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==Box–Jenkins model estimation==
Estimating the parameters for the Box–Jenkins models is a quite complicated non-linear estimation problem. For this reason, the parameter estimation should be left to a high quality software program that fits Box–Jenkins models. Fortunately, many statistical software programs now fit Box–Jenkins models.
The main approaches to fitting Box–Jenkins models are non-linear least squares and maximum likelihood estimation. Maximum likelihood estimation is generally the preferred technique. The likelihood equations for the full Box–Jenkins model are complicated and are not included here. See (Brockwell and Davis, 1991) for the mathematical details.
The Autoregressive Moving Average(ARMA) model is a form of the Box-Jenkins model:
y[t] = c + Ø[1]y[t-1]+ Ø[2]y[t-2]+ ...... Ø[p]y[t-p]+ θ[1]e[t-1] + θ[2]e[t-2] + ..... θ[q]e[t-q] + e[t]
where the p represents the order of the autoregressive part and the q is the order of the moving average part.
Another form of the Box-Jenkins model is the Autoregressive Integrated Moving Average model(ARIMA), which is combines the ARMA model with differencing to achieve further stationary.
==Box–Jenkins model diagnostics==
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