课程名称︰计量经济学一 课程性质︰经济系选修 课程教师︰刘锦添 开课学院：社会科学院 开课系所︰经济系 考试日期（年月日）︰100年1月13日 考试时限（分钟）：3小时 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 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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