Derivation of the Autocorrelation Function (ACF) for a Mixed ARMA(1,1) Process

Q: Why is the ACF for $ k \ge 2 $ different from $ k=1 $?

For $ k \ge 2 $, the MA(1) term $ \theta \epsilon_{t-1} $ becomes uncorrelated with $ Y_{t-k} $ and past $ \epsilon $ terms, simplifying the recursive relationship and making the ACF behave like a pure AR(1) process.

Q: How does $ \theta $ affect the ACF?

$ \theta $ influences the ACF primarily at lag 1. It modifies the strength of the relationship between $ Y_t $ and $ Y_{t-1} $ and also affects the variance $ \rho(0) $, reflecting the impact of the immediate past shock.

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The Formal Theorem

For a stationary ARMA(1,1) process

Y_t = \phi Y_{t-1} + \epsilon_t + \theta \epsilon_{t-1}

, where

\epsilon_t \sim WN(0, \sigma^2)

|\phi| < 1

, and

\epsilon_t

and

Y_s

for

s < t

are uncorrelated, the autocorrelation function

\rho(k)

is given by: For

k=0

\rho(0) = \frac{1 - \theta \phi}{1 - \phi^2}

For

k=1

\rho(1) = \frac{(1 - \theta \phi)\phi}{1 - \phi^2}

For

k \ge 2

\rho(k) = \phi \rho(k-1) = \phi^{k-1} \rho(1) = \phi^{k-1} \frac{(1 - \theta \phi)\phi}{1 - \phi^2}

Analytical Intuition.

Imagine a detective investigating a crime scene, trying to understand the influence of past events on the present. The Autocorrelation Function (ACF) for an ARMA(1,1) process is like this detective's meticulous report. It quantifies how the value of a time series at a certain point,

Y_t

, is related to its past values,

Y_{t-k}

. The AR(1) component,

\phi Y_{t-1}

, signifies a direct, decaying memory of the previous observation. The MA(1) component,

\epsilon_t + \theta \epsilon_{t-1}

, introduces an influence of past random shocks. The ACF elegantly combines these effects, showing a primary decay driven by

\phi

but with an initial boost and modification due to

\theta

, reflecting the complex interplay between the autoregressive and moving average influences.

CAUTION

Institutional Warning.

The key difficulty lies in correctly handling the interplay between the AR(1) and MA(1) terms, especially when calculating the autocovariance for lags $k \ge 1$ and ensuring stationarity conditions are met.

Academic Inquiries.

What does it mean for an ARMA(1,1) process to be stationary?

A stationary process has a constant mean, constant variance, and its autocovariance depends only on the time lag, not the specific time point. For an ARMA(1,1), the stationarity condition is $|\phi| < 1$ .

Why is the ACF for $k \ge 2$ different from $k=1$ ?

For $k \ge 2$ , the MA(1) term $\theta \epsilon_{t-1}$ becomes uncorrelated with $Y_{t-k}$ and past $\epsilon$ terms, simplifying the recursive relationship and making the ACF behave like a pure AR(1) process.

How does $\theta$ affect the ACF?

$\theta$ influences the ACF primarily at lag 1. It modifies the strength of the relationship between $Y_t$ and $Y_{t-1}$ and also affects the variance $\rho(0)$ , reflecting the impact of the immediate past shock.

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 Autocorrelation Function (ACF) for a Mixed ARMA(1,1) Process: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-autocorrelation-function--acf--for-a-mixed-arma-1-1--process

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The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

What does it mean for an ARMA(1,1) process to be stationary?

Why is the ACF for k≥2 k \ge 2 k≥2 different from k=1 k=1 k=1?

How does θ \theta θ affect the ACF?

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.

Why is the ACF for $k \ge 2$ different from $k=1$ ?

How does $\theta$ affect the ACF?