For MA model in Arima, why the order references acf, but not pacf? The emphasize is why not PACF.

In https://towardsdatascience.com/significance-of-acf-and-pacf-plots-in-time-series-analysis-2fa11a5d10a8

It mentions

Why would that be? Since our series is linear combination of the residuals and none of time series own lag can directly explain its present (since its not an AR), which is the essence of PACF plot as it subtracts variations already explained by earlier lags, its kind of PACF losing its power here

I don't still get why PACF cannot be used here, anyone can give a better intuition why it should be ACF but not PACF for giving the order for MA model?


1 Answer 1


They are saying that an AR is something like $y_{t}=y_{t-1}b+e_{t}$ so that each $y_{t-1}$ has an influence on $y_{t}$. Therefore, it make sense to look at the corr between $y_{t-2}$ and $y_{t}$ which is not explained by $y_{t-1}$. In a MA instead you never have $y_{t-1}$ in the independent variables (right hand side). In other words, you can think also about the following: one of the most likely use of the PACF is that it helps you identify the order of a model (like the q in an AR(q)). But for a MA(q) the ACF goes to 0 after q lags, so you immediately see that:

  • the process is not autoregressive otherwise you would have exponential decay,

  • and you immediately get an intuition of the order.

So why drawing a PACF?

To answer your comment: ACF instead captures MA because MA(1) for example is $y_{t}=e_{t-1}b+e_{t}$ so $cov(y_{t}, y_{t-1})= cov(e_{t-1}, be_{t-1})=bVar(e_{t-1})$ so it will be non-zero up to lag 1, for lag 2 then ACF decays to 0 as you have no common white noise terms.

  • $\begingroup$ Thanks, now I see why PACF does not captures MA (as we don't have $y_{t-1}$ in the right hand side variable - as you mentioned), but why ACF captures MA? $\endgroup$
    – william007
    Aug 19, 2019 at 14:14
  • $\begingroup$ @william007 edited the question to answer your last comment $\endgroup$
    – Fr1
    Aug 19, 2019 at 14:20
  • $\begingroup$ Thanks, it is getting clearer now, but $cov(y_{t}, y_{t-1})= cov(e_{t-1}, be_{t-1})=bVar(e_{t-1})$ this formula I can't see immediately how to establish it, is there any reference for it? $\endgroup$
    – william007
    Aug 19, 2019 at 14:25
  • $\begingroup$ Try to search “ACF for MA process” and you should find plenty of things including the full explantation of the latter $\endgroup$
    – Fr1
    Aug 19, 2019 at 14:29

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