University of Toronto
Department of Statistical Sciences
STA437/2005 Methods of Multivariate Data
Handout: Homework 5, Chapters 7 Date: 21 Nov 2020
1. Problem 7.1 from textbook.
2. Problem 7.3 from textbook.
3. Prove that in univariate linear regression βˆ0 and βˆ1 are given by
βˆ0 = y¯ − βˆ1z¯
βˆ1 =
∑n
j=1(yj − y¯)(zj − z¯)∑n
j=1 (zj − z¯)2
4. Under the general framework
Yn×1 = Zn×(r+1)β (r+1)×1 + n×1
with E[] = 0 and Cov() = σ2I and βˆ = (Z ′Z)−1Z ′y, prove that
(a) E
[
ˆ
]
= 0
(b) Cov
(
ˆ
)
= σ2 (I −H ), where H denotes the hat matrix.
5. Let
Yn×1 = Zn×(r+1)β (r+1)×1 + n×1
where Z has full rank r + 1 and is distributed as Nn(0, σ2I ).
Show that, βˆML (i.e., the maximum likelihood estimate of β) coincides with the least square
estimate βˆLSE in this case.
1

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