given the cross-elasticity of the demand for the price of liquor that you have
obtained, is liquor a complementary or substitute good? Explain your answer.

       d) Given the cross-elasticity of the demand for the price of other goods that you
have obtained, are “other goods” a complementary or substitute good for beer? Explain
your answer.
       e) Given the income elasticity of demand for beer that you obtained, is beer a
superior good, an inferior good, or a unitary elasticity of income good? Explain your

         f) We are going to test the following hypothesis using a so-called “subset” F-test.
We are going to test that the household that is investigated by the current data set has a
beer demand curve that is (1) independent of the prices of other goods and is (2)
independent of income. That is we want to test H 0 : β 4 = β 5 = 0 versus H 1 : β 4 ≠ 0 or
 β 5 ≠ 0 or both. To conduct this test, add the following code below your Proc Reg model
statement: test lpr = 0, li = 0; SAS gives you the answer F = ________. How many
numerator degrees of freedom does the F-statistic have? How many denominator degrees
of freedom does the F-statistic have? How were these degrees of freedom determined?
The p-value of this F-statistic is ________. What do you conclude, given this statistical

       g) Here we want to come to understand exactly how the computer calculated the
above F-statistic. We are going to be computing the F-statistic by hand using the
following formula and the restricted versus unrestricted model approach:

                             ( ESS r − ESS u ) / J      ( Ru2 − Rr2 ) / J
                        F=                         =                       .
                               ESS u /( N − K )      (1 − Ru2 ) /( N − K )

The unrestricted equation is represented by the regression equation (1) above. It
produces the numbers ESS u = _____________ and Ru2 = __________. The number of
restrictions implied by the null hypothesis in this case is J = _______. The number of
degrees of freedom of the errors in the unrestricted model are N – K = __________. The
restricted equation is the one that has only the intercept and the lpb and lpl variables in it.
Estimate this restricted regression equation by adding another model statement to your
Proc Reg step of the form: model lq = lpb lpl; It produces ESS r = ______________ and
 Rr2 = ________. Use both forms of the subset F-statistic above to calculate the F-
statistic of your test. Show your work. Do your calculations pretty well match up with
the result you got from using the “test” statement in part a) above?

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