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Derivation of the Autocorrelation Function (ACF) for a First-Order Autoregressive (AR(1)) Model

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The Formal Theorem

For a weakly stationary AR(1) process defined by the stochastic difference equation Xt=ϕXt−1+ϵt X_t = \phi X_{t-1} + \epsilon_t , where ϵt∼WN(0,σ2) \epsilon_t \sim WN(0, \sigma^2) and ∣ϕ∣<1 |\phi| < 1 , the Autocovariance Function (ACVF) is given by γ(k)=σ2ϕk1−ϕ2 \gamma(k) = \frac{\sigma^2 \phi^k}{1 - \phi^2} . Consequently, the Autocorrelation Function (ACF), defined as ρ(k)=γ(k)γ(0) \rho(k) = \frac{\gamma(k)}{\gamma(0)} , results in:
ρ(k)=ϕ∣k∣ \rho(k) = \phi^|k|

Analytical Intuition.

Imagine standing in a vast, echoing cathedral where every word you speak is a new 'innovation' ϵt \epsilon_t . You aren't speaking into a void; your current voice Xt X_t is a layered composite of this new sound and the fading resonance of your previous statement ϕXt−1 \phi X_{t-1} . This is the soul of the AR(1) model. The parameter ϕ \phi acts as the acoustic absorption coefficient of the stone walls. If ϕ \phi is near unity, the cathedral is cavernous, and your past words haunt the present with high fidelity. If ϕ \phi is near zero, the walls are damp, and the past dissolves almost instantly into silence. The Autocorrelation Function (ACF), ρ(k)=ϕk \rho(k) = \phi^k , is the mathematical score of this physical decay. It reveals that the correlation between the present and the past doesn't simply vanish—it erodes geometrically. Each step forward in time multiplies the existing correlation by ϕ \phi , creating a cinematic fade-out that defines the 'memory' of the system. Without the boundary condition ∣ϕ∣<1 |\phi| < 1 , the echoes would amplify into a deafening roar, destroying the stationarity of the soundscape.
CAUTION

Institutional Warning.

Students often struggle with the Yule-Walker recursive step. They might forget that since ϵt \epsilon_t is white noise occurring at time t t , it is inherently uncorrelated with any past observations Xt−k X_{t-k} . This orthogonality is the 'secret key' to solving the covariance equations.

Academic Inquiries.

01

Why does the AR(1) ACF decay geometrically rather than cutting off like an MA model?

The AR(1) model is recursive; the current value is built upon all previous values. This creates an 'infinite' chain of dependency where the impact of a shock ϵt−k \epsilon_{t-k} is scaled by ϕk \phi^k , lingering indefinitely but diminishing in strength.

02

What happens to the ACF if the coefficient ϕ \phi is negative?

If ϕ<0 \phi < 0 , the ACF ρ(k)=ϕk \rho(k) = \phi^k will alternate in sign between positive and negative values. Visually, this represents a process that 'flip-flops' or oscillates around the mean, creating a sawtooth pattern in the correlogram.

03

Is the ACF derivation valid if the process is non-stationary?

No. If ∣ϕ∣≥1 |\phi| \geq 1 , the variance of the process is not constant and grows with time. In such cases, the population autocovariance γ(k) \gamma(k) is not well-defined, and the standard ACF formula ceases to represent a stable statistical relationship.

Standardized References.

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

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Institutional Citation

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NICEFA Visual Mathematics. (2026). Derivation of the Autocorrelation Function (ACF) for a First-Order Autoregressive (AR(1)) Model: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/applied-statistics/derivation-of-the-autocorrelation-function--acf--for-a-first-order-autoregressive--ar-1---model

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