Autoregressive moving average model

In statistics, autoregressive moving average (ARMA) models are typically applied to time series data. Suppose we have at hand two time series, x₁, x₂, x₃, ..., and y₁, y₂, y₃, .... The series x is conventionally assumed to be unpredictable "shocks" which affect or modify y. We wish to predict y_t. If the prediction model contains only x terms, the model is called a moving average (MA) model. If the prediction model contains only y terms, the model is called an autoregressive (AR) model. If the prediction model contains both x and y terms, the model is called an autoregressive moving average (ARMA) model.

Table of contents

1 Moving average model 2 Autoregressive model 3 Autoregressive moving average model 4 Generalizations 5 References

Moving average model

The notation MA(q) means a moving average model with q terms. An MA(q) model can be written

for some coefficients θ₁, ..., θ_q. A moving average model is essentially a finite impulse response filter with some additional interpretation placed on it.

Autoregressive model

The notation AR(p) means an autoregressive model with p terms. An AR(p) model can be written

for some coefficients φ₁, ..., φ_p. An autoregressive model is essentially an infinite impulse response filter with some additional interpretation placed on it.

Autoregressive moving average model

The notation ARMA(p, q) means a model with p autoregressive terms and q moving average terms. This model subsumes the AR and MA models,

Generalizations

The dependence of y_t on past values of x or y is assumed to be linear unless specified otherwise. If the dependence is nonlinear, the model is specifically called a nonlinear moving average (NMA), nonlinear autoregressive (NAR), or nonlinear autoregressive moving average (NARMA) model. Autoregressive moving average models can be generalized in other ways. See also autoregressive conditional heteroskedasticity (ARCH) models and autoregressive integrated moving average (ARIMA) models.

References

• George E.P. Box and F.M. Jenkins. Time Series Analysis: Forecasting and Control, second edition. Oakland, CA: Holden-Day, 1976.

Category:Statistics