How are the ACF and PACF patterns characterized in an AR model?

In an Autoregressive (AR) model, the patterns of the Autocorrelation Function (ACF) and the Partial Autocorrelation Function (PACF) are quite specific and can be used to identify the order of the model.

ACF and PACF Patterns in AR Models:

1. ACF (Autocorrelation Function):
   • In an AR model of order ppp (denoted as AR(p)), the ACF typically decays exponentially or shows a slow decay as the lag increases.
   • In particular, for an AR model, the ACF tends to decrease in magnitude as lag increases but does not necessarily cut off at any specific lag.
   • There is no clear, abrupt cutoff like you would see in an MA (Moving Average) model.
2. PACF (Partial Autocorrelation Function):
   • The PACF for an AR model is more informative in terms of identifying the order of the model.
   • For an AR(p) model, the PACF shows a sharp cutoff after lag ppp. This means that for lags greater than ppp, the partial correlations tend to be very close to zero (or negligible).
   • For example, in an AR(1) model, the PACF would show a significant correlation at lag 1, and the partial correlations for lags greater than 1 would be close to zero.
   • For an AR(2) model, you would see significant partial correlations at lags 1 and 2, and the PACF would drop off sharply after lag 2.

To Summarize:

• ACF: Exhibits a slow, exponential decay with no sharp cutoffs for AR models.
• PACF: Exhibits a sharp cutoff after the order of the AR model (lag ppp), with significant partial autocorrelations up to lag ppp.

These patterns are useful in model selection when identifying the appropriate order for an AR model.