Derivation of the Method of Moments Estimators for an AR(1) Model with a Constant Term

Q: What happens if the stationarity condition $ |\phi| < 1 $ is violated?

If $ |\phi| \ge 1 $, the AR(1) process is non-stationary. The population moments (mean, variance, autocovariance) would not be constant over time, making the derivation of theoretical moments invalid. The MoM estimators derived would not consistently estimate the true parameters, and standard statistical inference would not apply. Such processes exhibit explosive behavior or unit roots, requiring different modeling approaches.

Q: How does the constant term $ c $ affect the mean of the AR(1) process?

The constant term $ c $ directly determines the long-run mean of the stationary AR(1) process. As derived, $ \mu = \frac{c}{1 - \phi} $. If $ c=0 $, the mean $ \mu $ would be zero. A positive $ c $ pushes the mean upwards, and a negative $ c $ pulls it downwards, with the effect amplified by $ \frac{1}{1-\phi} $. It represents the baseline level around which the series fluctuates.

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The Formal Theorem

Consider a stationary Autoregressive model of order 1 (AR(1)) with a constant term, defined as:

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

where

Y_t

is the time series at time

t

c

is a constant,

\phi

is the autoregressive parameter, and

\epsilon_t

is a white noise error term such that

E[\epsilon_t] = 0

Var[\epsilon_t] = \sigma^2

, and

Cov(\epsilon_t, \epsilon_s) = 0

for

t \neq s

. Additionally,

\epsilon_t

is uncorrelated with

Y_s

for

s < t

. For stationarity, we assume

|\phi| < 1

. The Method of Moments (MoM) estimators for

c

\phi

, and

\sigma^2

are derived by equating the first three theoretical (population) moments to their corresponding sample moments. **1. Population Mean (

\mu

):** For a stationary process,

E[Y_t] = E[Y_{t-1}] = \mu

. Taking the expectation of the AR(1) equation:

E[Y_t] = E[c + \phi Y_{t-1} + \epsilon_t]

\mu = c + \phi \mu + 0

Solving for

\mu

\mu(1 - \phi) = c \implies \mu = \frac{c}{1 - \phi}

**2. Population Variance (

\gamma_0

):** For a stationary process,

Var[Y_t] = Var[Y_{t-1}] = \gamma_0

. Taking the variance of the AR(1) equation:

Var[Y_t] = Var[c + \phi Y_{t-1} + \epsilon_t]

Since

c

is a constant and

Y_{t-1}

is uncorrelated with

\epsilon_t

\gamma_0 = \phi^2 Var[Y_{t-1}] + Var[\epsilon_t]

\gamma_0 = \phi^2 \gamma_0 + \sigma^2

Solving for

\gamma_0

\gamma_0(1 - \phi^2) = \sigma^2 \implies \gamma_0 = \frac{\sigma^2}{1 - \phi^2}

**3. Population Autocovariance at Lag 1 (

\gamma_1

):**

\gamma_1 = Cov(Y_t, Y_{t-1}) = E[(Y_t - \mu)(Y_{t-1} - \mu)]

We can rewrite the AR(1) model in terms of deviations from the mean:

Y_t - \mu = \phi (Y_{t-1} - \mu) + \epsilon_t

. (To see this, substitute

\mu = \frac{c}{1-\phi}

into

Y_t - \mu

)

\gamma_1 = E[ (\phi (Y_{t-1} - \mu) + \epsilon_t) (Y_{t-1} - \mu) ]

\gamma_1 = E[ \phi (Y_{t-1} - \mu)^2 + \epsilon_t (Y_{t-1} - \mu) ]

Since

\epsilon_t

is uncorrelated with

Y_{t-1}

E[\epsilon_t (Y_{t-1} - \mu)] = E[\epsilon_t] E[Y_{t-1} - \mu] = 0 \cdot 0 = 0

\gamma_1 = \phi E[(Y_{t-1} - \mu)^2] = \phi Var[Y_{t-1}]

For a stationary process,

\gamma_1 = \phi \gamma_0

**Method of Moments Estimators:** We equate the theoretical moments to their sample counterparts for a given time series

\{Y_1, \dots, Y_N\}

\hat{\mu} = \bar{Y} = \frac{1}{N} \sum_{t=1}^N Y_t

\hat{\gamma}_0 = s_Y^2 = \frac{1}{N} \sum_{t=1}^N (Y_t - \bar{Y})^2

\hat{\gamma}_1 = s_1 = \frac{1}{N} \sum_{t=2}^N (Y_t - \bar{Y})(Y_{t-1} - \bar{Y})

