I'm working to develop a forecasting model for a quarterly seasonal variable (quarterly estimated individual income tax payments) using several candidates for non-seasonal independent variables (quarterly average value of S&P 500, dividend income, interest income, etc.).

I understand that I need to prewhiten the dependent and independent variables in order to generate a cross-correlation function to identify the appropriate rational transfer function for the independent variables, but the seasonal dependent variable and candidate non-seasonal independent variables require different differencing operations to be stationary. Using year-over-year differencing for the non-seasonal variables leads to over-differencing, often with AR and/or MA terms near 1.0.

Is the only solution to create a synthetic seasonally adjusted series for the dependent variable before prewhitening? Or is there another way to prewhiten variables that require different differencing opterations?


1 Answer 1


there is no need for the pre-whitening differencing operators to be the same for all variables in the tf model. If your software requires that then perhaps you need more generalized software. See https://autobox.com/pdfs/PREFERRED.pdf .

For example a panel from AUTOBOX ( a piece of software that I have helped to develop) shows the generality enter image description here

Step by Step:

1) deterimine order of differencing for both Y and candidate X

2) apply differencing operators

3) identify an arma filter for the suitably differenced candidate X

4) apply that arma filter to both the suitably differenced X and suitably differenced Y

5) compute cross-correlations using results from step 4 to identify TF model

  • $\begingroup$ Thank you. For clarification, I'm attempting to use the cross-correlation plots from SAS Proc ARIMA to identify an appropriate transfer function. Would the CCF plots produce valid results if I, for instance, first use a differencing operator of 1 for quarterly-average S&P 500 and an AR(1) model to produce white-noise residuals and then use the resulting filtered series for the seasonal dependent variable (estimated payments) using a differencing operator of 4 for both series? $\endgroup$ Oct 18, 2019 at 14:27
  • $\begingroup$ I definitely don't think so as sequential approaches are usually inferior ..but only your data knows for sure. The SAS proc is deficient in many way including being blind-sided by anomalies in any series and also doesn't wholly treat the concern for homegeneity of the tf model's error process. $\endgroup$
    – IrishStat
    Oct 18, 2019 at 14:38

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