課程名稱︰迴歸分析
課程性質︰應數所數統組必修
課程教師︰鄭明燕
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2017/1/9
考試時限（分鐘）：15:30~18:20
試題 :
1. Consider the linear regression medel Y=Xβ+ε.
(a)(6 pts) Give the standardized residuals d , i=1,...,n, and explain a use
i
of them in model diagnostics.
(b)(6 pts) Give the studentized residuals r , i=1,...,n, and explain a use
i
of them in model diagnostics.
(c)(6 pts) Give the PRESS residuals e , i=1,...,n, and explain a use
(i)
of them in model diagnostics.
2. (8 pts)Consider the linear model Y=Xβ+ε. Suppose the ε's are uncorrelated
i
2 2
with zero mean and variances Var(ε)=c σ where c ,...,c are known positive
i i 1 n
2
constants. Five the BLUE of β and an unbiased estimator of σ .
T
3. Consider the model y =x β+ε, i=1,...,n, where ε=ρε + u with |ρ|<1
i i i i i-1 i
2
and u 's being uncorrelated with mean zero and variance σ . Assume that
i
2 2
E(ε)=0, Var(ε)=σ /(1-ρ ) and ε is independent of u ,...,u .
1 1 1 2 n
(a)(5 pts) Find the generalized least squares estimator of β.
^ n n 2
(b)(5 pts) Show that ρ=Σ e e /Σ e is asymptotically unbiased for ρ.
i=2 i i-1 i=1 i
(c)(8 pts) Find the estimated generalized least squares estimator of β and
2
an asymptotically unbiased estimator of σ .
4. Consider the simple linear regression model y = β+βx +ε, where Var(ε)
i 0 1 i i i
2 2
= σ x , i=1,...,n.
i
(a)(8 pts) Suppose that we use the transformations y'=y/x and x'=1/x. Is
this a variance-stabilizing transformation? What are the final estimators
2
of β,β, and σ ?
0 1
(b)(5 pts) Suppose we use the weighted least squares method with weights
2
w = 1/x , i=1,...,n. Is this equivalent to the transformations introduced
i i
in part (a)?
5. Consider the model y = x'β+ε, i=1,...,n, where the ε's are uncorrelated
i i i i
with zero mean.
(a)(6 pts) Describe problems caused by influential points, and how we can
cope with the problems.
(b)(9 pts) Define three measures to detect influential points.
6. Consider the one-way analysis of variance model y = μ+α+ε , i=1,...,3,
ij i ij
j=1,...,n, where y is the jth observation for the ith treatment level, μ
ij
is the grand mean, α is effect of the ith treatment, and ε's are
i ij
2
independent N(0,σ ) errors.
(a)(5 pts) Write down a linear regression model of the form Y=Xβ+ε for
the data, using indicator variables.
(b)(8 pts) Find the ordinary least squares estimators of μ,α,α, and α.
1 2 3
(c)(10 pts) Write down the One-Way Analysis of Variance table.
7. Consider the regression model Y = Xβ+ε. Describe the following three
possible remedies to multicollinearity.
(a)(5 pts) Incomplete principal component regression.
(b)(5 pts) Ridge regression.
(c)(5 pts) Add additional observations.