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Introduction to Time Series: Definitions of Stationarity and Autocorrelation Functions (ACF/PACF)

Master time series fundamentals: stationarity definitions, ACF/PACF intuition, and their crucial role in statistical modeling.

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The Formal Theorem

A time series {Xt}t∈Z \{X_t\}_{t \in \mathbb{Z}} is strictly stationary if for every integer k≥1 k \ge 1 and every sequence of indices t1,…,tk t_1, \dots, t_k , the joint distribution of (Xt1,…,Xtk) (X_{t_1}, \dots, X_{t_k}) is identical to the joint distribution of (Xt1+h,…,Xtk+h) (X_{t_1+h}, \dots, X_{t_k+h}) for all integers h h . A time series is weakly stationary (or covariance stationary) if: 1. E[∣Xt∣]<∞ E[|X_t|] < \infty for all t t 2. E[Xt]=μ E[X_t] = \mu for all t t (constant mean) 3. Var(Xt)=E[(Xt−μ)2]=σ2<∞ \text{Var}(X_t) = E[(X_t - \mu)^2] = \sigma^2 < \infty for all t t (constant variance) 4. Cov(Xt,Xt+h)=E[(Xt−μ)(Xt+h−μ)]=γ(h) \text{Cov}(X_t, X_{t+h}) = E[(X_t - \mu)(X_{t+h} - \mu)] = \gamma(h) depends only on the lag h h , not on t t . The autocorrelation function (ACF) is defined as ρ(h)=γ(h)γ(0) \rho(h) = \frac{\gamma(h)}{\gamma(0)} . The partial autocorrelation function (PACF) at lag k k is the correlation between Xt X_t and Xt+k X_{t+k} after removing the linear dependence on Xt+1,…,Xt+k−1 X_{t+1}, \dots, X_{t+k-1} .

Analytical Intuition.

Imagine observing a river's flow over time. Is its behavior predictable in a fundamental way, or does it constantly surprise you with new patterns? Stationarity is our measure of this fundamental predictability. A stationary river doesn't drastically change its average flow, its variability, or how its flow at one moment relates to another, regardless of when you start observing. Autocorrelation functions (ACF and PACF) then act like detective tools, revealing the hidden dependencies between the river's flow at different points in time – how much does today's flow predict tomorrow's, or the flow a week from now?
CAUTION

Institutional Warning.

Students often confuse strict and weak stationarity, overlooking that strict stationarity implies identical probability distributions, not just identical first and second moments.

Academic Inquiries.

01

What is the practical implication if a time series is not stationary?

If a time series is not stationary, standard statistical inference based on assumptions of constant mean, variance, and covariance will be invalid. Models fitted to non-stationary data often produce misleading results and poor forecasts.

02

Can a time series be strictly stationary but not weakly stationary?

No, strict stationarity implies weak stationarity. If the entire joint distribution is invariant, then the marginal distributions (for mean and variance) and pairwise joint distributions (for covariance) must also be invariant.

03

How do ACF and PACF help in identifying stationarity?

For stationary series, the ACF typically decays to zero relatively quickly. Non-stationary series often exhibit an ACF that decays very slowly or remains high even for large lags, indicating persistent dependence.

04

What is the difference between Autocovariance and Autocorrelation?

Autocovariance γ(h) \gamma(h) measures the linear relationship between Xt X_t and Xt+h X_{t+h} in the original scale of the data, while Autocorrelation ρ(h) \rho(h) is a standardized version, measuring the correlation on a scale from -1 to 1.

Standardized References.

  • Definitive Institutional SourceBrockwell, Peter J., and Richard A. Davis. Introduction to Time Series and Forecasting. Springer, 2016.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Introduction to Time Series: Definitions of Stationarity and Autocorrelation Functions (ACF/PACF): Visual Proof & Intuition. Retrieved from https://www.nicefa.org/library/general-linear-models-/introduction-to-time-series--definitions-of-stationarity-and-autocorrelation-functions--acf-pacf-

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