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Knowing that a time series is over-differenced or under-differenced, and adding an AR term to the model means that we are partially differencing the time series if it's underdifferenced till a white noise remains, and adding MA term means that we are removing the effect of over-difference time series till we get also to white noise. Thus as adding MA terms has an effect that oppose to the effect of adding AR terms to our model why are combining the two into ARMA model as the series is either over-differenced or under-differenced??

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    $\begingroup$ Why is a time series over- or underdifferenced, is that an assumption? And why do you think that AR or MA terms work as differencing or anti-differencing (they do not, at least in terms of the common definition of differencing)? Could you give some background for that? $\endgroup$ – Richard Hardy Nov 29 '18 at 15:38
  • $\begingroup$ people.duke.edu/~rnau/411arim3.htm Rule 6 and 7 in the precedent link answer your question $\endgroup$ – Chaymae Ahmed Nov 29 '18 at 16:20
  • $\begingroup$ Thank you! OK, I see these are written in quotation marks, that makes sense. You could loosely say that some forms of AR and MA do something like differencing or antidifferencing. But these are rather constrained cases and the argument could even be reversed by reversing the signs of AR and MA coefficients. But OK, thanks for the reference! $\endgroup$ – Richard Hardy Nov 29 '18 at 16:46
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The whole idea is to combine AR , DIFF and MA structure in a parsimonious way incorporating pulses,level/step,seasonal pulse and local time trends as needed following http://docplayer.net/12080848-Outliers-level-shifts-and-variance-changes-in-time-series.html and others .

Careful iterative steps should be taken to insure that no unnecessary and/or redundant structure is incorporated. Model building is not a 1 stop process but rather a sequence of steps culminating in an equation that may have needed complexity but doesn't have unnecessary structure. Comments like "the significance of variables is irrelevant" is self-serving due to software limitations which doesn't' enable necessity and sufficiency checking and appropriate remedies and flies in the face of "good statistical practice"

Software that simply tries to "fit without testing significance" and avoiding the treatment of deterministic structure often over-parameterizes requiring some of the steps you alluded to in your question ARIMA Time series analysis forecasting and How to determine order of sarima?.

So the suggested "logic" goes .. STEP 1:... use auto.arima to identify the ARIMA portion assuming no interventions ( one form of deterministic structure) are present in the data which erroneously misidentifies the noise if deterministic structure is present. This explains the unwarranted and extremely complex over-parameterized model selection typically reported. See Coefficient Significance in Regression with Arima Errors for a very recent example of this.

The model should be complex enough but not too complex. The residuals from the final model should be free of autocorrelation and have a constant mean and variance throughout time.

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