I have a system of two inputs and one output that I'd like to model using the following Box-Jenkins transfer function ("dynamic regression") structure:

$$y_t=\frac {\beta_1(B)}{\nu_1(B)}\omega_1Bx_{1,t}+\frac{\beta_2(B)}{\nu_2(B)}\omega_2Bx_{2,t}+\frac {\theta(B)}{\phi(B)}z_t+c$$

(using notation to conform with this Rob Hyndman post: http://robjhyndman.com/hyndsight/arimax/ where $B$ is the backshift operator, numerator polynomials like $\theta(B)$ stand for $(1+\theta_1 B+\theta_2 B^2+\cdots +\theta_nB^n)$, and denominator polynomials like $\phi(B)$ stand for $(1-\phi_1 B-\phi_2 B^2-\cdots -\phi_nB^n)$ )

My problem is this:

My $y$ is a stationary variable. However, both of my $x$s are not. (or to be precise, unlike my $y$, my $x$s are only very weakly stationary and have significant autocorrelation.) In discussions of pre-whitening that I've read, it's recommended to pre-whiten the $x$ variable (and then apply that same filter to $y$) before looking at the correlation structure. However, those examples only cover cases with only one $x$. On the other hand, this section of Hyndman's book: https://www.otexts.org/fpp/9/1 recommends repeatedly applying a difference to $y$ and all $x$s until all of the variables are stationary.

My question is:

Should I difference $y$ and all $x$s once before looking at the cross-correlations, even though this will make $y$ over-differenced? Or, should I pre-whiten $x_1$ and $x_2$, then apply each filter to the other variables, and then look at the cross-correlations? Or is another method recommended?

  • $\begingroup$ If your input data requires differencing then differencing should occur BEFORE prewhitening. In SAS it is very easy to do check this link $\endgroup$
    – forecaster
    Commented Sep 29, 2014 at 1:21
  • $\begingroup$ Yes... the question is, how do I handle the fact that the x variables need to be differenced, but that differencing the y variable would result in it being over-differenced? (Also, I'm using R and don't have access to SAS.) Thanks. $\endgroup$
    – ene100
    Commented Sep 30, 2014 at 17:52
  • $\begingroup$ unfortunately that is the drawback of using ARIMA model. There is a separate class of models called Unobserved components models which doesn't require stationary assumptions so its free from differencing. Also, R has very poor capabilities in terms of dynamic regression vs. other commercially available software packages. $\endgroup$
    – forecaster
    Commented Sep 30, 2014 at 18:19
  • $\begingroup$ Looking at this explanation of unobserved components models: support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/… it seems that they're quite similar to ARIMAX models, but with the optional addition of seasonal terms. Am I missing something? My data don't have seasonality issues, so I think I'll stick with the transfer function framework for now. $\endgroup$
    – ene100
    Commented Oct 6, 2014 at 21:43
  • $\begingroup$ unobserved components model (UCM) doesn't require your data to be stationary which is a big plus also they model trend and seasonality explicitly. You can ignore seasonality if your danot have it. However, UCM does not have the flexibility in terms a parsimoniously specifying regressor variables vs. transfer function. I would recommend do the analysis with both UCM and ARIMAX and pick a model that gives best predictive power. $\endgroup$
    – forecaster
    Commented Oct 6, 2014 at 22:00

2 Answers 2


I have programmed both no differencing and with differencing suggested by the pre-whitening filters. I would suggest that one compare the two final models to determine the best approach for your data.


I've decided that in my case at least, not doing any further differencing is probably correct. If I difference my dependent variable, it develops a one-period autocorrelation of around ~ -0.5, which seems far too high. Your mileage may vary. I'm planning on solving my transfer equation by:
1. Not doing any differencing of either inputs or outputs
2. Choosing a filter for $y$ and all $x$s such that the regression errors have as little serial correlation as possible.
3. Choosing additional filters for each X so as to maximize the regression $R^2$.


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