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2.2.5.2.2 Algorithm for Prewhitening

Contents

Fit an ARIMA Model

First fit an ARIMA model for the 1st time series (X series). NAG function nag_tsa_multi_inp_model_estim (g13bec) is used to fit an ARIMA model. For more details, refer to Algorithm of ARIMA. The residuals of X series is prewhtened X series.

Filter Y Series

From a given 2nd series (\(y_t\)), a new series \(b_1,b_2,b_3,...,b_n\) is calculated using a supplied (filtering) ARIMA model. The appropriate differencing of \(y_t\)

\(w_t=\nabla ^d \nabla_s^D y_t\),

both the seasonal and non-seasonal inverted autoregressive operations are then applied,

\(u_t=w_t- \Phi_1 w_{t-s}-...- \Phi_p w_{t-s\times p}\),
\(v_t=u_t- \phi_1 u_{t-1}-...- \phi_p u_{t-p}\),

followed by the inverted moving average operations,

\[z_t = v_t+\Theta_1z_{t-s}+...+\Theta_Qz_{t-s\times Q}\]
\[b_t = z_t+\theta_1b_{t-1}+...+\theta_qb_{t-q}\]

Reference

  1. nag_tsa_multi_inp_model_estim (g13bec)
  2. nag_tsa_arma_filter (g13bac)