Derivation of the Variance for a Stationary AR(1) Process

Q: Why is the stationarity condition $ |\phi| < 1 $ crucial for this derivation?

Without $ |\phi| < 1 $, the AR(1) process is not stationary. This means its statistical properties, including variance, would change over time. The fundamental step $ Var(X_t) = Var(X_{t-1}) $ relies directly on stationarity. If $ |\phi| \ge 1 $, the variance would either be infinite or non-constant, and the derived formula would be invalid.

Q: How does the constant term $ c $ disappear from the variance formula?

The variance of a random variable is defined as $ Var(Y) = E[(Y - E[Y])^2] $. When we add a constant to a random variable, it shifts its mean but does not affect its spread or variability. Mathematically, $ Var(Y + c) = Var(Y) $. Thus, in $ X_t = c + \phi X_{t-1} + \epsilon_t $, the constant $ c $ does not contribute to $ Var(X_t) $.

Q: What is the implication of a large $ |\phi| $ (close to 1) for $ Var(X_t) $?

As $ |\phi| $ approaches 1 (from below), the denominator $ 1 - \phi^2 $ approaches 0. This causes $ Var(X_t) $ to become very large, tending towards infinity as $ \phi \to \pm 1 $. A large $ |\phi| $ indicates strong persistence and 'memory' in the process, meaning past shocks have a long-lasting impact, leading to amplified fluctuations and high volatility.

Q: Why is the assumption $ Cov(X_{t-1}, \epsilon_t) = 0 $ valid?

This is a standard assumption in AR models. $ \epsilon_t $ represents a white noise process, meaning it is uncorrelated with its own past values and, crucially, with past values of the process $ X_t $. $ \epsilon_t $ captures the 'new' unpredictable information at time $ t $ and thus should not be correlated with the state of the process from the previous time step, $ X_{t-1} $.

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The Formal Theorem

Consider a stationary Autoregressive process of order 1 (AR(1)) defined by

X_t = c + \phi X_{t-1} + \epsilon_t

, where

X_t

is the process at time

t

c

is a constant,

\phi

is the autoregressive parameter, and

\epsilon_t

is a white noise error term with

E[\epsilon_t] = 0

and

Var(\epsilon_t) = \sigma^2_\epsilon

. For the process to be strictly stationary, the condition

|\phi| < 1

must hold. Under these conditions, assuming

Cov(X_{t-1}, \epsilon_t) = 0

, the variance of the process

X_t

is given by:

Var(X_t) = \frac{\sigma^2_\epsilon}{1 - \phi^2}

Analytical Intuition.

Imagine a vibrant, bustling city, where each day's energy level

X_t

is a direct consequence of yesterday's

X_{t-1}

, modulated by a societal rhythm

\phi

, plus a sudden, unpredictable spark

\epsilon_t

– perhaps a viral trend or a new technological breakthrough. For the city to remain stable and not descend into chaos or stagnation, the societal rhythm

\phi

must be controlled (i.e.,

|\phi| < 1

). The city's overall 'flicker' or variability, its variance

Var(X_t)

, isn't just due to the daily sparks

\epsilon_t

. Instead, it's a profound interplay. If

\phi

is near zero, the city quickly forgets its past, and its daily variability

Var(X_t)

largely mirrors the raw variability of the sparks

\sigma^2_\epsilon

. But as

\phi

approaches 1, yesterday's energy profoundly echoes into today, amplifying even small sparks into wide, persistent fluctuations, causing the city's overall variability to surge dramatically as the denominator

1 - \phi^2

shrinks.

CAUTION

Institutional Warning.

Students frequently overlook the critical stationarity condition $|\phi| < 1$ , which permits the assumption $Var(X_t) = Var(X_{t-1})$ . They might also forget that the covariance $Cov(X_{t-1}, \epsilon_t) = 0$ is a crucial assumption, or that the constant $c$ vanishes during variance calculation.

Academic Inquiries.

Why is the stationarity condition $|\phi| < 1$ crucial for this derivation?

Without $|\phi| < 1$ , the AR(1) process is not stationary. This means its statistical properties, including variance, would change over time. The fundamental step $Var(X_t) = Var(X_{t-1})$ relies directly on stationarity. If $|\phi| \ge 1$ , the variance would either be infinite or non-constant, and the derived formula would be invalid.

How does the constant term $c$ disappear from the variance formula?

The variance of a random variable is defined as $Var(Y) = E[(Y - E[Y])^2]$ . When we add a constant to a random variable, it shifts its mean but does not affect its spread or variability. Mathematically, $Var(Y + c) = Var(Y)$ . Thus, in $X_t = c + \phi X_{t-1} + \epsilon_t$ , the constant $c$ does not contribute to $Var(X_t)$ .

What is the implication of a large $|\phi|$ (close to 1) for $Var(X_t)$ ?

As $|\phi|$ approaches 1 (from below), the denominator $1 - \phi^2$ approaches 0. This causes $Var(X_t)$ to become very large, tending towards infinity as $\phi \to \pm 1$ . A large $|\phi|$ indicates strong persistence and 'memory' in the process, meaning past shocks have a long-lasting impact, leading to amplified fluctuations and high volatility.

Why is the assumption $Cov(X_{t-1}, \epsilon_t) = 0$ valid?

This is a standard assumption in AR models. $\epsilon_t$ represents a white noise process, meaning it is uncorrelated with its own past values and, crucially, with past values of the process $X_t$ . $\epsilon_t$ captures the 'new' unpredictable information at time $t$ and thus should not be correlated with the state of the process from the previous time step, $X_{t-1}$ .

Standardized References.

Definitive Institutional SourceBox, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. Time Series Analysis: Forecasting and Control. 5th Edition.

Intermediate

Proof that Autocovariance Depends Only on Lag for Weakly Stationary Processes

Explore the rigorous proof that autocovariance for weakly stationary processes depends only on lag, understanding its deep implications for time series analysis.

Foundational

Derivation of the Autocorrelation Function (ACF) for a White Noise Process

Derive the Autocorrelation Function (ACF) for white noise. Understand its theoretical properties and intuitive meaning in Time Series Analysis.

Intermediate

Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1)

Unravel the stationarity condition for AR(1) processes. Rigorous proof, cinematic intuition, and essential insights for time series analysis.

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 for a Stationary AR(1) Process: Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/time-series-analysis/derivation-of-the-variance-for-a-stationary-ar-1--process

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The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

Why is the stationarity condition ∣ϕ∣<1 |\phi| < 1 ∣ϕ∣<1 crucial for this derivation?

How does the constant term c c c disappear from the variance formula?

What is the implication of a large ∣ϕ∣ |\phi| ∣ϕ∣ (close to 1) for Var(Xt) Var(X_t) Var(Xt​)?

Why is the assumption Cov(Xt−1,ϵt)=0 Cov(X_{t-1}, \epsilon_t) = 0 Cov(Xt−1​,ϵt​)=0 valid?

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 stationarity condition $|\phi| < 1$ crucial for this derivation?

How does the constant term $c$ disappear from the variance formula?

What is the implication of a large $|\phi|$ (close to 1) for $Var(X_t)$ ?

Why is the assumption $Cov(X_{t-1}, \epsilon_t) = 0$ valid?