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 \(X_1(t)\)= NiƱo3.4, 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:
\(\)
Following \cite{Shumway2017} this model is initially fitted by least squares and the ARIMA model associated to the estimated residuals \(\hat{\eta}_t\) is identified. As a second step the model to refitted assuming autocorrelated errors. Figure 10 shows the autocorrelation and partial autocorrelation function of the estimated residuals, revealing a white noise structure with no additional refitting required.