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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 . It is given by:
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.
An ARCH(q) model can be estimated using ordinary least squares. A method for testing whether the residuals ϵ t {\displaystyle \epsilon _{t}} exhibit time-varying heteroskedasticity using the Lagrange multiplier test was proposed by Engle (1982).
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An ARMA(p, q) model's partial autocorrelation geometrically decays to 0 but only after lags greater than p. The behavior of the partial autocorrelation function mirrors that of the autocorrelation function for autoregressive and moving-average models.
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