課程名稱︰計量經濟理論一
課程性質︰必修
課程教師︰李宗穎
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）︰2014/11/12
考試時限（分鐘）：90分鐘
試題 :
1. Let X1 and X2 have the joint pdf
f(x1,x2)=6x2 , 0<x2<x1<1
=0 , elsewhere
(a)(8 points)Find the conditional probability density function (pdf)
of X2 given X1=x1.
(b)(4 points)Find E(X2|X1=x1).
2.(8 points) Let X have the pdf : f(x)=(1/4)*(1+x) , -2<x<2.
Find the pdf of Y=X^2.
3. Let X1,X2,...,Xn be independent Bernoulli(p) random variables and
n
let Yn=(1/n)ΣXi
i=1
(a)(10 points)Find the asymptotic distribution of Yn.
(b)(10 points)Let p≠(1/2). Find the asymptotic distribution of Yn(1-Yn).
4.(20 points) The score function is defined as s(θ)=∂log(f(X|θ)/∂θ
where f(X|θ)is the density function of X given parameter θ. Under
the regularity conditions, E(s(θ))=0,
Show that E(s(θ)^2)=-E(∂s(θ)/∂θ)
5. Let X1,X2,...,Xn be i.i.d with (marginal) pdf
f(x|θ)=θx^(θ-1),0≦x≦1,0<θ<∞.
(a)(10 points)Find the MLE of θ.
(b)(10 points)Find the method of moments estimator of θ.
6. Suppose X1,X2,...,Xn are independent with pdf's
f(xi|θ)=λi*e^(-λi*xi),θ=(λ1,λ2,...,λn)'.
We want to test H0: λ1=λ2=...=λn.
& H1: λi are not all equall.
(a)(15 points)Find the likelihood-ratio test statisitc ξLR.
(b)(5 points)What is the decision rule using ξLR with significance
level α=5% (Be clear on the critical value of the test.)