课程名称︰计量经济理论一 课程性质︰必修 课程教师︰张胜凯 开课学院：社会科学院 开课系所︰经济学研究所 考试日期（年月日）︰2011/1/12 考试时限（分钟）：120mins 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Problem1.(25 points ((7,8),10)) (1)For the model yi=zi*β + εi where zi∈R is considered endogenous.You consider using x1 and x2 for instrumentation to estimate β through GMM estimation. (a)What are the assumptions required to justify this choice of instruments? (b)Describe the estimator of β. Is the model over-identified? ︿ (2)True or False. The two-stage least square estimator β2SLS is unchanged if the original N╳L matrix of instrumental variables Z is replaced by a new * * matrix Z of instruments if Z =ZH ? where H is an invertible L╳L matrix. Exlain. Problem2.(30 points (10,10,10)) Consider a probit model P(yi=1|xi)=Φ(xi*θ), where Φ(.) denotes the standard normal cumulative distribution function (cdf). Both xi and θ are scalars, also let ψ(.) be the standard normal probability density function (pdf). (1)Write down log likelihood li(θ) for observation i. (2)Find the score function, si(θ) for each i. (3)Show that E(si(θ)|xi)=0. Problem3. (15 points) The model is yi=zi'β+ ei and E(xi*ei)=0. An economist wants to obtain the 2SLS estimates and standard errors for β. He uses the following steps: ︿ (i) Regresses zi on xi, obtains the predicted values zi. ︿ ︿ (ii)Regresses yi on zi, obtains the coefficient estimate β and standard error ︿ s(β) from this regression. Is this correct? Does this produce the 2SLS estimates and standard errors? Why or why not. Problem4. (30 points (5,7,8,10)) Take the model yi=zi*β+ ei and E(zi*ei)≠0, where the observations (yi,zi) are iid random samples. zi is scalar(1×1) and E(ei)=0. (1)Do we say that zi is "exogenous" or "endogenous" for β? ︿ n n (2)Is the OLS estimator β=(Σ zi*yi)/(Σ zi^2) consistent for β ? i=1 i=1 ～ n n (3)Consider an alternative estimator β=(Σ yi)/(Σ zi). Is there a condition( ～ i=1 i=1 other than E(xi*ei)=0) under which β is consistent for β ? ～ (4)Explain your finding in (3) by showing that you can write β as a valid IV estimator. Explain the identifying retriction. --

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