Penalty Function Methods

Since the early 1970s, some estimation-type identification procedures have been proposed. They are to choose the orders k and i minimizing $$P(k,i) = {\text{ln}}{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\sigma }}\mathop{{k,i}}\limits^{2} + (k + i)\frac{{C(T)}}{T}$$, where σ _k,i 2 is an estimate of the white noise variance obtained by fitting the ARMA(k, i) model to the observations. Because σ _k,i 2 decreases as the orders increase, it cannot be a good criterion to choose the orders minimizing it. If the orders increase, the bias of the estimated model will decrease while the variance increases. Therefore, we should compromise between them. For this purpose we add the penalty term, (k + i)C(T)/T, into the model selection criterion The penalty function identification methods are regarded as objective.