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AAEC 5307 Take Home Exam

创建日期：2024-04-23 18:04:49

所属分类：数学mathematics

标签： AAEC 5307

Take Home Exam

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AAEC 5307 Take Home Exam

1. Data on U.S. gasoline consumption for the years 1953 to 2004 are given in
File Takehomeex1.xls. Note, the consumption data appear as total expendi-
ture. To obtain the per capita quantity variable, divide GASEXP by GASP
times Pop. The other variables do not need transformation.
(a) Compute the multiple regression of per capita consumption of gasoline
on per capita income, the price of gasoline, all the other prices and a
time trend. Report all results. Do the signs of the estimates agree with
your expectations?
(b) Test the hypothesis that at least in regard to demand for gasoline, con-
sumers do not differentiate between changes in the prices of new and
used cars.
(c) Estimate the own price elasticity of demand, the income elasticity, and
the cross-price elasticity with respect to changes in the price of public
transportation. Do the computations at the 2004 point in the data.
(d) Re-estimate the regression in logarithms so that the coefficients are
direct estimates of the elasticities. (Do not use the log of the time
trend.) How do your estimates compare with the results in the previous
question?Which specification do you prefer?
(e) Compute the simple correlations of the price variables. Would you
conclude that multicollinearity is a problem for the regression in part
a or part d?
(f) Notice that the price index for gasoline is normalized to 100 in 2000,
whereas the other price indices are anchored at 1983 (roughly). If you
were to re-normalize the indices so that they were all 100.00 in 2004,
then how would the results of the regression in part a change? How
would the results of the regression in part d change?
1
(g) This exercise is based on the model that you estimated in part d. We are
interested in investigating the change in the gasoline market that oc-
curred in 1973. First, compute the average values of log of per capita
gasoline consumption in the years 1953-1973 and 1974-2004 and re-
port the values and the difference. If we divide the sample into these
two groups of observations, test whether the errors are homoscedastic
or not.
2. Christensen and Greene (1976) estimated a generalized Cobb.Douglas cost
function for electricity generation of the form
lnC = α+β lnQ+ γ[
1
2
(lnQ)2]+δk lnPk+δl lnPl+δ f lnPf + e
where Pk, Pl , andPf indicate unit prices of capital, labor, and fuel, respec-
tively, Q is output and C is total cost.To conform to the underlying theory
of production, it is necessary to impose the restriction that the cost function
be homogeneous of degree one in the three prices. This is done with the
restriction δk+δl+δ f = 1, or δ f = 1−δk−δl . Inserting this result in the
cost function and rearranging produces the estimating equation,
ln(
C
Pf
) = α+β lnQ+ γ[
1
2
(lnQ)2]+δk ln(
Pk
Pf
)+δl ln(
Pl
Pf
)+ e
We are interested in the efficient scale, which is the output at which the cost
curve reaches its minimum. That is the point at which( ∂ lnC∂ lnQ)|Q=Q∗ = 1 or
Q∗= exp[1−βγ ]
(a) Data on 158 firms extracted from Christensen and Greene’s study are
given in File Takehomeex2.xls. Using all 158 observations, compute
the estimates of the parameters in the cost function and the estimate of
the parameter covariance matrix.
(b) Note that the cost function does not provide a direct estimate of δ f .
Compute this estimate from your regression results, and estimate the
standard error.
(c) Compute an estimate of Q∗ using your regression results and then
form a confidence interval for the estimated efficient scale.
2
(d) Examine the raw data and determine where in the sample the effi-
cient scale lies. That is, determine how many firms in the sample have
reached this scale, and whether, in your opinion, this scale is large in
relation to the sizes of firms in the sample. Christensen and Greene
approached this question by computing the proportion of total output
in the sample that was produced by firms that had not yet reached ef-
ficient scale. (Note: There is some double counting in the data set.
more than 20 of the largest firms in the sample we are using for this
exercise are holding companies and power pools that are aggregates
of other firms in the sample. We will ignore that complication for the
purpose of our numerical exercise.)
(e) Test for the presence of heteroscedasticity using White’s general test.
Do your results suggest the nature of the heteroscedasticity?
(f) Use the Breusch-Pagan test to test for heteroscedasticity.
(g) Re-estimate the model assuming thatσ2i = σ2Q2i . The generalized
Cobb-Douglas cost function examined above is a special case of the
translog cost function,
