How do I interpret the phi and theta in the $SARIMA$ model? I know that they are both parameters of the model, but I am having a hard time trying to interpret them.

For example, the phi in the $AR(1)$ model equals $0.3467$, can I interpret it as that for every 1 unit increase in $X_{t-1}$, $X_t$ would increase by $0.3467$?

If Yes, how should I interpret the MA model coefficients as the pure $MA$ model depends only on the errors of the previous forecast?

If that's not the case, how should I interpret them?

[1]: https://i.sstatic.net/GPYGK.png

Thanks so much.


2 Answers 2


"For example, the phi in the AR(1) model equals 0.3467, can I interpret it as that for every 1 unit increase in Xt-1, Xt would increase by 0.3467?"

only if there is no differencing , no ma structure and no power transform .

You need to 1) either do a lot of algebra and re-present the model as a pure AR MODEL OR 2) use software that delivers that feature to you.

  • 1
    $\begingroup$ So how should I interpret the phi and theta then? $\endgroup$
    – dddd_y
    Mar 6, 2020 at 1:36
  • $\begingroup$ All ar models can be expressed as ma models. All ma models can be expressed as ar models. Sometimes it is parsimonious to incorporate both thus minimizing the # of coefficients in the model. All models can be expressed as a pure ar model or a pure ma model with enough coefficients. I would not be concerned about interpreting the ar and the ma coefficients BUT if I were you I would focus on the pure ar representation which details the form of the weighted average. See stats.stackexchange.com/… for more on this. $\endgroup$
    – IrishStat
    Mar 6, 2020 at 8:08
  • $\begingroup$ you should look up the details on sarima by loading the relevant package and then doing ?sarima. or, if ts in base R, you don't even to load a package. Note that, with the model in the form it's in, it's not valid to interpret the coefficient that way. As IrishStat mentioned, to make an int\erpretation, it needs to be in pure AR form or pure MA form. That may be what the psi weights are doing but I don't say that with total confidence. $\endgroup$
    – mlofton
    Mar 18, 2021 at 0:35

ARMA model coefficient Interpretation

Regarding the AR part, in my view, them have a purely correlational interpretation only. Therefore them are transformation of total or partial linear correlation coefficients and maintain them interpretation too. In the case of $MA$ part a non observable series is involved (errors) and I'm not sure if the same interpretation hold. However remember that any stationary and invertible $ARMA$ have a pure $AR$ representation too.

For example, the phi in the $AR(1)$ model equals $0.3467$, can I interpret it as that for every 1 unit increase in $X_{t-1}$, $X_t$ would increase by $0.3467$?

This statement is usually affirmed for any regression. It can be literally correct but can be misleading too. The word "increase" is ambiguous, what it mean? It is an observed fact or imply some intervention?

If you intend an observed fact the interpretation is correct and it is purely correlational. If you have in mind an intervention the interpretation is surely wrong because $AR$ model are "free of theory" model, it avoid causal reasoning. This my question is related: Structural equation and causal model in economics


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