課程名稱︰計量經濟理論一 課程性質︰必修 課程教師︰張勝凱 開課學院：社會科學院 開課系所︰經濟研究所 考試日期（年月日）︰ 2010/1/13 考試時限（分鐘）：120 --> 150 min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : M6140 Econometric Theory I Final Exam (1/13/2010) Problem 1 (25 points (5,5,5,5,5)) True or False: 1. If yi = xi'β + εi, xi belongs to R, and E(εi|xi) = 0, then E(xi^2εi)=0. 2. Let yi = xi'β + εi, xi belongs to R. If βcap is a consistent estimator of β, the βcap is an unbiased estimator of β. 3. If yi = xi'β + εi, xi belongs to R, E(εi|xi) = 0, and eicap is the OLS residual from the regression of yi on xi, then Σ(xi^2)eicap = 0. 4. From the attached Table 17-1, it shows that the education has much larger impact on the married women labor participation decision from the Logit and Probit estimation than from the OLS estimation since 0.221 and 0.131 are much larger than 0.038. 5. From the attached Table 17-2 for Tobit model, it indicates that one more year if education for married women will on average increase roughly 80.65 annual hours worked. Problem 2 (20 points (7,8,5)) For the model yi = ziβ + εi where zi belongs to R is considered endogenous. You consider using z1 and x2 for instrumentation to estimate β through GMM estimation. (1) What are the assumptions required to justify this choice of instruments? (2) Describe the estimator of β. (3) Is the model over-identified? Problem 3 (25 points(8,8,9)) Suppose that price and quantity are determined by the interactions of the linear demand and supply curves. For the following models (1) Demand: Q = a0 + a1P + a2Y + e1, and Supply: Q = b0 + b1P + b2W + e2 where Y and W are determined outside the market. In this model, which parameters are identified, and which parameters are not identified? Why? (2) Demand: Q = a0 + a1P + e1, and Supply: Q = b0 + b1P + b2W + b3Y +e2, where Y and W are determined outside the market. In this model, which parameters are identified, and which parameters are not identified? Why. (3) Demand: Q = a0 + a1P + e1, and Supply: Q = b0 + b1P + b2W + e2, where W is determined outside the market. In this model, which parameters are identified and which parameters are not identified? Why. Problem 4 (30 points) (8,8,7,7) In the linear model estimated by GMM with general weight matrix W, the asymtotic variance of βcapGMM is V = (Q'WQ)^-1Q'WΩWQ(Q'WQ)^-1 (1) Let V0 be this matrix when W = Ω^-1. Show that V0 = (Q'Ω^-1Q)^-1. (2) We want to show that for any W, V-V0 is positive semi-definite. From matrix algebra, we know that V-V0 is positive semi-definite. If and only if V0^-1 - V^-1 = A is positive semi-definite. Write out the matrix A. (3) Since Ω is positive semi-definite, there exists a non-singular matrix C such that C'C = Ω^-1. Letting H = CQ and G = C'^-1WQ, verify that A can be written as A = H'(I-G(G'G)^-1G')H. (4) Show that A is positive and semi-definite. (Hint: The matrix I-G(G'G)^-1G') is symmetric and idempotent, and therefore positive semi-definite. -- Nobody ever saw a dog make a fair and deliberate exchange of one bone for another with another dog. --

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