课程名称︰计量经济学 课程性质︰系内选修 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）︰2011/01/05 考试时限（分钟）：110(0910-1200) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. In the model Yi=β2*Xi+u_i (Note: there is no intercept) 2 2 you are told that var(u_i)=σ Xi. Show that 2 4 ︿ σ ΣXi Var(β_2)=------------- 2 2 (ΣXi) (10%) 2. Suppose a sample of adults is classified into groups 1,2 and 3 on the basis of whether their education stopped in (or at the end of) elementary school, high school, or university, respectively. The relationship y=β1+β2D2+β3D3+ε is specified, where y is income, Di=1 for those in group i and zero for all others. (a) In terms of the parameters of the model, what is the expected income of those whose education stopped in university. (b) In terms of the parameters of the model, what is the null hypothesis that going on to university after high school makes no contribution to adult income? (c) Can the specified model be expressed in a simpler, equivalent from y=α0+α1χ+ε, whereχis the years of education completed? Explain. (d) Suppose that the dummy variables had been defined as D4=1 if attended high school, zero otherwise; D5=1 if attended university, zero otherwise and y=α3+α4Ｄ4+α5Ｄ5+ε was wstimated. Answer parts (a) and (b) above for this case. (15%) 3. Suppose y=β0+β1Ｄ+ε where Ｄ is a dummy for sex (male=1). The average y value for the 20 males is 3, and for the 30 females is 2, and you know that ε is distributed as N(0,10) (a) What are the OLS estimates of the β0 & β1 ? (b) What is the value of the test statistic for testing 3β0+2β1=3 ? (c) How is tis statistic distributed? (20%) 4. Suppose we have observations y1=1, y2=2, and y3=5 from the linear model y=β+ε (i.e., only an intercept) with V(ε) diagonal elements 1.0, 0.5, and 0.2. Calculate: OLS GLS (a) β and β ; OLS GLS (b) V(β ) and V(β ); OLS 2 -1 (c) the tradition estimate of V(β ), namely s (X'X) ; GLS (d) the estimate of V(β ), assuming you only know that V(ε) is proportional to the varience-covariance matrix specified above. (20%) 5. Suppose the linear applies to y=βx+ε expected that ε is first-order autocorrelated with autocorrelation coefficient ρ=0.5 and variance 9. You have to observations on x and y; the first observations are x=1 and y=4, and the second observations are x=2 and y=10. (a) What is the OLS estimate of β? (b) What is the GLS estimate of β? (c) What are the variances of these to estimates? (20%) 6. In 1985, neither Florida nor Georgia had laws banning open alcol containers in vehicle passenger compartments. By 1990, Florida had passed such a law, but Georgia had not. (a) Suppose you can collect random samples of the driving-age population in both states, for 1985 and 1990. Let arrest be a binary variable equal to unity if a person was arrested for drunk driving during the year. Without controlling for any other factor, write down a linear probability model (OLS) that allows you to test whether the open container law reduced the probability of being arrested for drunk driving. Which coefficient in your model measures the effect of the law? (b) Why might you want to control for other factors in the model? What might some of these factors be? (c) Now, suppose that you can only collect data for 1985 and 1990 at the county level for the two states. The dependent variable would be the fraction of licensed drivers arrested for drunk driving during the year. How does this data structure differ from the individual-level data described in part (a)? What econometric method would you use? (15%) --

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