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.

Moving average model

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

<math> y_t = x_t + \theta_1 x_{t-1} + \cdots + \theta_q x_{t-q}

<math>

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

<math> y_t = \phi_1 y_{t-1} + \cdots + \phi_p y_{t-p} <math>

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,

<math> y_t = \phi_1 y_{t-1} + \cdots + \phi_p y_{t-p}

  + x_t + \theta_1 x_{t-1} + \cdots + \theta_q x_{t-q}

<math>

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.

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