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The notation ARMAX(p, q, b) refers to a model with p autoregressive terms, q moving average terms and b exogenous inputs terms. The last term is a linear combination of the last b terms of a known and external time series d t {\displaystyle d_{t}} .
Non-seasonal ARIMA models are usually denoted ARIMA(p, d, q) where parameters p, d, q are non-negative integers: p is the order (number of time lags) of the autoregressive model, d is the degree of differencing (the number of times the data have had past values subtracted), and q is the order of the moving-average model.
Different authors have different approaches for identifying p and q. Brockwell and Davis (1991) [3] state "our prime criterion for model selection [among ARMA(p,q) models] will be the AICc", i.e. the Akaike information criterion with correction. Other authors use the autocorrelation plot and the partial autocorrelation plot, described below.
The acronyms "ARFIMA" or "FARIMA" are often used, although it is also conventional to simply extend the "ARIMA(p, d, q)" notation for models, by simply allowing the order of differencing, d, to take fractional values.
[1] [2] The moving-average model specifies that the output variable is cross-correlated with a non-identical to itself random-variable. Together with the autoregressive (AR) model, the moving-average model is a special case and key component of the more general ARMA and ARIMA models of time series, [3] which have a more complicated stochastic ...
Together with the moving-average (MA) model, it is a special case and key component of the more general autoregressive–moving-average (ARMA) and autoregressive integrated moving average (ARIMA) models of time series, which have a more complicated stochastic structure; it is also a special case of the vector autoregressive model (VAR), which ...
The partial autocorrelation for an AR(p) model is nonzero for lags less than or equal to p and 0 for lags greater than p. Moving-average model: If , >, the partial autocorrelation oscillates to 0. If , <, the partial autocorrelation geometrically decays to 0. Autoregressive–moving-average model: An ARMA(p, q) model's partial autocorrelation ...
In that case, the GARCH (p, q) model (where p is the order of the GARCH terms and q is the order of the ARCH terms ), following the notation of the original paper, is given by y t = x t ′ b + ϵ t {\displaystyle y_{t}=x'_{t}b+\epsilon _{t}}