課程名稱︰計量經濟學一
課程性質︰經濟系選修
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）︰100年1月13日
考試時限（分鐘）：3小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Consider the binomial variable y, which takes on the values zero or one
according to the probability 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 "failure"(y = 0) is given by f(0) = 1-θ. Verify that E(y) = θ
, and Var(θ) = 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 asymtotic variance of the MLE estimator.
2. Suppose the Berkeley Unified School District introduces a new after-school
tutoring program, while the Oakland Unified School District doesn't. For
both school districts, you observe the avarage score on a standardized Math
exam in the year before and the year after Berkeley introduces the program:
Average score(on a maximum of 100)
┌───────┬────┬────┐
│ │Berkeley│Oakland │
├───────┼────┼────┤
│Before Program│ 60 │ 55 │
├───────┼────┼────┤
│After Program │ 70 │ 58 │
└───────┴────┴────┘
a.Calculate the differences-in-differences estimate of the effect of the
tutoring program.
b.Suppose you estimate the following model for a student exam result:
score = β_1 + β_2berkeley + β_3after + β_4berkeley × after + u
where score is the score obtained by the student on the math exam, berkeley
is a dummy variable equal to one if the student attends a Berkeley school,
0 if he attends an Oakland school, and after is a dummy variable equal to
one for the test taken after Berkeley introduces the program, and 0 for
the test taken before. Which parameter yields the
differences-in-differences estimate of the effect of the tutoring program?
c.What estimates do you expect for the parameters β_1, β_2, and β_3 in
the regression in part b.?
3. Consider the relation S_t = α + βY_t + u_t, where S is household saving
and Y is household income. You also know the age group the head of the
household falls into, but not the actual age. To follow for age-group
differences, you define three dummy variables, D_1 = 1 for age < 25,
D_2 = 1 for age 51-65, and D_3 = 1 for age > 65. the age-group 26-50 is the
control. You have data on 50 households.
a.It is very likely that both coefficients are difference across age groups
Derives the most general model (U) that incorporates that belief.
b.You want to test the null hypothesis, "There is no difference in the
relation between age group < 25 and age > 65." Carefully write down the
null hypothesis for this test.
c.Next derive the restricted model.
d.Describe the steps tp carry out this test.
e.You suspect that random error term u_t is heteroscedastic with a variance
σ_t^2 that depends on the size of the family P_t. Write down an
appropriate version of the auxiliary equation for the error variance.
f.Next state the null hypothesis that there is no heteroscedasticity.
g.Describe the regression(s) to be run using Model U as the basis.
h.Write down the test statistic, its distribution, and the numerical value
of its degrees of freedom.
i.Describe the criterion for rejection of the null hypothesis.
j.Suppose that you find that there is heteroscedasticity and want to use the
weighted least squres procedure to estimate the parameters. Your research
assisdant is a good programmer, but does not know any econometrics.
Describe step-by-step hoe your R.A. should proceed to estimate Model U by
weighted least squares. Note that your description must be specific to the
model. For simplicity, assume that there is no negative or zero variance
problem.
4. Consider the following model of patents and R&D(research and development)
expenditures:
PATENTS_t = α + βR&D_t + u_t
Where PATENTS is the number of patent applications field in a given year and
R&D is the expenditures on the research and development in that year. The
model was using annual U.S. data for the years 1960 through 1993(n = 34).
You want to use the Durbin-Watson statistic, which had the value of
d = 0.234, tp test the model for first order serial correlation, that is for
AR(1).
a.Write down the auxiliary equation for the error term implied by the
presence of AR(1).
b.Write down the null hypothesis for no autocorrelation.
c.Carry out the Durbin-Watson test. Is there significant autocorrelation or
not? Show all the details of your pricedure.
d.Describe the auxiliary regression needed to carry out the LM test for
AR(1).
e.Suppose this regression had error sum of squares ESS = 793.09 and total
sum of squares TSS = 3985.38. Compute the numerical value of the test
statistic.
f.Write down its distribution and degrees of freedom.
g.Write down the critical value for a test at the level of 0.001.
h.State the decision rule and the conclusion.
i.Based on your conclusion, are OLS estimators of the parameters of the
model unbiased, consistent, efficient? Are test valid?
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