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According to the answer in this question by IrishStat, the reason that you pre-whiten X is to identify a filter that can transform Y and X into y and x where x is white noise.

Assume that X is following ARIMA(p, q, d):

$\left( 1 - \sum_{i=1}^p \phi_i L^i \right) (1-L)^d X_t = \left( 1 + \sum_{i=1}^q \theta_i L^i \right) \varepsilon_t $

The L.H.S. is the filter of ARI(p, q) and the R.H.S is a linear combination of i.i.d. white noise, which should also be a white noise. So by applying the filter of ARI(p, q), we can then achieve the objective - transform X into white noise. Is my understanding correct?

Ken T
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    *a linear combination of i.i.d. white noise <...> should also be a white noise* – no. An MA process is autocorrelated. You are combining a given white noise process with its lags, so they are dependent. (A combination of independent white noise processes would remain white noise.) – Richard Hardy May 07 '18 at 07:57
  • @RichardHardy You are right. If x is still autocorrelated, the CCF between y and x still can’t enjoy the cut-off property. Then what is the filter for X with moving averaging component? – Ken T May 07 '18 at 12:45
  • I don't know much about filters. – Richard Hardy May 07 '18 at 14:20

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