課程名稱︰ 計量經濟學一
課程性質︰ 大學部選修
課程教師︰ 劉錦添
開課學院： 社會科學院
開課系所︰ 經濟系
考試日期（年月日）︰ 2007/11/22
考試時限（分鐘）： 9:10~12:00
是否需發放獎勵金： ok
試題 :
(*) 助教特別提醒: 2,3,4題中的ESS是"Error" Sum of Squares,
不是課本中的ESS(Explained sum of squares)!
1. Consider the binomial variable y, which takes on the values of zero or one
according to the probability density function (pdf)
f(y) = (Θ^y) * ((1-Θ)^(1-y)) 0≦Θ≦1 y=0,1
Thus the probability of a "success" (y=1) is given by f(1)=Θ, and the
probability of a "failure" (y=0) is given by f(0)=1-Θ. Verify that
E(y)=Θ, and Var(y)=Θ(1-Θ). If a random sample of n observations is drawn
from this distribution, find the MLE of Θ and the variance of its sampling
distribution. Find the asymptotic variance of the MLE estimator.
2. Consider the simple regression model Yt = α + βXt + Ut in which Yt is
total expenditure on travel and Xt is total income for the n-th state.
Including the District of Columbia, you have data for 51 observations. Both
variables are measured in billions of dollars. The following is a partial
computer output for the data:
Variable Coefficient Standard Error
--------------------------------------
Constant 0.4981 0.5355
Income 0.0556 0.0033
Error Sum of Squares (ESS) 417.11
Total Sum of Squares (TSS) 2841.33
a. What is the economic interpretation of the estimated coefficient for
income? Does the numerical value appear reasonable?
b. Test individually whether the coefficients for the constant term and
income are significantly different from 0 at the 5% level. Be sure to
state the null and alternative hypotheses, the test statistic and its
distribution, the critical value (or range), and the criterion. What is
your conclusion?
c. Compute the measure of goodness of fit.
d. Test the model for goodness of fit at the 1% level of significance. Show
all your derivations. What is your conclusion?
e. Suppose the data on X and Y are converted to thousands and a new model
is estimated as Yt(*) = α(*) + β(*)Xt(*) + Ut(*), where the variables
with asterisks (*) are the transformed ones. In the following table,
fill in the blanks, indicated by underlined items, that give the values
for the transformed model. Show your derivations.
Variable Estimate Standard Error
-----------------------------------------------------
Constant(*) _______ ______
Income(*) _______ ______
Error Sum of Squares (ESS) _______
Total Sum of Squares (TSS) _______
R-squared _______
3. The following table presents estimates are related statistics for three
models relating the number of private housing units authorized by building
permits and its determinants. The data refer to 40 cities in the United
States. The model is (omitting the t-subscript)
HOUSING = β1 + β2*VALUE + β3*INCOME + β4*LOCALTAX + β5*STATETAX
+ β6*POPCHANG + u
Where HOUSING is the actual number of building permits issued, VALUE is the
median price of owner-ocuppied homes (in hundreds of dollars), LOCALTAX is
average local tax per capita (in dollars), STATETAX is average state tax
per capita (in dollars), and POPCHANG is the percent increase in population
between 1980 and 1982.
Values in parentheses are p-values for a two-sided alternative. Note that
you are not given the standard errors or t-statistics and they cannot be
calculated from the information given.
a. For each regression coefficient in model A, test whether it is
significantly different from zero at the 10% level (ignore the constant
term). Based on your test, which variables are candidates to be dropped
from the model and why?
Use model A as the unrestricted model and Model C as the restricted model,
and carry out the following steps for an appropriate test:
b. Write down the null and alternative hypotheses in term of the βs.
c. Compute the test statistic.
d. State the distribution of your test statistic, under the null, including
the degrees of freedom.
e. Write down the critical value (or range) for the test, and state whether
or not you would reject the null hypothesis at the 10 percent level.
Which of the models is the "best"? Explain what the criteria you used to
choose the best model.
g. For each regression coefficient (ignore the constant term for this) in
your best model, state whether some of the coefficients are "wrong" in
sign. State which sign you would expect and why. Then identify whether
it has the right sign or not.
