Derivation of the Explicit Formula for the l-step Ahead Forecast $ \hat{Z}_t(l) $ for a Stationary AR(1) Model

Q: Why do we set $ E[\epsilon_{t+l} | \mathcal{F}_t] = 0 $ for $ l > 0 $?

The white noise assumption means $ \epsilon_t $ are independent and identically distributed with zero mean. Since future errors $ \epsilon_{t+l} $ (for $ l > 0 $) are independent of the information available at time $ t $ (denoted $ \mathcal{F}_t $), their conditional expectation is simply their unconditional mean, which is 0.

Q: What happens to the formula if the AR(1) model is mean-centered, i.e., $ Z_t - \mu = \phi (Z_{t-1} - \mu) + \epsilon_t $?

If the model is mean-centered, $ Z_t = \mu + \phi (Z_{t-1} - \mu) + \epsilon_t $. Comparing this to $ Z_t = c + \phi Z_{t-1} + \epsilon_t $, we have $ c = \mu (1 - \phi) $. Substituting this into the explicit formula gives $ \hat{Z}_t(l) = \mu (1 - \phi) \frac{1 - \phi^l}{1 - \phi} + \phi^l Z_t = \mu (1 - \phi^l) + \phi^l Z_t = \mu + \phi^l (Z_t - \mu) $. This is the standard form for a mean-centered AR(1) forecast.

Q: Why does the forecast converge to $ \frac{c}{1 - \phi} $ as $ l \to \infty $?

For a stationary AR(1) process, $ |\phi| < 1 $. As the forecast horizon $ l $ tends to infinity, $ \phi^l $ approaches zero. Thus, the term $ \phi^l Z_t $ vanishes, and $ \frac{1 - \phi^l}{1 - \phi} $ approaches $ \frac{1}{1 - \phi} $. The forecast therefore converges to $ \frac{c}{1 - \phi} $, which is the unconditional mean $ E[Z_t] $ of the stationary process. This reflects that far-ahead forecasts rely less on the current observation and more on the process's long-run average.

Exploring the cinematic intuition of Derivation of the Explicit Formula for the l-step Ahead Forecast $ \hat{Z}_t(l) $ for a Stationary AR(1) Model.

Visualizing...

Our institutional research engineers are currently mapping the formal proof for Derivation of the Explicit Formula for the l-step Ahead Forecast $ \hat{Z}_t(l) $ for a Stationary AR(1) Model.

Apply for Institutional Early Access →

The Formal Theorem

Given a stationary AR(1) process defined by

Z_t = c + \phi Z_{t-1} + \epsilon_t

, where

c

is a constant,

\phi

is the autoregressive coefficient such that

|\phi| < 1

, and

\epsilon_t

is a white noise process with

E[\epsilon_t] = 0

and

Var(\epsilon_t) = \sigma^2

. The

l

-step ahead forecast of

Z_{t+l}

at time

t

, denoted

\hat{Z}_t(l)

, is given by the explicit formula:

\hat{Z}_t(l) = c \frac{1 - \phi^l}{1 - \phi} + \phi^l Z_t

where

Z_t

is the last observed value of the series at time

t

, and

l

is the forecast horizon (

l \ge 1

Analytical Intuition.

Imagine yourself as a seasoned navigator on a vast, unpredictable ocean (our time series

Z_t

). You're charting a course

l

steps into the future, but your knowledge is strictly limited to what you've observed up to the present moment, time

t

(your information set

\mathcal{F}_t

). The AR(1) model tells you that tomorrow's position

Z_{t+1}

depends primarily on today's position

Z_t

, nudged by a constant drift

c

and a random, unpredictable gust of wind

\epsilon_{t+1}

. When forecasting, we essentially ignore these future 'gusts' because they average out to zero (their expectation is zero). So, your best guess for tomorrow,

\hat{Z}_t(1)

, is simply

c + \phi Z_t

. Now, how about the day after,

\hat{Z}_t(2)

? It depends on

Z_{t+1}

, which we don't know, so we substitute our best guess

\hat{Z}_t(1)

. This creates a chain reaction, a recursive dependency. As you project further and further out (increasing

l

), the initial observed value

Z_t

's influence diminishes exponentially due to the

\phi^l

term (since

|\phi| < 1

). Eventually, the forecast converges to the long-run average,

c/(1-\phi)

, the equilibrium point of your oceanic journey, as the memory of

Z_t

fades into the horizon.

