Calculation of the First Few $\psi$-weights for a Specific ARMA(1,1) Model

Q: Does the value of $ \sigma_{\epsilon}^2 $ affect the $\psi$-weights?

No, the $\psi$-weights themselves are determined by the AR and MA parameters ($ \phi $ and $ \theta $) and represent the structure of the dependence. The variance $ \sigma_{\epsilon}^2 $ scales the magnitude of these weights when calculating the variance of $ Y_t $ or covariances.

Q: What happens if $ |\phi| \ge 1 $?

If $ |\phi| \ge 1 $, the ARMA(1,1) process is not stationary. In this case, the MA(infinity) representation might not converge, and the $\psi$-weights would not be well-defined in the usual sense, or they would grow infinitely.

Exploring the cinematic intuition of Calculation of the First Few $\psi$-weights for a Specific ARMA(1,1) Model.

Visualizing...

Our institutional research engineers are currently mapping the formal proof for Calculation of the First Few $\psi$-weights for a Specific ARMA(1,1) Model.

Apply for Institutional Early Access →

The Formal Theorem

Consider an ARMA(1,1) process defined by

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

, where

\{ \epsilon_t \}

is a white noise process with

E[\epsilon_t] = 0

and

Var(\epsilon_t) = \sigma_{\epsilon}^2

, and

|\phi| < 1

for stationarity. The

\psi

-weights,

\psi_j

for

j \ge 0

, are defined by the MA(

\infty

) representation

Y_t = \sum_{j=0}^{\infty} \psi_j \epsilon_{t-j}

. The first few

\psi

-weights are given by:

\psi_0 = 1

\psi_1 = \phi + \theta

\psi_j = \phi \psi_{j-1}

for

j \ge 2

Analytical Intuition.

Imagine a complex ripple in a pond, where the current state

Y_t

is influenced by its immediate past

Y_{t-1}

and a 'shock'

\epsilon_t

that also lingers from the previous moment

\epsilon_{t-1}

. The

\psi

-weights are like a chain reaction, quantifying how each past shock

\epsilon_{t-j}

contributes to the current observation

Y_t

\psi_0

is the direct impact of the current shock (always 1, representing the shock itself).

\psi_1

captures the combined effect of the current shock and the 'memory' of the previous shock carried forward by the AR(1) component. For subsequent shocks, the influence fades geometrically, dictated solely by the AR(1) parameter

\phi

, like an echo diminishing over time.

CAUTION

Institutional Warning.

Students often struggle to see why $\psi_j$ for $j \ge 2$ only depends on $\phi$ and not $\theta$ , forgetting that $\theta$ 's influence is fully captured in $\psi_0$ and $\psi_1$ .

Academic Inquiries.

What is the MA(infinity) representation and why is it important?

The MA(infinity) representation expresses any stationary ARMA process as an infinite moving average of past white noise innovations. It's crucial because it allows us to understand the unconditional dependence structure of the process and forms the basis for calculating forecast error variances and impulse response functions.

How is the ARMA(1,1) process related to the MA(infinity) representation?

By repeatedly substituting the AR(1) part of the ARMA(1,1) equation into itself, we can express $Y_t$ solely as a linear combination of current and past error terms $\epsilon$ , effectively transforming it into an infinite moving average process.

Does the value of $\sigma_{\epsilon}^2$ affect the $\psi$ -weights?

No, the $\psi$ -weights themselves are determined by the AR and MA parameters ( $\phi$ and $\theta$ ) and represent the structure of the dependence. The variance $\sigma_{\epsilon}^2$ scales the magnitude of these weights when calculating the variance of $Y_t$ or covariances.

What happens if $|\phi| \ge 1$ ?

If $|\phi| \ge 1$ , the ARMA(1,1) process is not stationary. In this case, the MA(infinity) representation might not converge, and the $\psi$ -weights would not be well-defined in the usual sense, or they would grow infinitely.

Standardized References.

Definitive Institutional SourceBrockwell, P. J., & Davis, R. A. (2016). Introduction to Time Series and Forecasting. Springer.

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). Calculation of the First Few \(\psi\)-weights for a Specific ARMA(1,1) Model: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/calculation-of-the-first-few---psi--weights-for-a-specific-arma-1-1--model

Dominate the Logic.

"Abstract theory is just a movement we haven't seen yet."

Subscribe for Full Proofs Early Access

Visualizing...

The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

What is the MA(infinity) representation and why is it important?

How is the ARMA(1,1) process related to the MA(infinity) representation?

Does the value of σϵ2 \sigma_{\epsilon}^2 σϵ2​ affect the ψ\psiψ-weights?

What happens if ∣ϕ∣≥1 |\phi| \ge 1 ∣ϕ∣≥1?

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.

Does the value of $\sigma_{\epsilon}^2$ affect the $\psi$ -weights?

What happens if $|\phi| \ge 1$ ?