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I am presenting some ACF and PACF plots to colleagues. I can explain how to interpret the plots and how to determine p and q based on what the plots look like, but I cannot find a simple intuitive explanation of what a PACF plot actually means.

I have read the explanation here, but am finding it a tad long winded: https://people.duke.edu/~rnau/411arim3.htm

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An intuitive description of PACF can be "the amount of correlation with each lag that is not accounted for by more recent lags".

Autocorrelation satisfies a property that we could call dampened transitivity. If $x_t$ is correlated with $x_{t-1}$ by some amount $\rho<0$, then $x_{t-1}$ is correlated with $x_{t-2}$ by $\rho$. This implies that $x_t$ is correlated with $x_{t-2}$, although by some amount smaller than $\rho$.

Partial autocorrelation computes the "pure" correlation between $x_t$ and $x_{t-2}$ by removing the "transitive" correlation, that is, the amount of correlation explained by the first lag, and recomputing. For the partial autocorrelation between $x_t$ and $x_{t-3}$, we will remove the correlation with both $x_{t-1}$ and $x_{t-2}$ and recompute, and so on.

You can add some geometric flavour to the explanation. You can picture your time series at each lag as a vector in space. A highly autocorrelated series would look something like this.

enter image description here

The time series with lag 0 could be the vector at the bottom, for instance, the one above the series at lag 1, and the other one is lag 2. The autocorrelation translates to this setting as a large projection of each vector onto each other.

However, what happens if we remove from the original series the projection onto lag 1?

enter image description here

The projection of the remaining length of series 0 onto series 2 is very small. This corresponds to the PACF at lag 2.

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  • $\begingroup$ (+1) Could you explain a bit more how to interpret those figures? I'm unused to thinking in those terms and it seems useful, but for one thing, I don't know what the axes are supposed to represent. $\endgroup$ – mkt - Reinstate Monica Apr 13 '18 at 9:05
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    $\begingroup$ If you think of the time series as a random variable, you can think of an observation of the time series as a vector in high-dimensional space, and each lagged series will be a different vector, say $v_0, \dots, v_k$ up to lag $k$. The correlation between a pair of these is then equivalent to the length of the projection of one onto the other, normalized by their respective lengths, simply because of how it is computed. The PAC at lag $k$ is the equivalent to 1) removing from $v_0$ the projection onto the span of $v_1, \dots, v_{k-1}$ 2) measuring the projection of the residual onto $v_k$. $\endgroup$ – broncoAbierto Apr 13 '18 at 9:15
  • $\begingroup$ Thanks so much, this was an excellent explanation! :-) $\endgroup$ – JassiL Nov 15 '18 at 19:06

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