4.2 Partial autocorrelation function (PACF) and its interpretation

Partial Autocorrelation Function (PACF)

The Partial Autocorrelation Function (PACF) measures the correlation between a time series observation and a lagged version of itself, after removing the effects of all intermediate lags. While the regular ACF shows the total correlation at each lag (which includes indirect effects passed through shorter lags), the PACF isolates the direct relationship at each lag. This distinction makes the PACF essential for identifying how many autoregressive (AR) terms belong in an ARIMA model.

PACF Definition

The core idea: when you calculate the correlation between $y_t$ and $y_{t-k}$, some of that correlation might just be because $y_t$ is correlated with $y_{t-1}$, which is correlated with $y_{t-2}$, and so on down to $y_{t-k}$. The PACF strips away those indirect, "passed-along" correlations.

• The PACF at lag $k$ is the correlation between $y_t$ and $y_{t-k}$ after removing the linear effects of lags $1, 2, \ldots, k-1$
• Think of it like a regression: if you regressed $y_t$ on $y_{t-1}, y_{t-2}, \ldots, y_{t-k}$, the PACF at lag $k$ would be the coefficient on $y_{t-k}$
• This "controlling for intermediate lags" is what makes the PACF useful for pinpointing the order of an AR model

PACF Plot Interpretation

A PACF plot shows the partial autocorrelation value on the y-axis for each lag on the x-axis, along with confidence bands (typically at the 95% level). Here's how to read it:

• Sharp cutoff pattern: For a pure AR($p$) process, the PACF will show significant spikes at lags $1, 2, \ldots, p$ and then drop to approximately zero for all lags beyond $p$. The lag where it cuts off tells you the AR order. For example, if the PACF is significant at lags 1 and 2 but insignificant from lag 3 onward, that points to an AR(2) model.
• Gradual decay pattern: If the PACF doesn't cut off sharply but instead tapers off gradually (exponentially or in a damped sinusoidal pattern), this suggests a moving average (MA) component rather than a pure AR process. You'd then turn to the ACF plot to identify the MA order.
• Significance bands: Spikes that fall within the confidence bands (the shaded region or dashed lines) are treated as not significantly different from zero. Only spikes that extend beyond these bands count as meaningful.

ACF vs. PACF Comparison

Using ACF and PACF plots together is how you identify the structure of an ARIMA model. Each plot reveals different components:

The ACF is better at identifying the MA order ($q$) because it cuts off sharply for pure MA processes. The PACF is better at identifying the AR order ($p$) because it cuts off sharply for pure AR processes. When both plots show gradual decay, you're likely dealing with a mixed ARMA model, and determining the exact orders requires more careful analysis (such as comparing information criteria like AIC or BIC across candidate models).

PACF Behavior for MA Models

The section heading says "moving average models," so here's the key point: for a pure MA process, the PACF does not cut off sharply. Instead, it decays gradually, often showing an exponential or oscillating tail-off pattern. This is the mirror image of how the ACF behaves for pure AR models.

• If you see gradual decay in the PACF combined with a sharp cutoff in the ACF at lag $q$, that suggests an MA($q$) model
• If the PACF decays gradually and the ACF also decays gradually, consider a mixed ARMA model
• In practice, you'd use both plots together: the ACF cutoff identifies the MA order, while the PACF cutoff identifies the AR order

A practical example: suppose the ACF cuts off after lag 1 and the PACF shows damped exponential decay. This pattern suggests an MA(1) model. If you instead saw the PACF cut off after lag 2 while the ACF decayed gradually, that would suggest an AR(2) model.