Econ 427 lecture 13 slides ARMA Models Byron

Econ 427 lecture 13 slides ARMA Models Byron Gangnes

• • • Auto. Regressive 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 movingaverage (ARMA) model. An ARMA(1, 1) is: Byron Gangnes

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

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

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

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) = 1. 089 (inv=0. 91) and =-1. 95 (inv=-. 51) This is approx the model from page 158 in book (CANEMPL) Byron Gangnes