What does it mean for a time series to be weakly stationary?

## Why the concept exists You get exactly one realisation of a time series: one path, one history, one ordering. Yet you want to estimate a mean, a variance and a set of lag correlations from it. That is only defensible if those quantities are constant along the path, so that the first third of the data and the last third are informative about the same underlying numbers. Stationarity is the formal version of "the rules generating this series are not changing". ## The three conditions A series `X_1, X_2, ..., X_T` is **weakly stationary** (equivalently covariance stationary, or second-order stationary) when: 1. **Constant mean.** `E[X_t] = mu` for every `t`. The level does not drift upward, downward, or around a repeating pattern. 2. **Constant, finite variance.** `Var(X_t) = sigma^2 < infinity` for every `t`. The spread around the level is the same at the start and at the end. 3. **Lag-only autocovariance.** `Cov(X_t, X_{t+k}) = gamma(k)` for every `t`. How strongly two points move together depends on the distance `k` between them, not on the calendar date. Setting `k = 0` in condition 3 reproduces condition 2, which is why some textbooks state only two conditions. The substance is the same. ## What weak stationarity does *not* require - **Not independence.** Stationary does not mean the observations are unrelated. An autoregressive series where each value is 0.7 times the previous one plus noise is stationary and heavily autocorrelated at the same time. - **Not a flat line.** A stationary series moves, sometimes a lot. What holds still is the *statistical description* of the movement, not the values. - **Not normality.** Weak stationarity constrains only the first two moments; the shape of the distribution is unconstrained as long as the variance is finite. - **Not the absence of long excursions.** A persistent stationary process can wander away from its mean for a long time and still be stationary, which is exactly why eyeballing a plot is not conclusive. ## Weak versus strict stationarity **Strict stationarity** is the stronger property: the entire joint distribution of `(X_t, X_{t+1}, ..., X_{t+k})` is unchanged when you shift the whole block in time. It constrains every moment and every dependence structure, not just the first two. The relationships are worth memorising. Strict stationarity plus finite second moments implies weak stationarity. Weak stationarity does **not** imply strict stationarity in general — the third and fourth moments are free to move. The one clean exception is a Gaussian process: a normal distribution is fully described by its mean and covariance, so a weakly stationary Gaussian series is also strictly stationary. Almost all applied time-series work uses the weak version, because it is what estimation actually needs. ## Recognising a violation Each condition fails in a recognisable way. A series whose level climbs year over year violates the constant mean. A series whose swings are small in early years and large in later years violates the constant variance. A series whose pattern repeats on a fixed annual cycle also violates the constant mean, because the expected value in one month differs systematically from another. And a series that accumulates its shocks — where today's value is yesterday's value plus fresh noise — violates conditions 2 and 3 together, because the variance keeps growing and the covariance between two points depends on how far along the path they are. ## The canonical illustration Take daily closing prices of a stock. The level in one year has no reason to equal the level five years earlier, the spread of prices widens as the price level rises, and the price today is essentially the accumulation of every past move. It is not weakly stationary in any useful sense. Now take the daily **log returns**, `log(P_t) - log(P_{t-1})`. Those hover around a near-zero mean with roughly the same unconditional spread throughout, and their correlation structure is close to lag-only. That is the object you can actually model. The usual caveat is that returns show volatility clustering — calm stretches and turbulent stretches — so the *conditional* variance moves even when the unconditional variance is treated as constant. ## Why interviewers ask Every classical forecasting model estimates one fixed parameter set from the entire history. If the mean, variance or dependence structure changes along the way, those parameters describe no actual period of the data, and the forecast intervals they produce are not trustworthy. Recognising non-stationarity is therefore the first step in any time-series workflow, before a single model is fitted.