Interpretation of the partial autocorrelation function for a pure MA process

Question

I have been working with some time-series theory and I noticed something that I can understand "mathematically", but not based on the intuitive explanations of what the partial auto-correlation function (PACF) is supposed to represent: The correlation between points of a given lag with the effects of smaller correlations removed.

For pure autoregressive time series (AR), the autocorrelation function (ACF) shows the expected slow decay while the PACF is truncated at the largest lag involved in the AR process. This all makes sense to me intuitively.

Now...when we get to a pure moving average process (MA), the behaviors are "reversed" with the ACF being truncated and the PACF decaying. However, under the intuitive interpretation of the ACF and PACF, I cannot (inutuively) see why the PACF would show positive autocorrelations beyond the lags involved in the construction of the MA time series?

For example, take an MA(2) process:

$Z_{t+1}=\varepsilon_{t-1} + \varepsilon+{t}\;\; [\varepsilon_i \sim \mathcal{N}(0,1)]$

Now, with this, I can see why the ACF would truncate for lags > 1. Howver, the PACF will slowsly decay, so the partial autocorreleation will be positive even for lags that do not contribute to the same $Z_t$...therefore, what is the PACF really telling us for an MA process? It can't be telling us "the relationship between the lags not accounted for by earlier lags"

My initial thought is that since an AR and MA process are dual, the actual results of the ACF and PACF calculations are also dual, such that applying the PACF calculations to an MA process is equivalent to applying the ACF calculations to the dual AR process. Therefore, the names are really only intuitively correct for AR processes. Could this be it?

Answer 1

The main reason for the "reversal" you are looking at when you deal with AR and MA processes, is that these processes generally have the property that they are invertible to the form of the other process (so long as the coefficients in the models are within the unit circle). So a finite AR process can be represented as an infinite MA process, and a finite MA process can be represented as an infinite AR process. For a general MA(q) process you have:

$$Z_t = \Bigg( 1 - \sum_{i=1}^q \theta_i B^i \Bigg) \epsilon_t = \prod_{i=1}^q (1 - \tau_i B) \epsilon_t,$$

where $B$ is the backshift operator. If $\max|\tau_i| < 1$ (so that all the coefficients are inside the unit circle) then the process is invertible and we have:

$$\epsilon_t = \prod_{i=1}^q (1 - \tau_i B)^{-1} Z_t = \prod_{i=1}^q \Bigg( \sum_{k=0}^\infty \tau_i^k B^k \Bigg) Z_t.$$

Re-arranging this expression gives the AR($\infty$) process:

$$Z_t = \Bigg[ \prod_{i=1}^q \Bigg( \sum_{k=0}^\infty \tau_i^k B^k \Bigg) -1 \Bigg] Z_t + \epsilon_t.$$

Now, the PACF is giving you the conditional correlation for a given lag, conditional on knowledge of the values of the intervening times. For an AR process, this measures the autocorrelations in the process. Hence, for an invertible MA process, the PACF will measure the autocorrelations in the AR($\infty$) process that corresponds to that process. The measured PACF values will decay gradually because the AR process being measured is infinite.