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What is an autoregressive moving average model (ARMA)?

Module by: Brandon Hodgson

What is an autoregressive moving average model (ARMA)?

Given a time seriesof data Xt, the autoregressive moving average model (ARMA), sometimes called the Box-Jenkins model after George Box and G.M. Jenkins, is used as a tool for understanding and possibly predicting future values in the time series (Wikipedia 2006). The ARMA is typically applied to time series data (Wikipedia 2006).

The model typically consists of two parts, an autoregressive (AR) part and a moving average (MA) part (Wikipedia 2006). The model is usually then referred to as an ARMA(p,q) model where p is the order of the autoregressive part and q is the order of the moving average part (Wikipedia 2006).

Autoregressive model

The notation AR(p) refers to an autoregressive model of order p (Wikipedia 2006). Thus, an AR(p) model is written as

Figure 1
Figure 1 (85265c25c745bb6c33b76c18587db1d9)
(1)(Wikipedia 2006)

where

Figure 2
Figure 2 (5d2f048220952fe1926f47c1874422c4)
are the parameters of the model, c is a constant and εt is an error term (Wikipedia 2006). The constant term is omitted by many authors for simplicity (Wikipedia 2006).

Example: An AR(1) model is given by

Figure 3
Figure 3 (a72aecefcd54a589c2a27ebef4f73586)
(2)(Wikipedia 2006).

An autoregressive model is essentially an infinite impulse response filter with some additional interpretation placed on it (Wikipedia 2006).

Some constraints are necessary on the values of the parameters of this model in order that the model remains stationary (Wikipedia 2006).

Example: In an AR(1) model, if |φ1| > 1 then the model will not be well behaved (Wikipedia 2006).

Moving average model

The notation MA(q) refers to a moving average model of order q (Wikipedia 2006). This is given by

Figure 4
Figure 4 (b205cb9b4f4713ecc9e08b50640d02c7)
(3)(Wikipedia 2006).

where the θ1, ..., θq are the parameters of the model and the εt, εt-1,... are as in the AR model, the error terms (Wikipedia 2006). A moving average model is essentially a finite impulse response filter with some additional interpretation placed on it (Wikipedia 2006).

Autoregressive moving average model

Taking the AR model and the MA model, we get the ARMA model. The notation ARMA(p, q) refers to a model with p autoregressive terms and q moving average terms (Wikipedia 2006). This model subsumes the AR and MA models,

Figure 5
Figure 5 (fa351b01dc01ba11909cef42b5823680)
(4)(Wikipedia 2006).

The error terms εt are generally assumed to be independent identically-distributed random variables sampled from a normal distribution with zero mean: εt ~ N(0,σ2) where σ2 is the variance(Wikipedia 2006). These assumptions may be weakened but doing so will change the properties of the model (Wikipedia 2006). In particular, a change to the iid assumption would make a rather fundamental difference (Wikipedia 2006).ARMA models in general can, after choosing p and q, be fitted by least squares regression to find the values of the parameters which minimise the error term (Wikipedia 2006). It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data (Wikipedia 2006). For a pure AR model then the Yule-Walker equations may be used to provide a fit (Wikipedia 2006).Exercise: The dependence of Xt on past values and the error terms εt is assumed to be linear unless specified otherwise (Wikipedia 2006). What happens if the dependence is non-linear? AnswerThe dependence of Xt on past values and the error terms εt 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.References:Wikipedia. "Autoregressive moving average model", Wikimedia Foundation Inc,  http://en.wikipedia.org/wiki/Autoregressive_moving_average_model, Last accessed 14 February 2006.

Brandon Hodgson

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