CAUTION

Institutional Warning.

Students often confuse $E[Z_t | \mathcal{F}_t] = Z_t$ with a general forecast. The main pitfall in derivation is incorrect index management during recursive unrolling, leading to errors in the geometric series bounds or the final power of $\phi$ multiplying $Z_t$ . Careful handling of $l$ vs. $l-1$ is key.

Academic Inquiries.

Why do we set $E[\epsilon_{t+l} | \mathcal{F}_t] = 0$ for $l > 0$ ?

The white noise assumption means $\epsilon_t$ are independent and identically distributed with zero mean. Since future errors $\epsilon_{t+l}$ (for $l > 0$ ) are independent of the information available at time $t$ (denoted $\mathcal{F}_t$ ), their conditional expectation is simply their unconditional mean, which is 0.

What happens to the formula if the AR(1) model is mean-centered, i.e., $Z_t - \mu = \phi (Z_{t-1} - \mu) + \epsilon_t$ ?

If the model is mean-centered, $Z_t = \mu + \phi (Z_{t-1} - \mu) + \epsilon_t$ . Comparing this to $Z_t = c + \phi Z_{t-1} + \epsilon_t$ , we have $c = \mu (1 - \phi)$ . Substituting this into the explicit formula gives $\hat{Z}_t(l) = \mu (1 - \phi) \frac{1 - \phi^l}{1 - \phi} + \phi^l Z_t = \mu (1 - \phi^l) + \phi^l Z_t = \mu + \phi^l (Z_t - \mu)$ . This is the standard form for a mean-centered AR(1) forecast.

Why does the forecast converge to $\frac{c}{1 - \phi}$ as $l \to \infty$ ?

For a stationary AR(1) process, $|\phi| < 1$ . As the forecast horizon $l$ tends to infinity, $\phi^l$ approaches zero. Thus, the term $\phi^l Z_t$ vanishes, and $\frac{1 - \phi^l}{1 - \phi}$ approaches $\frac{1}{1 - \phi}$ . The forecast therefore converges to $\frac{c}{1 - \phi}$ , which is the unconditional mean $E[Z_t]$ of the stationary process. This reflects that far-ahead forecasts rely less on the current observation and more on the process's long-run average.

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 Explicit Formula for the l-step Ahead Forecast \( \hat{Z}_t(l) \) for a Stationary AR(1) Model: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-explicit-formula-for-the-l-step-ahead-forecast---hat-z--t-l---for-a-stationary-ar-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.

Why do we set E[ϵt+l∣Ft]=0 E[\epsilon_{t+l} | \mathcal{F}_t] = 0 E[ϵt+l​∣Ft​]=0 for l>0 l > 0 l>0?

What happens to the formula if the AR(1) model is mean-centered, i.e., Zt−μ=ϕ(Zt−1−μ)+ϵt Z_t - \mu = \phi (Z_{t-1} - \mu) + \epsilon_t Zt​−μ=ϕ(Zt−1​−μ)+ϵt​?

Why does the forecast converge to c1−ϕ \frac{c}{1 - \phi} 1−ϕc​ as l→∞ l \to \infty l→∞?

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 do we set $E[\epsilon_{t+l} | \mathcal{F}_t] = 0$ for $l > 0$ ?

What happens to the formula if the AR(1) model is mean-centered, i.e., $Z_t - \mu = \phi (Z_{t-1} - \mu) + \epsilon_t$ ?

Why does the forecast converge to $\frac{c}{1 - \phi}$ as $l \to \infty$ ?