Description
Problem 1: (10pts)
Let y = Xβ + be a linear model where X is of size n × p and the error terms
are independent, normally distributed with mean 0 and variance σ
2
. Suppose
furthermore that the columns of X can be partitioned as
X =
W Z
where W is of size n × q and is of full-column rank and Z is of size n × (p − q)
and is of full-column rank, for some q satisfying 1 ≤ q ≤ p, and that WT Z = 0.
We now partition β as β =
h
β1
β2
i
where β1 is of size q × 1 and β2 is of size
(p − q) × 1. Let βˆ =
h
βˆ1
βˆ2
i
be the least square estimate of β.
(a) Show that βˆ
1 = (WTW)
−1WT y and βˆ
2 = (Z
T Z)
−1Z
T y.
(b) Show that βˆ
1 and βˆ
2 are independent.
(c) Let a be a q×1 vector and b be a (q−p)×1 vector. Let (l1, u1) and (l2, u2)
be the individual 95% confidence intervals for a
T β1 and b
T β2 based on
βˆ
1 and βˆ
2, respectively. Is the confidence interval (l1, u1) independent of
the confidence interval (l2, u2) ? Justify your answer.
Problem 2: (10pts)
Let W be a n×p matrix and X be a n×q matrix and that C(W) ⊆ C(X). Denote
by PX and PW the symmetric idempotent matrices projecting onto C(X) and
1