推 miraclecity :已收至精華區：） 01/21 22:20
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︰經濟學研究所
考試日期（年月日）︰2011/1/12
考試時限（分鐘）：120mins
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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