where each Aj (an n x n matrix) is the autoregression coefficient. Zt is the column vector of length n, denoting the values of the time series variables at time t. p is the order of the filter which is generally much less than the length of the series. The noise term or residual, Et, is almost always assumed to be Gaussian white noise.
In a more general case, we can consider that the values Zt-1, Zt-2 ... Zt-p are themselves noisy values. Adding the noise values to Eq. 2, we get the ARMA (autoregressive moving average) equation:
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