Week 2 Flashcards
AR(p)
Autoregressive Model of order p
Stochastic process of below form
XT is stationary Φ1,…,Φp are real constants and Φp != 0
Backshift operator
Used to make writing down AR model cleaner
Stationarity from associated polynomial of AR(p)
If all roots are larger in size than 1 then model is stationary, otherwise non stationary
MA(q)
Backshift Operayor
Associated polynomial
Moving Average model of order q
MA(q) models are always
Stationary
MA(1)
MA(2)
For AR(1) if |Φ1 | < 1
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
Autocovariance of stationary solution of AR(1) model
for AR(1), if Φ1 > 0 ?
Expo decrease
If < 0 this => oscillatory decrease
As γ(h) = Φ 1h
What are Yule Walker equations
Set of linear equations formed from γ(h) and ρ(h)
Used to estimate parameters of AR model
Using Yule Walker equations
ARMA(1, 1)
Causality and invertibility of ARMA(p, q)
Check causality of AR(1)
If φ(z) != 0 for |z| <= 1
We must have
AR(2) model is causal if
Rewrite 2nd order polynomial
Parameter redundancy
For AR(1) ?
Result of Yule Walker Estimators for AR(2)
Method of moments for MA(1)
Important equality for YW equations
ρ(0)=1
ρ(h) = ρ(-h)
Whenevr asked to show that something equals something else
State definitions as this will often reveal method
Conditions for ARMA model to be stationary
Roots of characteristic polynomial for AR part given by φ must be outside unit circle
MA part is always stationary
