Econometrics(I) HW1
Q1
Let a be an n × 1 non random vector and let u be an n × 1 random vector with E [uu′ ] = In . Show
that
E [tr(au′ ua′ )] = n
n
∑
a2i .
i=1
Q2

1


′
−1
′
Let P = X(X X) X be the projection matrix. If A = 1

1

 
0
6

 

 
1 and b = 0 .

 
2
0
a. Find the projection matrix onto the column space of A.
b. Find the projection of b and the error.
c. Verify the error is perpendicular to the projection.
Q3
Find the rank and the four eigenvalues/vectors (independent) of A and C. Also, check that the sum
of eigen values equals the trace of the matrix, while the product equals the determinant.


1 1


1 1
A=

1 1

1 1
1
1
1
1


1


1


1

1
1 2


C = 0 4

0 0
1
3


5

6
Q4
Let X be an n × k matrix partitioned as
(
X = X1
)
X2
where X1 is n × k1 and X2 is n × k2 .
(a) Show that

X1′ X1
X ′X = 
X2′ X1
X1′ X2
X2′ X2


What are the dimensions of each of the matrices? Hint: The best practice is to discuss the contents
of the four corners respectively.
(b) Let b be a k1 × 1 vector, partitioned as
 
b1
b= 
b2
where b1 is k1 × 1 and b2 is k2 × 1. Show that


′
′
(X
X
)b
+
(X
X
)b
1
1
2
2
1
1
.
(X ′ X)b = 
′
′
(X2 X1 )b1 + (X2 X2 )b2
Q5
The scalar α ia given by the quadratic form
α = x′ Ax,
where x is n × 1, A is n × n, and x is a function of the vector z, while A does not depend on z.
Show that
∂α
∂x
= x′ (A + A′ ) .
∂z
∂z
If A is symmetric, conclude that
∂α
∂z
= 2x′ A ∂x
∂z .
2