Abstract
A penalty function identification of ARMA processes is to choose the orders minimizing $${\text{ln}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }}\mathop{{k,i + }}\limits^{2} (k + i)\frac{{C(T)}}{T}$$ among k = 0,…, K and i = 0,…, I. Here σ k,i 2 is an estimate of the innovation variance obtained by fitting the ARMA(k, i) model to the observations and K and I are determined a priori as upper bounds of the orders. Because there are (K+1)×(I+1) possible ARMA models to be estimated, it is computationally onerous to apply ML estimation methods. Even though many algorithms have been presented to obtain the exact ML estimates as mentioned in Chapter 1, there are still many problems in applying them to all possible ARMA models. Especially if the MA part exists, then the ML estimates are not always on the stationary and invertible region. They are sensitive to the quality of starting values for the algorithms.
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