From

\gamma_1 = \phi \gamma_0

, we get the estimator for

\phi

\hat{\phi} = \frac{\hat{\gamma}_1}{\hat{\gamma}_0} = \frac{\frac{1}{N} \sum_{t=2}^N (Y_t - \bar{Y})(Y_{t-1} - \bar{Y})}{\frac{1}{N} \sum_{t=1}^N (Y_t - \bar{Y})^2}

From

\mu = \frac{c}{1 - \phi}

, we solve for

c

and substitute the estimators:

c = \mu(1 - \phi)

\hat{c} = \bar{Y}(1 - \hat{\phi})

From

\gamma_0 = \frac{\sigma^2}{1 - \phi^2}

, we solve for

\sigma^2

and substitute the estimators:

\sigma^2 = \gamma_0 (1 - \phi^2)

\hat{\sigma}^2 = \hat{\gamma}_0 (1 - \hat{\phi}^2) = \left( \frac{1}{N} \sum_{t=1}^N (Y_t - \bar{Y})^2 \right) \left( 1 - \left( \frac{\hat{\gamma}_1}{\hat{\gamma}_0} \right)^2 \right)

These three equations provide the Method of Moments estimators for

\phi

c

, and

\sigma^2

for an AR(1) model with a constant term.

Analytical Intuition.

Imagine you're a forensic detective in a high-stakes cinematic thriller, tasked with understanding the hidden 'operating system' of a seemingly chaotic sequence of events – our time series

Y_t

. The AR(1) model with a constant term,

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

, is the complex 'blueprint' we suspect is at play, but the true parameters

c

\phi

, and

\sigma^2

are unknown, like encrypted codes. The Method of Moments is your ingenious decryption tool. Instead of trying to crack the entire code directly, you look for its 'fingerprints' – specific characteristics or moments – in the observed data. The 'mean' tells you the average level of activity, the 'variance' indicates its overall volatility, and the 'autocovariance at lag 1' reveals how strongly the current event is influenced by the immediate past. You calculate these same 'fingerprints' from your observed data (sample moments) and then, in a moment of cinematic revelation, you equate them to the theoretical 'fingerprints' predicted by the model's blueprint (population moments). By solving these equations, you reverse-engineer the unknown parameters,

\hat{c}

\hat{\phi}

, and

\hat{\sigma}^2

, effectively 'unlocking' the model and revealing its underlying dynamics. It's about matching the model's expected statistical signature with the signature found in reality.

CAUTION

Institutional Warning.

Students often confuse population moments (theoretical values from the model's distribution) with sample moments (calculated directly from data). Another common pitfall is the correct handling of $\sum$ limits and denominators ( $N$ vs. $N-k$ ) when defining sample autocovariances, which can impact estimator bias, though MoM typically uses $N$ for direct analogy.

Academic Inquiries.

Why use the Method of Moments (MoM) instead of Maximum Likelihood Estimation (MLE) for AR(1) models?

MoM estimators are generally simpler to compute and require fewer assumptions about the error distribution (e.g., $\epsilon_t$ only needs to be white noise, not necessarily Gaussian). However, MLE estimators are often asymptotically more efficient (have smaller variance) if the distributional assumptions are correct, particularly for large sample sizes. For AR(1) with Gaussian errors, MLE is preferred, but MoM provides a good starting point and can be robust.

What happens if the stationarity condition $|\phi| < 1$ is violated?

If $|\phi| \ge 1$ , the AR(1) process is non-stationary. The population moments (mean, variance, autocovariance) would not be constant over time, making the derivation of theoretical moments invalid. The MoM estimators derived would not consistently estimate the true parameters, and standard statistical inference would not apply. Such processes exhibit explosive behavior or unit roots, requiring different modeling approaches.

How does the constant term $c$ affect the mean of the AR(1) process?

The constant term $c$ directly determines the long-run mean of the stationary AR(1) process. As derived, $\mu = \frac{c}{1 - \phi}$ . If $c=0$ , the mean $\mu$ would be zero. A positive $c$ pushes the mean upwards, and a negative $c$ pulls it downwards, with the effect amplified by $\frac{1}{1-\phi}$ . It represents the baseline level around which the series fluctuates.

Standardized References.

Definitive Institutional SourceShumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications: With R Examples. 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). Derivation of the Method of Moments Estimators for an AR(1) Model with a Constant Term: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/derivation-of-the-method-of-moments-estimators-for-an-ar-1--model-with-a-constant-term

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The Formal Theorem

Analytical Intuition.

Institutional Warning.

Academic Inquiries.

Why use the Method of Moments (MoM) instead of Maximum Likelihood Estimation (MLE) for AR(1) models?

What happens if the stationarity condition ∣ϕ∣<1 |\phi| < 1 ∣ϕ∣<1 is violated?

How does the constant term c c c affect the mean of the AR(1) process?

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 happens if the stationarity condition $|\phi| < 1$ is violated?

How does the constant term $c$ affect the mean of the AR(1) process?