1. Econ 427 lecture 13 slides
ARMA Models
Byron Gangnes

2. AutoRegressive Moving-Average (ARMA) models
We have looked at autoregressive (AR) and moving average models.
The two kinds of dynamics can be combined into what is called an autoregressive moving-average (ARMA) model.
An ARMA(1,1) is:
Byron Gangnes

3. ARMA(1,1) model How would I write this is lag operator form?
What’s required for covariance stationarity?
If abs val of phi < 1, it has a moving average form (as do all cov stat processes):
What’s required for invertibility?
If abs val of theta < 1, we can write in autoregressive form:
Byron Gangnes

4. Why do this?
Because very parsimonious ARMA models can capture fairly sophisticated dynamic behavior.
Byron Gangnes

5. Higher Order ARMA models
Higher order ARMA processes involve additional lags of y and epsilon:
What are the conditions for invertibility and covariance stationarity?
For covariance stationarity, inverse of all roots of the lag polynomial, Phi, must be inside the unit circle
For invertibility, inverse of all roots of the lag polynomial, Theta, must be inside the unit circle
Byron Gangnes

6. Roots of a lag polynomial
Suppose y is given by:
This is a quadratic in the lag operator:
{-1.43+/-sqrt[(1.43)2-4(.47)]} / 2(.47)
= (inv=0.91) and
=-1.95 (inv=-.51)
This is approx the model from page 158 in book (CANEMPL)
Byron Gangnes