英语b级试题Econometrics710
Midterm Exam
March6,2001
1.The model is
y i=x0iβ+e i,E(e i|x i)=0,E³e2i|x i´=σ2i.
Assume theσ2i are known.Let D=diag{σ21,...,σ2n}.Letˆβbe the OLS estimator of β,and let˜βbe the(infeasible)GLS estimator ofβ.
(a)Show that Cov³ˆβ−˜β,˜β|X´=0.
(b)Deduce that Cov³ˆβ,˜β|X´=V ar³˜β|X´.
(c)Deduce that V ar³ˆβ−˜β|X´=V ar³ˆβ|X´−V ar³˜β|X´.
(d)Write V n=V ar³ˆβ−˜β|X´as a function of X and D.
2.The model is
y i=x0iβ+e i E(e i|x i)=0.
内向英文An econometrician is worried about the impact of some unusually large values of the regressors.The model is thus estimated on the subsample for which|x i|≤c,for some Þxed c.Let˜βdenote the OLS estimator on this subsample.It equals
˜β=Ãn X i=1x i x i1(|x i|≤c)!−1Ãn X i=1x i y i1(|x i|≤c)!
i miss u missing me
where1(·)denotes the indicator function.
(a)Show that˜β→pβ.
分钟用英语怎么说
(b)Find the asymptotic distribution of√n³˜β−β´.
(c)Bonus Question:Suppo instead the model is
y i=x0iβ+e i E(x i e i)=0.
Does result(a)change?
3.Let(y1,...,y n)be a real-valued random sample from distribution F with meanµ=
E(y i)and varianceσ2=V ar(y i).Letˆµ=y be the sample mean and let T n=ˆµ−µ.
remind的用法Lethattrick
τn=E(T n)=Z T n dF
四级作文常用句型be the bias ofˆµforµ.Let
lileiτ∗n=Z T n dF n
walkbe the bootstrap estimate of bias,where F n(x)is the empirical distribution function of the data(y1,...,y n).(Note:This is distinct fromˆτ∗n,the simulation estimate ofτ∗n).
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(a)Show thatτn=0andτ∗n=0
(b)Now considerθ=µ2.Letˆθ=ˆµ2,t T n=ˆθ−θ=ˆµ2−µ2.Findτn=E(T n),
the bias ofˆθforθ.
(c)Bonus Question:Findτ∗n=R T n dF n,the bootstrap estimate of bias.
