Autoregressive moving average model

In statistics, autoregressive moving average (ARMA) models, sometimes called Box-Jenkins models after George Box and F. M. Jenkins, are typically applied to time series data.

Given a time series of data X_t then the ARMA model is a tool for understanding and, perhaps, predicting future values in this series. The model consists of two parts, an autoregressive (AR) part and a moving average or (MA) part. 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 (as defined below).

Contents

1 Autoregressive model 1.1 Example: An AR(1)-Process 2 Moving average model 3 Autoregressive moving average model 4 Note about the error terms 5 Specification in terms of lag operator 6 Fitting models 7 Generalizations 8 References

Autoregressive model

The notation AR(p) refers to an autoregressive model of order p. An AR(p) model can be written

<math> X_t = c + \sum_{i=1}^p \phi_i X_{t-i}+ \epsilon_t .\,<math>

where <math>\phi_1, \ldots \phi_p<math> are referred to as the parameters of the model, <math>c<math> is a constant and <math>\epsilon_t<math> is an error term (see below). The constant term is omitted by many authors for simplicity.

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

It should be noticed that some constraints are necessary on the values of the parameters of this model in order that the model remains stationary. For example, in an AR(1) model, if |φ₁| > 1 then the model will not be well behaved.

Example: An AR(1)-Process

An AR(1)-Process is given by

<math>X_t = c + \phi X_{t-1}+\epsilon_t,\,<math>

where <math>\epsilon_t<math> is a white noise process with zero mean and variance <math>\sigma^2<math>. (Note: The subscript on <math>\phi_1<math> has been dropped.) The process is covariance-stationary if <math>|\phi|<1<math>. If <math>\phi=1<math> then <math>X_t<math> exhibits a unit root and can also be considered as a random walk. The calculation of the expectation of <math>X_t<math> is straightforward. Assuming covariance-stationarity we get

<math>\mbox{E}(X_t)=\mbox{E}(c)+\phi\mbox{E}(X_{t-1})+\mbox{E}(\epsilon_t)\Rightarrow \mu=c+\phi\mu+0<math>.

thus:

<math>\mu=\frac{c}{1-\phi},<math>

where <math>\mu<math> is the mean. The variance is found to be:

<math>\textrm{var}(X_t)=E(X_t^2)-\mu^2=\frac{\sigma^2}{1-\phi^2}<math>

The autocorrelation function is given by:

<math>B_n=E(X_{t+n}X_t)-\mu^2=\frac{\sigma^2}{1-\phi^2}\,\,\phi^{|n|}<math>

It can be seen that the autocorrelation function decays with a decay time of <math>\tau=-1/\ln(\phi)<math>. The spectral density function is the Fourier transform of the autocorrelation function. In discrete terms this will be the discrete-time Fourier transform:

<math>\Phi(\omega)=

\frac{1}{\sqrt{2\pi}}\,\sum_{n=-\infty}^\infty B_n e^{-i\omega n} =\frac{1}{\sqrt{2\pi}}\,\left(\frac{\sigma^2}{1+\phi^2-2\phi\cos(\omega)}\right) <math>

This expression contains aliasing due to the discrete nature of the <math>X_j<math>. If we assume that the sampling time is much smaller than the decay time (<math>\tau\ll 1<math>), then we can use a continuum approximation to <math>B_n<math>:

<math>B(t)\approx \frac{\sigma^2}{1-\phi^2}\,\,\phi^{|t|}<math>

which yields a Lorentzian profile for the spectral density:

<math>\Phi(\omega)=

=\frac{1}{\sqrt{2\pi}}\,\frac{\sigma^2}{1-\phi^2}\,\frac{\gamma}{\pi(\gamma^2+\omega^2)}<math>

where <math>\gamma=1/\tau<math> is the angular frequency associated with the decay time <math>\tau<math>.

An alternative expression for <math>X_t<math> can be derived by first substituting <math>c+\phi X_{t-2}+\epsilon_{t-1}<math> for <math>X_{t-1}<math> in the defining equation. Continuing this process N times yields:

<math>X_t=c\sum_{k=0}^{N-1}\phi^k+\phi^NX_{t-N}+\sum_{k=0}^{N-1}\phi^k\epsilon_{t-k}<math>

For N approaching infinity, <math>\phi^N<math> will approach zero and:

<math>X_t=\frac{c}{1-\phi}+\sum_{k=0}^\infty\phi^k\epsilon_{t-k}<math>

It is seen that <math>X_t<math> is white noise convolved with the <math>\phi^k<math> kernel plus the constant mean. By the central limit theorem, the <math>X_t<math> will be normally distributed as will any sample of <math>X_t<math> which is much longer than the decay time of the autocorrelation function.

Moving average model

The notation MA(q) refers to a moving average model of order q.

<math> X_t = \varepsilon_t + \sum_{i=1}^q \theta_i \varepsilon_{t-i}\,<math>

where the θ₁, ..., θ_q are referred to as the parameters of the model and the ε_t, ε_t-1,... are again, the error terms. A moving average model is essentially a finite impulse response filter with some additional interpretation placed on it.

Autoregressive moving average model

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

<math> X_t = \varepsilon_t + \sum_{i=1}^p \phi_i X_{t-i} + \sum_{i=1}^q \theta_i \varepsilon_{t-i}.\,<math>

Note about the error terms

The error terms ε_t are generally assumed to be iid variables sampled from a normal distribution with zero mean: ε_t ~ N(0,σ²) where σ² is the variance. These assumptions may be weakened but doing so will change the properties of the model. In particular, a change to the iid assumption would make a rather fundamental difference.

Specification in terms of lag operator

In some texts the models will be specified in terms of the lag operator L. In these terms then an AR(p) model is given by

<math> \varepsilon_t = \left(1 - \sum_{i=1}^p \phi_i L^i\right) X_t = \phi X_t\,<math>

where φ represents polynomial

<math> \phi = 1 - \sum_{i=1}^p \phi_i L^i.\,<math>

An MA(q) model is given by

<math> X_t = \left(1 + \sum_{i=1}^q \theta_i L^i\right) \varepsilon_t = \theta \varepsilon_t\,<math>

where θ represents the polynomial

<math> \theta= 1 + \sum_{i=1}^q \theta_i L^i.\,<math>

Finally, the combined ARMA model is given by

<math> \left(1 - \sum_{i=1}^p \phi_i L^i\right) X_t = \left(1 + \sum_{i=1}^q \theta_i L^i\right) \varepsilon_t\,<math>

or more concisely,

<math> \phi X_t = \theta \varepsilon_t.\,<math>

Fitting models

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. It is generally considered good practice to find the smallest values of p and q which provide an acceptable fit to the data. For a pure AR model then the Yule-Walker equations may be used to provide a fit.

Generalizations

The dependence of X_t on past values and the error terms ε_t 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. If multiple time series are to be fitted then a vectored ARIMA (or VARIMA) model may be fitted. If the time-series in question exhibits long memory then fractional ARIMA (FARIMA, sometimes called ARFIMA) modelling is appropriate. If the data is thought to contain seasonality then a SARIMA model should be used.

References

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

su:Autoregressive moving average model