lnC = α+β lnQ+δk lnPk+δl lnPl+δ f lnPf +φkk[.5(lnPk)2]+
φll[.5(lnPl)2]+φ f f [.5(lnPf )2]+φkl[lnPk][lnPl]
+φk f [lnPk][lnPf ]+φl f [lnPl][lnPf ]+ γ[0.5(lnQ)2]
+θQk[lnQ][lnPk]+θQl[lnQ][lnPl]+θQf [lnQ][lnPf ]+ e
The theoretical requirement of linear homogeneity in the factor prices
imposes the following restrictions: δk+δl+δ f = 1,φkk+φkl+φk f =
0,φkl + φll + φl f = 0φk f + φl f + φ f f = 0,θQk+ θQl + θQf = 0 . Note
that although the underlying theory requires it, the model can be es-
timated (by least squares) without imposing the linear homogeneity
restrictions. (Thus, one could test the underlying theory by testing
the validity of these restrictions. See Christensen, Jorgenson, and Lau
(1975).) We will repeat this exercise in part b. A number of additional
restrictions were explored in Christensen and Greene (1976) study.
The hypothesis of homotheticity of the production structure would add
the additional restrictions θQk = 0,θQl = 0,θQf = 0. Homogeneity of
the production structure adds the restriction γ = 0. The hypothesis that
3
all elasticities of substitution in the production structure are equal to
−1 is imposed by the six restrictions φi j = 0 for alli and j .
(h) Write the restrictions in the Rβ = r format. What is R? What isβ?
What isr?
(i) Test the theory of production using all 158 observations. Use an F test
to test the restrictions of linear homogeneity. Christensen and Greene
enforced the linear homogeneity restrictions by building them into the
model.You can do this by dividing cost and the prices of capital and
labor by the price of fuel. Terms with f subscripts fall out of the
model, leaving an equation with 10 parameters. Compare the sums of
squares for the two models to carry out the test. Of course, the test
may be carried out either way and will produce the same result.
(j) Test the hypothesis homotheticity of the production structure under the
assumption of linear homogeneity in prices.
(k) Test the hypothesis of the generalized Cobb-Douglas cost function
against the more general translog model suggested here, once again
(and henceforth) assuming linear homogeneity in the prices.
(l) The simple Cobb-Douglas function appears in the first line of the
model above. Test the hypothesis of the Cobb-Douglas model against
the alternative of the full translog model.
(m) Test the hypothesis of the generalized Cobb-Douglas model against
the homothetic translog model.
(n) Which of the several functional forms suggested here do you conclude
is the most appropriate for these data?
3. In this exercise we consider the hedonic price model where sale prices of
houses are function of their characteristics. The attached data contain sales
prices of 546 houses, sold during April, May, and June of last year in one of
the cities of Texas. Along the house prices, the data include house attributes.
The following characteristics are available: the lot size of the property in
square feet, the numbers of bedrooms, full bathrooms and garage places,
and the number of stories. In addition there are dummy variables for the
presence of a driveway, recreational room, full basement and central air
conditioning, for being located in a preferred area and for using gas for hot
water heating.
4
(a) Provide a table with the descriptive statistics of the data used in this
analysis.
(b) Estimate a model that explains the log of the sale price from the log of
the lot size, the numbers of bedrooms and bathrooms and the presence
of air conditioning.
(c) What is the expected sale price of a house with four bedrooms, one
full bathroom, a lot size of 5000 sq. ft?
(d) Use a the RESET to test the functional form of this specification.
(e) Estimate the model by including all other variables. Comment on your
results.
(f) Is there any multicollinearity problem?
(g) Are all seven additional variables jointly significant?
(h) What is the expected sale price of a 2-story house on a lot of 10 000
square feet, located in a preferred neighbourhood of the city, with 4
bedrooms, 1 bathroom, 2 garage places, a driveway, recreational room,
air conditioning and a full and finished basement, using gas for water
heating.
(i) Estimate a linear specification of the hedonic price model.
(j) Which model is appropriate? The linear or the log?
(k) Given the data is a cross-sectional one, what potential issues should
you test for?
(l) One way of quickly checking for heteroscedasticity is to plot the resid-
uals against the fitted values. If there is no heteroskedasticity, we can
expect that the dispersion of residuals does not vary with different lev-
els of the fitted values. Plot the residuals against the fitted values.
What do you conclude?
(m) (m) Formally test for heteroscedasticity. Is your conclusion in accor-
dance with the one you drew from the graph?

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