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Derivation of the Partial Autocorrelation Function (PACF), specifically \( \phi_{22} \), for an AR(2) Model

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The Formal Theorem

Let \ Y_t \ be a stationary time series. The Partial Autocorrelation Function (PACF) at lag \ k \, denoted \ \\phi_{kk} \, is the correlation between \ Y_t \ and \ Y_{t-k} \ after removing the linear dependence of \ Y_t \ on \ Y_{t-1}, \\dots, Y_{t-k+1} \. For a stationary AR(2) process, \ Y_t = \\phi_1 Y_{t-1} + \\phi_2 Y_{t-2} + \\epsilon_t \, where \ \\epsilon_t \ is white noise, the PACF at lag 2, \ \\phi_{22} \, is precisely the coefficient \ \\phi_2 \ from the underlying AR(2) model. This coefficient is derived from the Yule-Walker equations and expressed in terms of the autocorrelation function (ACF) at lags 1 and 2, \ \\rho_1 \ and \ \\rho_2 \, as:
phi22=fracrho2−rho121−rho12\begin{aligned} \\phi_{22} = \\frac{\\rho_2 - \\rho_1^2}{1 - \\rho_1^2} \end{aligned}

Analytical Intuition.

Imagine a complex dance routine where each dancer's move influences the next. The Autocorrelation Function (ACF) is like watching dancer \ Y_t \ and dancer \ Y_{t-k} \ and seeing how much their moves are correlated, considering ALL the dancers in between. It’s a direct, unfiltered connection. Now, imagine a master choreographer wants to understand the *direct* influence of dancer \ Y_{t-k} \ on dancer \ Y_t \, *isolating* it from the chain reactions set off by dancers \ Y_{t-1}, \\dots, Y_{t-k+1} \. This isolation is the PACF. For an AR(2) model, \ Y_t \ is primarily influenced by \ Y_{t-1} \ and \ Y_{t-2} \. So, \ \\phi_{22} \ quantifies the *unique*, direct impact of \ Y_{t-2} \ on \ Y_t \, after we've meticulously accounted for the path \ Y_t \ takes through \ Y_{t-1} \. It's like removing the middleman to see the raw connection.
CAUTION

Institutional Warning.

Students often confuse \ \\phi_{kk} \ with \ \\rho_k \ or the \ k^{th} \ coefficient of an AR(p) process where \ p \ is not \ k \. \ \\phi_{kk} \ is *specifically* the last coefficient of an AR(k) process fitted to the data, not just any \ \\phi_k \.

Academic Inquiries.

01

Why is \ \\phi_{22} \ derived from Yule-Walker equations?

The Yule-Walker equations relate the autocorrelations \ \\rho_k \ of a stationary AR(p) process to its coefficients \ \\phi_1, \\dots, \\phi_p \. The PACF \ \\phi_{kk} \ is defined as the last coefficient \ \\phi_k \ of an AR(k) model. Thus, by solving the Yule-Walker equations for an AR(k) process, we can find \ \\phi_{kk} \. For an AR(2) model, \ \\phi_{22} \ is simply \ \\phi_2 \ in the AR(2) Yule-Walker system.

02

What is the key difference between ACF and PACF?

ACF measures the *total* correlation between \ Y_t \ and \ Y_{t-k} \, including indirect effects through intermediate lags. PACF measures the *direct* correlation between \ Y_t \ and \ Y_{t-k} \ *after* removing the linear dependence due to \ Y_{t-1}, \\dots, Y_{t-k+1} \. It isolates the unique contribution of \ Y_{t-k} \.

03

How does \ \\phi_{22} \ help in identifying an AR(2) model?

For an AR(p) process, the PACF theoretically cuts off at lag \ p \. This means \ \\phi_{pp} \ will be non-zero, but \ \\phi_{kk} \ for \ k > p \ will be zero. If we observe a significant \ \\phi_{22} \ but subsequent PACF values like \ \\phi_{33}, \\phi_{44} \ are close to zero, it strongly suggests an AR(2) model is appropriate.

04

Can \ \\phi_{22} \ be negative?

Yes, \ \\phi_{22} \ can be negative. The range of \ \\phi_{kk} \ is between -1 and 1, just like any correlation coefficient. A negative value implies an inverse relationship between \ Y_t \ and \ Y_{t-2} \ after accounting for \ Y_{t-1} \.

Standardized References.

  • Definitive Institutional SourceBox, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time Series Analysis: Forecasting and Control. Wiley.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Derivation of the Partial Autocorrelation Function (PACF), specifically \( \phi_{22} \), for an AR(2) Model: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-partial-autocorrelation-function--pacf---specifically------for-an-ar-2--model

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