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︰ 经济研究所 考试日期（年月日）︰2009/11/11 考试时限（分钟）：100 --> 120 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Econ 323 M6140 Midterm Exam November 11, 2009 You have 100 minutes. Please write clearly and show all your work. Answers without and explanation will not earn any points. Good luck !! Problem 1 (25 points (5,5,10,5)) The model is yi = xi'β + εi, E(εi|xi) = 0, and E(εi^2|xi) = 0. Assume that σ^2 are known. Let Ω = diag(σ1^2, σ2^2, σ3^2,...,σn^2). Let βcap be the OLS estimator of β, and let βtiota be the GLS estimator of β. (a) Are βcap and βtiota unbiased estimators of β? Why or why not. (b) Are βcap and βtiota consistent estimators of β? Why or why not. (c) Find Var(βcap|X) and Var(βtiota|X). (d) Find a consistent estimator of asymptotic variance of βcap. Problem 2 (25 points) (2A) (18 points (10,8)) Suppose that a time series process yt is generated by yt = z + et, for all t = 1,2,..., where et is an i.i.d sequence with mean zero and variance σe^2. The random variable z does not change over time; it has mean zero and variance σz^2. Assume that each et is uncorrelated with z. (a) Find the expected value, variance of yt, and Cov(yt, yt+h) for any t and h. Is yt co-variance stationary? (b) Find Corr(yt, yt+h) for any t and h = 1,2,3. (2B) (7 points) (True or False) If the errors in a regression model contain Autoregressive Conditional Heteroskedasticity (ARCH) , they must be serially correlated. Problem 3 (25 points (15,10)) Let Y be n X 1, X be n X k (rank k), Var(Y|X) = Ω and Ω is known. Let Z =XB, where B is k X k with rank k. (a) Let (β1cap, e1cap) denotes the OLS coefficients and residuals from regression of Y of X. Similarly, let (β1tiota, e1tiota) denotes these from OLS regression of Y on Z. Find the relationship between β1cap and β1tiota, and the relationship between e1cap and e1tiota. (b) Let β2cap denotes the GLS coefficients from regression of Y on X. Similarly, let β2tiota denotes the GLS coefficients from regression of Y on Z. Find the relationship between β2cap and β2tiota. Problem 4 (25 points (15,10)) 1. Let Y1,..., Yn be a random sample from a Bernoulli(θ) distribution. The p.m.f. of y is f(y|θ) = θ^y(1-θ)^(1-y). Find the Maximum Likelihood Estimator of θ. 2. Let X1,.., Xn be a random sample from a uniform distribution on the interval [0,θ], where the value of the parameter θ is unknown (θ > 0). The p.d.f. f(x|θ) = 1/θ , if 0 ≦xi≦θ, and f(x|θ) = 0 otherwise. Find the Maximum Likelihood Estimator of θ. -- Nobody ever saw a dog make a fair and deliberate exchange of one bone for another with another dog. --

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