Partial Autocorrelation for AR(p) Process

Property 1: For an AR(p) process y_i = φ₀ + φ₁ y_i-1 +…+ φ_p y_i-p + ε_i, PACF(k) = φ_k

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

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

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 ε_i ∼ N(0,1), and calculate ACF and PACF.

From Example 2 of Characteristic Equation of AR(p) Process, we know that this process is stationary.

Figure 2 – Simulated AR(2) process

This time we place the formula =5+0.4*0-0.1*0+B4 in cell C4, =5+0.4*C4-0.1*0+B5 in cell C5 and =5+0.4*C5-0.1*C4+B6 in cell C6, highlight the range C6:C103 and press Ctrl-D.

The ACF and PACF are shown in Figure 3.

Figure 3 – ACF and PACF for AR(2) process

As you can see, there isn’t a perfect fit between the theoretical and actual ACF and PACF values.