Derivation of the Variance of the l-step Ahead Forecast Error for a General ARMA(p,q) Process

Q: What are $ \psi_i $ in the variance formula?

$ \psi_i $ are the coefficients of the infinite MA representation of the ARMA process, i.e., $ X_t = \sum_{i=0}^{\infty} \psi_i Z_{t-i} $, where $ \psi_0 = 1 $. These coefficients dictate how past innovations affect the current value of the process.

Q: Is the forecast error variance the same as the process variance?

No. The process variance $ \gamma_0 = Var(X_t) $ is the variance of the current value of the process. The forecast error variance $ Var(\psi_l) $ is the variance of the difference between the actual future value and its forecast, and it generally increases with the forecast horizon $ l $.

Q: What is the significance of $ \sigma_Z^2 $ in the formula?

$ \sigma_Z^2 $ is the variance of the white noise innovation term $ Z_t $. It represents the fundamental 'shock' or randomness in the system. The forecast error variance is directly proportional to this fundamental uncertainty.

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The Formal Theorem

For a stationary ARMA(p,q) process defined by

\phi(B)X_t = \theta(B)Z_t

, where

\phi(B) = 1 - \sum_{i=1}^p \phi_i L^i

and

\theta(B) = 1 + \sum_{j=1}^q \theta_j L^j

, and

\{Z_t\}

is a white noise process with

E[Z_t] = 0

and

Var(Z_t) = \sigma_Z^2

, the variance of the

l

-step ahead forecast error, denoted

\psi_l = X_{t+l} - \hat{X}_{t+l|t}

, is given by:

Var(\psi_l) = \sigma_Z^2 \sum_{i=0}^{l-1} \psi_i^2

Analytical Intuition.

Imagine you're a detective trying to predict the next crime wave

X_{t+l}

based on past evidence

X_t, X_{t-1}, \dots

. The 'forecast error'

\psi_l

is how wrong your prediction turns out to be. The ARMA model breaks down this prediction uncertainty into components: the intrinsic randomness of the 'noise'

Z_t

and how this noise propagates through the system over time, influenced by the autoregressive (AR) and moving average (MA) coefficients. As we forecast further into the future (increasing

l

), more 'noise' has a chance to influence the outcome, and our uncertainty, the variance of this error, grows. The formula elegantly sums up the squared contributions of past and present innovations

Z_t

to the future error, reflecting this accumulation of uncertainty.

CAUTION

Institutional Warning.

Students often confuse the forecast error $\psi_l$ with the process itself $X_t$ or misinterpret the summation as being over $p$ or $q$ directly, rather than the innovations $Z_t$ .

Academic Inquiries.

What are $\psi_i$ in the variance formula?

$\psi_i$ are the coefficients of the infinite MA representation of the ARMA process, i.e., $X_t = \sum_{i=0}^{\infty} \psi_i Z_{t-i}$ , where $\psi_0 = 1$ . These coefficients dictate how past innovations affect the current value of the process.

How does the forecast error variance relate to the forecast horizon $l$ ?

The forecast error variance is a non-decreasing function of the forecast horizon $l$ . As $l$ increases, more future white noise terms $Z_t$ contribute to the error, leading to increased uncertainty.

Is the forecast error variance the same as the process variance?

No. The process variance $\gamma_0 = Var(X_t)$ is the variance of the current value of the process. The forecast error variance $Var(\psi_l)$ is the variance of the difference between the actual future value and its forecast, and it generally increases with the forecast horizon $l$ .

What is the significance of $\sigma_Z^2$ in the formula?

$\sigma_Z^2$ is the variance of the white noise innovation term $Z_t$ . It represents the fundamental 'shock' or randomness in the system. The forecast error variance is directly proportional to this fundamental uncertainty.

Standardized References.

Definitive Institutional SourceBrockwell, P. J., & Davis, R. A., 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 Variance of the l-step Ahead Forecast Error for a General ARMA(p,q) Process: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-variance-of-the-l-step-ahead-forecast-error-for-a-general-arma-p-q--process

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The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

What are ψi \psi_i ψi​ in the variance formula?

How does the forecast error variance relate to the forecast horizon l l l?

Is the forecast error variance the same as the process variance?

What is the significance of σZ2 \sigma_Z^2 σZ2​ in the formula?

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.

What are $\psi_i$ in the variance formula?

How does the forecast error variance relate to the forecast horizon $l$ ?

What is the significance of $\sigma_Z^2$ in the formula?