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I have a very basic question about MA(1) time-series processes, that I haven't found an answer for yet. I am trying to understand the intuition behind the typical exponential decay we see in the PACF of an MA(1) process.

It is clear why the ACF drops off after lag 1 ($Y_t$ and $Y_{t-2}$ are not correlated as the error terms are i.i.d.). However, if the ACF can be interpreted as direct + indirect effect combined and PACF as the direct effect only, it is not clear how at higher lags there can be no joint direct + indirect effect but there is an exponentially declining direct effect?

For AR(1) processes it is clear why the ACF declines exponentially and the PACF drops off after 1 lag.

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It is evident from looking at the Autocorrelation Function (ACF) of an MA(1) process that it stops after the initial lag. This occurs because in an MA(1) process, none of the stages before it affect the output directly; rather, it only depends on the error term of the prior step. There is no correlation between the time series values after the first lag since the error terms are taken to be independent and identically distributed.

The Partial Autocorrelation Function (PACF), on the other hand, measures the correlation between observations at different lags that is not explained by correlations at all lower-order lags. Now, for an MA(1) process, the PACF does not cut off abruptly after lag 1 like the ACF does. Instead, it declines more gradually. This might seem counterintuitive at first, because if there is no correlation after lag 1, one would expect the PACF to show the same.

The reason for this apparent discrepancy is that while ACF considers both direct and indirect effects of correlations, PACF aims to isolate the direct effect at each lag. The PACF at higher lags for an MA(1) process is attempting to account for the direct correlation between observations separated by more than one time period, assuming we could account for the effect at lag 1.

However, because an MA(1) process only directly incorporates one lag of error, the indirect effects that PACF is trying to account for don't actually exist. The appearance of a declining PACF is a consequence of the statistical methods used to estimate the PACF values. These methods effectively impose an artificial structure when trying to isolate the direct effects at higher lags in an MA(1) model, leading to the estimated PACF values that appear to decay exponentially rather than cutting off. This is somewhat of a mathematical artifact rather than a true indication of direct correlations at higher lags.

To further appreciate this, consider the PACF as an attempt to use all available data up to k-1 to explain the connection at lag k. As there is no further information available after lag 1 to explain the correlation in an MA(1) process, interpreting the PACF at higher lags turns into an exercise in extrapolating the observed decay pattern from the available data.

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    $\begingroup$ Many thanks, very helpful! In that respect I presume it is not really instructive to look at a PACF in the context of an MA process? I would like to understand this a bit better: "These methods effectively impose an artificial structure when trying to isolate the direct effects at higher lags in an MA(1) model, leading to the estimated PACF values that appear to decay exponentially rather than cutting off." Are there any relevant sources you could direct me to? $\endgroup$
    – Pilik
    Commented Nov 9, 2023 at 16:30
  • $\begingroup$ I disagree that the significance of the PACF is a mathematical artifact. The significance can be thought in terms of forecasting. Suppose you only know $y_{t-1}$ and you want to forecast the value $y_t$. If $\{y_t\}$ follows an AR(1) process, then you cannot do better: knowing more past values of the series (like $y_{t-2}$) won't improve your forecasting. However, in a MA(1) process, the more past values of the series you know, the better you can forecast $y_t$. In particular, the 2nd lag of the PACF measures the gain in the forecast accuracy when you know $y_{t-2}$ in addition to $y_{t-1}$. $\endgroup$ Commented Nov 9, 2023 at 22:41
  • $\begingroup$ Thank you, seems there is some disagreement about this. "The more past values of the series you know, the better you can forecast 𝑦𝑡" - Would you be able to explain this further? How I see it, those are not directly or indirectly correlated as the error terms of which they consist are iid. $\endgroup$
    – Pilik
    Commented Nov 10, 2023 at 15:39

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