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Why we usually do not exceed ARMA(5,5) models in practice? Is there any mathematical justification for this?


marked as duplicate by S. Kolassa - Reinstate Monica, kjetil b halvorsen, Community Mar 22 '18 at 11:22

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  • $\begingroup$ well, there are myriad examples with longer lag-length required: e.g. unployment rate in US. $\endgroup$ – Math-fun Jul 14 '16 at 8:56
  • $\begingroup$ @Math-fun: are you sure the unemployment rate is not more adequately captured via seasonal ARIMA, which the OP does not seem to be asking about? $\endgroup$ – S. Kolassa - Reinstate Monica Jul 14 '16 at 8:58
  • $\begingroup$ @StephanKolassa it is definitely seasonal. Removing seasonality we will be left with something that has a rather sticky autocorrelation function ... $\endgroup$ – Math-fun Jul 14 '16 at 9:00

I can think of two and a half reasons.

  1. The metaphysical reason: I understand why a process should depend on the previous realization, in an AR(1) model. Or even that $y_{t-2}$ may have an impact on $y_t$ above and beyond the influence from $y_{t-1}$ (which is already modeled by the first autoregressive parameter $\phi_1$), yielding an AR(2) model instead of AR(1). With higher and higher orders, I find it progressively harder to think of a decent justification for why there should still be an appreciable impact of the $p+1$st AR lag over and beyond what the previous $p$ lags already model.

    Of course, this does not apply to seasonal ARIMA, which you were not asking about.

  2. The statistical reason: more parameters = more degrees of freedom = more model variance but lower bias = beyond some point, worse forecasts. The bias-variance tradeoff. Even if your data-generating process is AR(7), fitting a misspecified AR(5) model may yield better forecasts, in the sense of lower mean squared error. Fitting high AR orders requires large amounts of data.

Essentially the same points apply to moving average (MA) terms.

The half reason I can think of is that it is hard to think of real-life moving average processes at all, apart from misspecification.

  • $\begingroup$ This is only true if the AR roots are away from unity. In financial contexts that's barely the case. $\endgroup$ – Math-fun Jul 14 '16 at 9:02
  • $\begingroup$ Thank you for accepting my answer, but you may want to think about un-accepting it and looking whether better answers come around. Questions with accepted answers get less attention. @Math-fun, for one, does not seem to agree with my arguments. $\endgroup$ – S. Kolassa - Reinstate Monica Jul 14 '16 at 9:23
  • $\begingroup$ Does the argument that AR(1) can be written as MA(\infty) and vice versa, has any relation to this as well? $\endgroup$ – Waqas Jul 17 '16 at 10:29
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    $\begingroup$ @Marko: yes in that case, too. With the same argument. That (seasonal or non-seasonal) ARIMA orders will usually be $\leq 5$ is accepted wisdom among forecasters, although I couldn't find a clear source for this in FPP2. $\endgroup$ – S. Kolassa - Reinstate Monica Nov 15 '17 at 7:38
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    $\begingroup$ @Marko: this might be helpful: stats.stackexchange.com/a/285099/1352 $\endgroup$ – S. Kolassa - Reinstate Monica Nov 15 '17 at 12:05

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