Partial Autocorrelation for AR(p) Process

Key Property 

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. See Partial Autocorrelation.

AR(1) Example

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.

AR(2) Example

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.

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

Alonso, A. M., Garcia-Martos, C. (2012) Time series analysis: autoregressive, MA and ARMA processes
https://www.academia.edu/35659911/Time_Series_Analysis_Autoregressive_MA_and_ARMA_processes

Greene, W. H. (2002) Econometric analysis. 5th Ed. Prentice-Hall
https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/reference/referencespapers.aspx?referenceid=1243286

Gujarati, D. & Porter, D. (2009) Basic econometrics. 5th Ed. McGraw Hill
http://www.uop.edu.pk/ocontents/gujarati_book.pdf

Hamilton, J. D. (1994) Time series analysis. Princeton University Press
https://press.princeton.edu/books/hardcover/9780691042893/time-series-analysis

Wooldridge, J. M. (2009) Introductory econometrics, a modern approach. 5th Ed. South-Western, Cegage Learning
https://cbpbu.ac.in/userfiles/file/2020/STUDY_MAT/ECO/2.pdf