Similarly, an ARIMA model was fitted to the El Niño3.4 series for the training period. An autoregressive model of order 2 (AR(2) ) was adequate for the series. Figure \ref{586715} b2) presents the sample crosscorrelation function (CCF) between El Niño 3.4 series (X) and the PDSI series (Y), and the sample CCF between the pre-whitened X series (residuals after fitting an ARIMA model) and the filtered Y series (after applying the AR(2) filter) in figure \ref{586715} b1).
From the sample CCF functions and according to Box and Jenkins (1970), a transfer function model of order \((r_1,s_1,d_1)=(1,1,1)\) for input series \(X_1(t)\)= Niño3.4, and order \((r_2,s_2,d_2)=(1,0,1)\) for input series \(X_2(t) =PDO\), was proposed for this data set. In this case \(\alpha_1(B)=(\delta_{01}+\delta_{11}B)B^1/(1-\omega_{11}B^1)\); and \(\alpha_2(B)=(\delta_{02})B^1/(1-\omega_{12}B^1)\) \(\).
The final model to be fitted is of the form: