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）︰2010/11/10 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 : Problem 1.(25 points (5,10,10)) ︿ ︿ ︿ Let θ=(θ1,θ2)' be a √N-asymptotically normal estimator for θ=(θ1,θ2)'. ︿ ︿ ︿ Let γ=2*θ1*θ2^2 be an estimator of γ=2*θ1*θ2^2. ︿ 1. Is γ a consistent estimator of γ ? Explain. ︿ ︿ 2. Find Avar(γ) in terms of θ and Avar(θ). ︿ ︿ 3. If, for a sample of data,θ=(1,2)' and Avar(θ) is estimated as ┌ ┐ │1 -0.5│ │-0.5 2 │ └ ┘ ︿ Find the asymptotic standard error of γ. Problem 2. (30 points (10,10,10)) Let E(y|x,q)= x*β+γ* q , thus y=x*β+γ* q+ v, where E(v|x,q)=0. 1. Suppose that we can observe y and both x and q, would OLS estimators of y on x and q provide an unbiased estimators of β ? Why or why not? Would OLS estimators of y on x and q provide an best linear unbiased estimator( BLUE) ofβ ? Why or why not? 2. Suppose we do not observe q, under what conditions , would OLS estimators of y on x provide an unbiased estimators ofβ ? Why? 3. Suppose there is a variable z where E(q|z)=δ*z. Suppose also that E(y|z,q,z)=E(y|x,q), would OLS estimators of y on x and z provide an unbiased estimators ofβ ? Why or why not? Problem 3. (45 points (8,8,5,8,8,8)) -1 Recall the long regression model y=X1*β1+X2*β2+ε. Let N1≡X1(X1'X1) X1', * * M1≡I-N1, X2 ≡M1*X2 and y ≡M1*y.Let b1 and b2 be an OLS estimatiors ofβ1 andβ2 in equation (1),e is the residual term correspondingly. That is ︿ y=X1*b1+X2*b2+e, y =X1*b1+X2*b2 1. Write down the normal equations (first order conditions) for OLS estimators b1 and b2. *' *-1 *' 2. Show that b2=(X2 X2 ) X2 y. 3. Show M1e=e. ︿ ︿ 4. Show y'y= y'y +e'e. * 5. Find the residual term of the regression y on X1. * * 6. Find the residual term of the regression y on X2 . --

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