ECONOMETRICS

Question 1

a) What is the dimension of 𝑥𝑖?

Answer 1

K x 1

Question 2

b) What is the dimension of 𝑦𝑖?

Answer 2

scalar, 1 x 1

Question 3

c) What is the dimension of 𝑏?

Answer 3

k x 1

Question 4

d) What is the dimension of 𝑒𝑖?

Answer 4

scalar 1 x 1

Question 5

e) What is the dimension of 𝑒?

Answer 5

n x 1

Question 6

f) What is the dimension of 𝑋?

Answer 6

n x k

Question 7

g) What is the dimension of 𝑦̂ ?

Answer 7

n x 1

Question 8

h) What is the dimension of 𝑅𝑆𝑆?

Answer 8

Scalar

Question 9

𝑒′𝑒 = 𝑦′𝑦 − 2𝑦′𝑋𝑏 + 𝑏′𝑋′𝑋𝑏

Answer 9

e = 𝑦 − 𝑋𝑏

Question 10

We have 𝑓(𝑏) = 𝑦′𝑦 − 2𝑦′𝑋𝑏 + 𝑏′𝑋′𝑋𝑏 where 𝑏 is 𝑘 × 1, 𝑦 is 𝑛 × 1 and 𝑋 is 𝑛 × 𝑘. Find the derivative
of 𝑓 with respect to 𝑏 (the gradient of 𝑓).

Answer 10

Derv of transpose becoomes 2X^Ty to keep same dimensions rule is CB = C^T

Question 11

Problem 13
We have 𝑓(𝑏) = 𝑦′𝑦 − 2𝑦′𝑋𝑏 + 𝑏′𝑋′𝑋𝑏 where 𝑏 is 𝑘 × 1, 𝑦 is 𝑛 × 1 and 𝑋 is 𝑛 × 𝑘. Find the stationary
point of 𝑓, the value of 𝑏 where all partial derivatives are zero. Assume that 𝑋′𝑋 is invertible.

Answer 11

Invertible leads to 𝑋′𝑋^-1 both sides cancel out on b side leads to value where b equals

Question 12

Definition of exogeneity

Answer 12

E(ε∣X)=0

Question 13

definition of homoskedastity

Answer 13

Var(ε∣X)=σ2I

Question 14

LIE

Answer 14

E(Y)=E(E(Y∣X)) E(X)=E(E(X∣Y))

Question 15

LIV

Answer 15

Var(X)=E(Var(X∣Y))+Var(E(X∣Y))