[試題] 98上 張勝凱 計量經濟理論一 期中考
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）︰2009/11/11
考試時限（分鐘）：100 --> 120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Econ 323 M6140 Midterm Exam
November 11, 2009
You have 100 minutes. Please write clearly and show all your work. Answers
without and explanation will not earn any points. Good luck !!
Problem 1 (25 points (5,5,10,5))
The model is yi = xi'β + εi, E(εi|xi) = 0, and E(εi^2|xi) = 0. Assume that
σ^2 are known. Let Ω = diag(σ1^2, σ2^2, σ3^2,...,σn^2). Let βcap be the
OLS estimator of β, and let βtiota be the GLS estimator of β.
(a) Are βcap and βtiota unbiased estimators of β? Why or why not.
(b) Are βcap and βtiota consistent estimators of β? Why or why not.
(c) Find Var(βcap|X) and Var(βtiota|X).
(d) Find a consistent estimator of asymptotic variance of βcap.
Problem 2 (25 points)
(2A) (18 points (10,8)) Suppose that a time series process yt is generated by
yt = z + et, for all t = 1,2,..., where et is an i.i.d sequence with mean zero
and variance σe^2. The random variable z does not change over time; it has
mean zero and variance σz^2. Assume that each et is uncorrelated with z.
(a) Find the expected value, variance of yt, and Cov(yt, yt+h) for any t and h.
Is yt co-variance stationary?
(b) Find Corr(yt, yt+h) for any t and h = 1,2,3.
(2B) (7 points) (True or False) If the errors in a regression model contain
Autoregressive Conditional Heteroskedasticity (ARCH) , they must be serially
correlated.
Problem 3 (25 points (15,10))
Let Y be n X 1, X be n X k (rank k), Var(Y|X) = Ω and Ω is known. Let Z =XB,
where B is k X k with rank k.
(a) Let (β1cap, e1cap) denotes the OLS coefficients and residuals from
regression of Y of X. Similarly, let (β1tiota, e1tiota) denotes these from OLS
regression of Y on Z. Find the relationship between β1cap and β1tiota, and
the relationship between e1cap and e1tiota.
(b) Let β2cap denotes the GLS coefficients from regression of Y on X.
Similarly, let β2tiota denotes the GLS coefficients from regression of Y on Z.
Find the relationship between β2cap and β2tiota.
Problem 4 (25 points (15,10))
1. Let Y1,..., Yn be a random sample from a Bernoulli(θ) distribution. The
p.m.f. of y is f(y|θ) = θ^y(1-θ)^(1-y). Find the Maximum Likelihood
Estimator of θ.
2. Let X1,.., Xn be a random sample from a uniform distribution on the interval
[0,θ], where the value of the parameter θ is unknown (θ > 0). The p.d.f.
f(x|θ) = 1/θ , if 0 ≦xi≦θ, and f(x|θ) = 0 otherwise. Find the Maximum
Likelihood Estimator of θ.
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