Variable Model A Model B Model C
------------------------------------------------
^
β1 Constant -420.323 -1071.982 -973.017
^ (0.80) (0.40) (0.44)
β2 VALUE -0.724 -0.864 -0.778
^ (0.15) (0.05) (0.07)
β3 INCOME 111.898 110.193 116.6
^ (0.08) (0.08) (0.06)
β4 LOCALTAX 0.503 0.491
^ (0.41) (0.41)
β5 STATETAX -0.636
^ (0.15)
β6 POPCHANG 28.257 29.662 24.857
(0.07) (0.05) (0.08)
------------------------------------------------
ESS 4.886 4.941 5.038
Unadjusted R^2 0.332 0.325 0.312
SIGMASQ 1.437 1.412 1.399
AIC 1.649 1.586 1.538
FPE 1.653 1.588 1.539
HQ 1.807 1.712 1.635
SCHWARZ 2.124 1.959 1.821
SHIBATA 1.588 1.544 1.511
GCV 1.691 1.613 1.555
RICE 1.745 1.647 1.574
4. The following table presents estimates and related statistic (p-value in
parentheses) for four models relating the number of private housing units
authorized by building permits and their determinants (if an entry is
blank, it means that the variable is absent from the model). The data refer
to 40 cities in the United States. The model is as follows:
HOUSING = β1 + β2*DENSITY + β3*VALUE + β4*INCOME + β5*POPCHANG
+ β6*UNEMP + β7*LOCALTAX + β8*STATETAX + u
HOUSING = Actual number of building permits issued
DENSITY = Population density per square mile
VALUE = Median value if owner-occupied homes (in hundreds of dollars)
INCOME = Median household income (in thousands of dollars)
POPCHANG = Percent increase in population between 1980 and 1992
UNEMP = Unemployment rate
LOCALTAX = Average local tax oer capita (in dollars)
STATETAX = Average state tax oer capita (in dollars)
a. For each regression coefficient in model A, test whether or not it is
zero at the 10 percent level (values in paratheses are p-values for a
two-sided alternative). Based on your test would you say that the
variable should be retained or should dropped from the model?
b. In model A, test the joint hypothesis H0: β2=β6=β7=β8=0 at the 10
percent level. Be sure to state the alternative hyphothesis, compute the
test statistic, state its distribution under the null, and the criterion
for acceptance or rejection. State yout conclusion.
c. Which of the models is the "best"? Explain what criteria you used to
choose the best model.
d. For each regression coefficient (ignore the constant term for this) in
your best model, state whether some of the coefficients are "wrong" in
sign. State which sign you would expect and why. Then identify whether
it has the right sign.
e. In Model D, suppose you measure HOUSING in thousands of units and at the
same time measure INCOME in hundredsof dollars. Write down the estimated
coefficients of the new model, the corresponding p-values, and the new
unadjusted R-square.
f. Interpret the results.
Variable Model A Model B Model C Model D
-----------------------------------------------------------
Constant 813 -392 -1279 -973
(0.74) (0.81) (0.34) (0.44)
DENSITY 0.075 0.062 0.042
(0.43) (0.32) (0.47)
VALUE -0.855 -0.873 -0.994 -0.778
(0.13) (0.11) (0.06) (0.07)
INCOME 110.411 133.025 125.705 116.597
(0.14) (0.04) (0.05) (0.06)
POPCHANG 26.766 29.185 29.406 24.857
(0.11) (0.06) (0.01) (0.08)
UNEMP -76.546
(0.48)
LOCALTAX -0.061
(0.95)
STATETAX -1.006 -1.004
(0.40) (0.37)
-----------------------------------------------------------
ESS 4.763e+7 4.843e+7 4.962e+7 5.038e+7
Unadjusted R^2 0.349 0.338 0.322 0.312
SIGMASQ 1.488e+6 1.424e+6 1.418e+6 1.299e+6
AIC 1.776e+6 1.634e+6 1.593e+6 1.538e+6
FPE 1.786e+6 1.638e+6 1.595e+6 1.539e+6
HQ 2.007e+6 1.791e+6 1.719e+6 1.635e+6
SCHWARZ 2.490e+6 2.105e+6 1.967e+6 1.821e+6
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