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Im currently reading The Foundation of Modern Time Series Analysis by Terrence C. Mills.

In the fourth chapter he discusses the concept of difference data to remove secular trend and the history behind it.

On page 36, section 4.7 it says:

Cave and Pearson then computed correlation coefficients for all pairs of pairs of indicies at each level of differencing. we shall content ourselves with reporting correlations ($\pm$ probable errors) between tobacco and savings for $d=0,1,....6,$ the $d=0$ correlation being the focus of concern...

\begin{array}{c|lcr} d & \text{Correlation} \\ \hline 0 & 0.984 \pm 0.005\\ 1 &0.766 \pm0.065\\ 2 &-0.044 \pm0.182\\ 3&-0.327\pm0.181\\ 4 &-0.380 \pm0.188\\ 5&-0.402\pm0.188\\ 6&-0.432\pm0.204\\ \end{array}

It is clear that the large positive correlation between tobacco and savings at d=0 does appear to be spurious: by $d=3$ the correlation is negative and by $d=6$ significantly so.

From this research quoted in the book, a true relationship was uncovered.

This is very interesting for me as a quantitative analyst and amateur social scientist in applying these methods to regression analysis.

However no limit for the "required" amount differences that need to be taken of both a the variables is given to uncover a true relationship.

Is there a limit for differencing? can you over difference data?

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You can induce structure in your Transfer Function Model (ADL/PDL) if you over-difference one or more of the series in your model. I am not totally sure that the one example you quoted is a gold-plated decision rule as I would expect different kinds of data might provide different measures of cross-correlation thus suggesting differing rules based upon the actual data example. Knowing Terry Mills as I do , I am sure he didn't imply that this was "the way " to determine a sufficient level of differencing for each and every series in the model but rather was being anecdotal.

For example if both X and Y are I(1) it is quite possible that Y and X can have a model of the form $Y(t)= \beta_0 + \beta_1X(t)$ OR $[1-B]Y(t)= \beta_0 + \beta_1[1-B]X(t)$ . Only the data and the DGF knows for sure and it is your task to find out which one is sufficient. False conclusions can easily be reached if you don't validate i.e. test the model assumptions viz . constant mean of the residuals and constant variance of the residuals and constant parameters over time.

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  • $\begingroup$ You know mills personally? can you get him to comment on this question? $\endgroup$ – EconJohn Nov 13 '17 at 18:07
  • $\begingroup$ As a time series person , I have interacted with similar folk over 50 years. I no longer have any contact info on Terry and my initial attempts via Google failed. It appears he has retired from teaching as he is now classified as an emeritus professor. $\endgroup$ – IrishStat Nov 13 '17 at 18:30
  • $\begingroup$ That's unfortunate. $\endgroup$ – EconJohn Nov 13 '17 at 18:39
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    $\begingroup$ I would reflect that model form should be customized to the data just as a tailor customizes your suit to your dimensions/data , There is no "smoking gun" or "magic wand" just an iterative attempt to separate signal and noise from the data providing a model with necessary structure while being sufficient. This can often arise from a good starting model that is theory based or alternatively from the data. Developing models from the data should not imply developing theory but rather " a useful model ". More often than not a good starting model provides a basis for "tuning" l $\endgroup$ – IrishStat Nov 13 '17 at 18:46

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