Now that you understand what the Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) represent and how to plot them, let's focus on how these plots guide us in selecting appropriate time series models. Specifically, we'll look for patterns that suggest Autoregressive (AR), Moving Average (MA), or combined ARMA characteristics in our stationary time series data. Remember, ACF and PACF interpretation is typically performed after ensuring the time series is stationary, possibly through differencing (as discussed in Chapter 2).

The core idea is that pure AR and MA processes have distinct theoretical signatures on their ACF and PACF plots. By matching the patterns observed in your data's plots to these theoretical signatures, you can hypothesize the order of the model needed.

Reading the Plots: Significance Bounds

Before interpreting patterns, recall that ACF/PACF plots usually include shaded areas representing significance bounds (commonly 95% confidence intervals). Values that fall outside these bounds are considered statistically significantly different from zero. Values inside the bounds are generally considered statistically insignificant (consistent with zero correlation). We focus on the lags where the correlation spikes outside these bounds.

Identifying AR(p) Models

An Autoregressive model of order $p$ , denoted AR(p), suggests the current value of the series depends linearly on its previous $p$ values.

Y_t = c + \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + \dots + \phi_p Y_{t-p} + \epsilon_t

where $\epsilon_t$ is white noise.

The theoretical signatures for an AR(p) process are:

ACF: Tails off gradually. The autocorrelation diminishes exponentially or via a damped sine wave pattern as the lag increases. It does not suddenly drop to zero.
PACF: Cuts off sharply after lag $p$ . The partial autocorrelation is significant for the first $p$ lags and then drops abruptly to statistical insignificance (within the bounds) for subsequent lags.

Therefore, if you observe:

An ACF plot where correlations decay slowly.
A PACF plot with significant spikes up to lag $p$ and then an immediate drop into the significance bounds.

This pattern strongly suggests an AR(p) model is appropriate. The order $p$ is determined by the last significant lag in the PACF plot.

Example PACF plot for an AR(2) process. Note the significant spikes at lags 1 and 2, followed by values within the significance bounds (dashed lines/shaded area). This suggests $p=2$ .

Identifying MA(q) Models

A Moving Average model of order $q$ , denoted MA(q), suggests the current value of the series depends linearly on the current and previous $q$ white noise error terms.

Y_t = c + \epsilon_t + \theta_1 \epsilon_{t-1} + \theta_2 \epsilon_{t-2} + \dots + \theta_q \epsilon_{t-q}

The theoretical signatures for an MA(q) process are essentially the opposite of an AR(p) process:

ACF: Cuts off sharply after lag $q$ . The autocorrelation is significant for the first $q$ lags and then drops abruptly to statistical insignificance.
PACF: Tails off gradually. The partial autocorrelation diminishes as the lag increases.

So, if your plots show:

An ACF plot with significant spikes up to lag $q$ and then an immediate drop into the significance bounds.
A PACF plot where correlations decay slowly.

This pattern strongly suggests an MA(q) model is appropriate. The order $q$ is determined by the last significant lag in the ACF plot.

Example ACF plot for an MA(1) process. A significant spike occurs only at lag 1, followed by insignificant values. This suggests $q=1$ .

Identifying ARMA(p,q) Models

An Autoregressive Moving Average model, ARMA(p,q), combines both AR and MA components.

Y_t = c + \phi_1 Y_{t-1} + \dots + \phi_p Y_{t-p} + \epsilon_t + \theta_1 \epsilon_{t-1} + \dots + \theta_q \epsilon_{t-q}

The theoretical signatures for an ARMA(p,q) process are less distinct:

ACF: Tails off gradually after lag $q$ .
PACF: Tails off gradually after lag $p$ .

If both the ACF and PACF plots show patterns of gradual decay (either exponential or sinusoidal), it suggests that an ARMA(p,q) model might be appropriate.

Identifying the specific orders $p$ and $q$ directly from the plots becomes more challenging compared to pure AR or MA models. The "tailing off" behavior doesn't give precise cut-off points. In practice, if both plots tail off, you might hypothesize low-order models like ARMA(1,1), ARMA(2,1), or ARMA(1,2) and then use model selection criteria (like AIC or BIC, discussed in Chapter 6) and residual analysis (Chapter 4) to choose the best fit.

Example ACF and PACF plots for an ARMA(1,1) process. Both functions exhibit a tailing-off behavior, making precise order identification difficult solely from these plots.

Summary of Interpretation Guidelines

Model	ACF Pattern	PACF Pattern
AR(p)	Tails off gradually	Cuts off after lag $p$
MA(q)	Cuts off after lag $q$	Tails off gradually
ARMA(p,q)	Tails off after lag $q$	Tails off after lag $p$

Practical Considerations

Stationarity is Required: These interpretations apply to stationary time series. A non-stationary series typically shows a very slow decay in the ACF plot that dominates any other pattern. Always ensure stationarity first.
Real Data is Noisy: Theoretical patterns are clean. Real-world data plots might show spikes that are slightly outside the bounds due to random chance, especially at higher lags. Look for clear cut-offs or distinct tailing-off patterns. Don't over-interpret small deviations.
Seasonality: If your data has seasonality (e.g., monthly sales data), you will often see significant spikes in the ACF/PACF at the seasonal lags (e.g., 12, 24, 36). This indicates the need for seasonal models like SARIMA, which we will cover in Chapter 5. The non-seasonal orders ( $p, q$ ) can still sometimes be inferred from the behavior at the initial non-seasonal lags.
Starting Point: Use these plots as a guide to propose candidate model orders ( $p, q$ ). The differencing order, $d$ , comes from the stationarity analysis (Chapter 2). Together, these form the basis for trying ARIMA(p, d, q) models, which we explore next in Chapter 4.

By carefully examining the ACF and PACF plots of your stationary time series, you gain valuable insights into its underlying structure, providing a strong foundation for building effective ARIMA forecasting models.