Partial Autocorrelation Function (PACF)

The Autocorrelation Function (ACF) offers a general overview of how a time series observation correlates with its past values. However, ACF does not differentiate between direct and indirect relationships. For example, the correlation between $y_t$ and $y_{t-2}$ measured by ACF includes the influence of $y_{t-1}$, because $y_t$ often correlates with $y_{t-1}$, which in turn correlates with $y_{t-2}$. To isolate the direct relationship between $y_t$ and a specific lag $y_{t-k}$, removing the influence of intermediate observations such as $y_{t-1}, y_{t-2}, \dots, y_{t-k+1}$, a different tool is needed. The Partial Autocorrelation Function (PACF) measures this direct relationship.

What is Partial Autocorrelation?

Think of partial autocorrelation at lag $k$ as the correlation between $y_t$ and $y_{t-k}$ that is not explained by their mutual correlations with the time series values at the intervening lags ($t-1, t-2, \dots, t-k+1$).

You can imagine calculating it by:

1. Predicting $y_t$ using the intermediate lags $y_{t-1}, \dots, y_{t-k+1}$. Let the error of this prediction be $e_t$.
2. Predicting $y_{t-k}$ using the same intermediate lags $y_{t-1}, \dots, y_{t-k+1}$. Let the error of this prediction be $e_{t-k}$.
3. The partial autocorrelation at lag $k$ is the correlation between these two prediction errors, $Corr(e_t, e_{t-k})$.

This process effectively removes the linear dependence associated with the intermediate lags, leaving only the direct connection between $y_t$ and $y_{t-k}$.

Difference between ACF and PACF when considering the relationship between $y_t$ and $y_{t-2}$. PACF isolates the direct link after accounting for $y_{t-1}$.

Why Use PACF? Identifying AR Models

The primary utility of the PACF lies in identifying the order ($p$) of an Autoregressive (AR) model. Recall that an AR(p) model expresses $y_t$ as a linear combination of its previous $p$ values plus an error term:

$$y_t = c + \phi_1 y_{t-1} + \phi_2 y_{t-2} + \dots + \phi_p y_{t-p} + \epsilon_t$$

By definition, in a pure AR(p) process, $y_t$ has a direct linear dependency on $y_{t-1}, \dots, y_{t-p}$. However, once you account for these $p$ lags, there should be no direct linear relationship between $y_t$ and further lags like $y_{t-p-1}, y_{t-p-2}$, etc. Any correlation seen at these further lags in the ACF is indirect, flowing through the first $p$ lags.

Therefore, for a stationary AR(p) process, we expect the PACF plot to show:

• Significant partial autocorrelations up to lag $p$.
• A sharp cut-off after lag $p$, with partial autocorrelations for lags $k > p$ being statistically insignificant (close to zero and within the significance bounds).

This distinct pattern contrasts with the ACF of an AR(p) process, which typically shows a more gradual decay towards zero.

Interpreting PACF Plots

Similar to ACF plots, PACF plots display the partial autocorrelation values for different lags on the y-axis against the lag number on the x-axis. They also typically include confidence intervals (often represented by a shaded area, usually at the 95% level).

• Significant Lags: Spikes extending past the confidence interval are considered statistically significant.
• Cut-off: A sharp drop to insignificance after a certain lag $p$ is characteristic of an AR(p) process.
• Decay: Unlike the ACF for AR models, the PACF for MA (Moving Average) models tends to decay gradually.

By examining the PACF plot alongside the ACF plot (as we'll discuss how to generate in the next section), you gain valuable clues about the underlying structure of your time series and can make informed decisions about which model type (AR, MA, or ARMA) might be appropriate and what order parameters to investigate. For identifying AR model orders, the PACF is particularly informative.