Week 2

Question 1

AR(p)

Answer 1

Autoregressive Model of order p
Stochastic process of below form
X_T is stationary Φ1,…,Φp are real constants and Φp != 0

Question 2

Backshift operator

Answer 2

Used to make writing down AR model cleaner

Question 3

Stationarity from associated polynomial of AR(p)

Answer 3

If all roots are larger in size than 1 then model is stationary, otherwise non stationary

Question 4

MA(q)
Backshift Operayor
Associated polynomial

Answer 4

Moving Average model of order q

Question 5

MA(q) models are always

Answer 5

Stationary

Question 6

MA(1)

Answer 6

Question 7

MA(2)

Answer 7

Question 8

For AR(1) if |Φ₁ | < 1

Answer 8

Stationary solution of AR(1) model

‘we know that’ through an application of the geometric series to the operator

therefore we can rewrite as a linear model

Question 9

Autocovariance of stationary solution of AR(1) model

Answer 9

Question 10

for AR(1), if Φ₁ > 0 ?

Answer 10

Expo decrease

If < 0 this => oscillatory decrease

As γ(h) = Φ ₁^h

Question 11

What are Yule Walker equations

Answer 11

Set of linear equations formed from γ(h) and ρ(h)
Used to estimate parameters of AR model

Question 12

Using Yule Walker equations

Answer 12

Question 13

ARMA(1, 1)

Answer 13

Question 14

Causality and invertibility of ARMA(p, q)

Answer 14

Question 15

Check causality of AR(1)

Answer 15

If φ(z) != 0 for |z| <= 1
We must have

Question 16

AR(2) model is causal if

Answer 16

Question 17

Rewrite 2nd order polynomial

Answer 17

Question 18

Parameter redundancy

Answer 18

Question 19

For AR(1) ?

Answer 19

Question 20

Result of Yule Walker Estimators for AR(2)

Answer 20

Question 21

Method of moments for MA(1)

Answer 21

Question 22

Important equality for YW equations

Answer 22

ρ(0)=1

ρ(h) = ρ(-h)

Question 23

Whenevr asked to show that something equals something else

Answer 23

State definitions as this will often reveal method

Question 24

Conditions for ARMA model to be stationary

Answer 24

Roots of characteristic polynomial for AR part given by φ must be outside unit circle

MA part is always stationary