Partial Autocorrelation for AR(p)

Property 1: For an AR(p) process y_i = φ₀ + φ₁y_i-1  + … + φ_py_p-1, PACF(k) = φ_k

Thus, for k > p it follows that PACF(k) = 0

Example 1: Chart PACF for the data in Example 1 of Autoregressive Process Basic Concepts.

Using the PACF function and Property 1, we get the result shown in Figure 1.

Figure 1 – Graph of PACF for AR(1) process

Observation: We see from Figure 1 that the PACF values for lags > 1 are close to zero, as is expected, although there is some random fluctuation from zero.

Example 2: Repeat Example 1 for the AR(2) process

where  and calculate ACF and PACF.

From Example 2 of , we know that this process is stationary.

Under construction