In the statistical analysis of time series, autoregressive–moving-average (ARMA) models are a way to describe a (weakly) stationary stochastic process using autoregression (AR) and a moving average (MA), each with a polynomial.
They are a tool for understanding a series and predicting future values.
AR involves regressing the variable on its own lagged (i.e., past) values.
The general ARMA model was described in the 1951 thesis of Peter Whittle, Hypothesis testing in time series analysis, and it was popularized in the 1970 book by George E. P. Box and Gwilym Jenkins.
[1][2] In order for the model to remain stationary, the roots of its characteristic polynomial must lie outside the unit circle.
[3] The augmented Dickey–Fuller test assesses the stability of IMF and trend components.
The final value is obtained by reconstructing the predicted outcomes of each time series.
The notation MA(q) refers to the moving average model of order q: where the
white noise error terms that are commonly normal random variables.
represents the polynomial Finally, the combined ARMA(p, q) model is given by or more concisely, or This is the form used in Box, Jenkins & Reinsel.
is the characteristic polynomial of the moving average part of the ARMA model, and
is the characteristic polynomial of the autoregressive part of the ARMA model.
[7][8] An appropriate value of p in the ARMA(p, q) model can be found by plotting the partial autocorrelation functions.
Both p and q can be determined simultaneously using extended autocorrelation functions (EACF).
[9] Further information can be gleaned by considering the same functions for the residuals of a model fitted with an initial selection of p and q. Brockwell & Davis recommend using Akaike information criterion (AIC) for finding p and q.
After choosing p and q, ARMA models can be fitted by least squares regression to find the values of the parameters which minimize the error term.
It is good practice to find the smallest values of p and q which provide an acceptable fit to the data.
For a pure AR model, the Yule-Walker equations may be used to provide a fit.
ARMA outputs are used primarily to forecast (predict), and not to infer causation as in other areas of econometrics and regression methods such as OLS and 2SLS.
The general ARMA model was described in the 1951 thesis of Peter Whittle, who used mathematical analysis (Laurent series and Fourier analysis) and statistical inference.
[12][13] ARMA models were popularized by a 1970 book by George E. P. Box and Jenkins, who expounded an iterative (Box–Jenkins) method for choosing and estimating them.
[14] ARMA is essentially an infinite impulse response filter applied to white noise, with some additional interpretation placed on it.
In digital signal processing, ARMA is represented as a digital filter with white noise at the input and the ARMA process at the output.
ARMA is appropriate when a system is a function of a series of unobserved shocks (the MA or moving average part) as well as its own behavior.
For example, stock prices may be shocked by fundamental information as well as exhibiting technical trending and mean-reversion effects due to market participants.
Autoregressive conditional heteroskedasticity (ARCH) models time series where the variance changes.
Autoregressive fractionally integrated moving average (ARFIMA, or Fractional ARIMA, FARIMA) model time-series that exhibits long memory.
Multiscale AR (MAR) is indexed by the nodes of a tree instead of integers.
Statistical packages implement the ARMAX model through the use of "exogenous" (that is, independent) variables.
Care must be taken when interpreting the output of those packages, because the estimated parameters usually (for example, in R[15] and gretl) refer to the regression: where