Derivation of the Partial Autocorrelation Function (PACF) for an AR(p) Process, demonstrating its Cut-off Property

Q: What is the practical significance of the PACF cut-off property?

The cut-off property is crucial for identifying the order $ p $ of an AR(p) model. By inspecting the sample PACF plot, one looks for the lag $ k $ after which the PACF values become statistically insignificant (close to zero). This lag $ k $ indicates the likely order $ p $ of the AR component.

Q: How does the PACF differ from the ACF for an AR(p) process?

For an AR(p) process, the ACF $ \rho_k $ typically decays exponentially or sinusoidally (tails off) as $ k $ increases, meaning all past observations have some indirect influence. In contrast, the PACF $ \phi_{kk} $ *cuts off* abruptly to zero after lag $ p $, indicating that only the most recent $ p $ observations have a *direct* impact on the current value.

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The Formal Theorem

Let

\{X_t\}

be a stationary Autoregressive process of order

p

, denoted AR(p), defined by the equation:

X_t = \sum_{i=1}^p \phi_i X_{t-i} + \epsilon_t

where

\phi_i

are the autoregressive coefficients with

\phi_p \neq 0

, and

\epsilon_t

is a white noise process with

E[\epsilon_t] = 0

and

\text{Var}(\epsilon_t) = \sigma^2

. The Partial Autocorrelation Function (PACF) at lag

k

, denoted

\phi_{kk}

, is the last coefficient of the best linear predictor of

X_t

based on

X_{t-1}, \dots, X_{t-k}

. These coefficients,

\phi_{k1}, \dots, \phi_{kk}

, are derived by solving the Yule-Walker equations for an AR(k) model:

\sum_{j=1}^k \phi_{kj} \rho_{|l-j|} = \rho_l \quad \text{for } l = 1, \dots, k

where

\rho_l = \text{Corr}(X_t, X_{t-l})

is the autocorrelation function (ACF) at lag

l

. For an AR(p) process, the true ACF satisfies the Yule-Walker equations for

l > p

\rho_l = \sum_{i=1}^p \phi_i \rho_{l-i} \quad \text{for } l > p

By substituting the vector

(\phi_1, \dots, \phi_p, 0, \dots, 0)^T

(with

k-p

zeros) as a candidate solution into the Yule-Walker system for an AR(k) process where

k > p

, it can be shown that these coefficients satisfy the system. This implies that for an AR(p) process, the PACF exhibits a sharp cut-off property:

\phi_{kk} = \begin{cases} \neq 0 & \text{for } k \le p \\ 0 & \text{for } k > p \end{cases}

Specifically,

\phi_{pp} = \phi_p

, demonstrating that the direct linear dependence of

X_t

X_{t-k}

, after accounting for intermediate observations, vanishes for lags beyond the true order

p

of the AR process.

Analytical Intuition.

Imagine a grand symphony orchestra. Each instrument,

X_{t-i}

, contributes to the current harmony,

X_t

. In an AR(p) process, the conductor has specifically scored parts for only the most recent

p

sections — say, the last

p

measures of music. The 'partial autocorrelation' at lag

k

\phi_{kk}

, is like asking: 'How much unique influence does the instrument from

k

measures ago,

X_{t-k}

, still have on the current note

X_t

, *after* we've already listened to and accounted for all the instruments playing in the intervening

k-1

measures (

X_{t-1}, \dots, X_{t-k+1}

)?' If our symphony truly only uses the last

p

measures (an AR(p) process), then any instrument from

k > p

measures ago, say

X_{t-(p+1)}

, would have its influence entirely 'filtered out' or 'explained away' by the

p

crucial preceding instruments. Its unique, direct contribution becomes zero, like a silent instrument beyond its assigned score. This vanishing act is the PACF's cinematic cut-off property.

CAUTION

Institutional Warning.

Students often confuse PACF with ACF. While ACF measures total correlation, PACF isolates the *direct* effect, net of intermediate influences. Misinterpreting sample PACF values beyond $p$ as small but significant rather than statistically zero is a common pitfall.

Academic Inquiries.

What is the practical significance of the PACF cut-off property?

The cut-off property is crucial for identifying the order $p$ of an AR(p) model. By inspecting the sample PACF plot, one looks for the lag $k$ after which the PACF values become statistically insignificant (close to zero). This lag $k$ indicates the likely order $p$ of the AR component.

How does the PACF differ from the ACF for an AR(p) process?

For an AR(p) process, the ACF $\rho_k$ typically decays exponentially or sinusoidally (tails off) as $k$ increases, meaning all past observations have some indirect influence. In contrast, the PACF $\phi_{kk}$ *cuts off* abruptly to zero after lag $p$ , indicating that only the most recent $p$ observations have a *direct* impact on the current value.

Is the cut-off property always perfectly observed in real-world data?

In theoretical AR(p) processes, the cut-off is exact. However, with real-world time series data, we work with *sample* PACF values. These will rarely be exactly zero due to sampling variability. We typically look for PACF values that fall within a confidence interval around zero for lags $k > p$ to declare a cut-off, not necessarily exact zeros.

Can PACF also be used for MA(q) processes?

Yes, for an MA(q) process, the PACF theoretically decays exponentially or sinusoidally (tails off), while the ACF exhibits a cut-off at lag $q$ . This complementary behavior of ACF and PACF helps distinguish between AR and MA components when identifying suitable ARIMA models.

Standardized References.

Definitive Institutional SourceBrockwell, Peter J., and Richard A. Davis. Introduction to Time Series and Forecasting.

Intermediate

Proof that Autocovariance Depends Only on Lag for Weakly Stationary Processes

Exploring the cinematic intuition of Proof that Autocovariance Depends Only on Lag for Weakly Stationary Processes.

Foundational

Derivation of the Autocorrelation Function (ACF) for a White Noise Process

Exploring the cinematic intuition of Derivation of the Autocorrelation Function (ACF) for a White Noise Process.

Intermediate

Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1)

Exploring the cinematic intuition of Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1).

Intermediate

Proof of the Invertibility Condition for an MA(1) Process (|θ| < 1)

Exploring the cinematic intuition of Proof of the Invertibility Condition for an MA(1) Process (|θ| < 1).

Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Derivation of the Partial Autocorrelation Function (PACF) for an AR(p) Process, demonstrating its Cut-off Property: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-partial-autocorrelation-function--pacf--for-an-ar-p--process--demonstrating-its-cut-off-property

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Visualizing...

The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

What is the practical significance of the PACF cut-off property?

How does the PACF differ from the ACF for an AR(p) process?

Is the cut-off property always perfectly observed in real-world data?

Can PACF also be used for MA(q) processes?

Standardized References.

Related Proofs Cluster.

Proof that Autocovariance Depends Only on Lag for Weakly Stationary Processes

Derivation of the Autocorrelation Function (ACF) for a White Noise Process

Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1)

Proof of the Invertibility Condition for an MA(1) Process (|θ| < 1)

Institutional Citation

Dominate the Logic.