What is the primary characteristic of a stationary time series?

Question

BlackTom AI · Accepted Answer

Reframing the question helps: we’re asked to identify the main trait of a stationary time series.
Option 1: 'It must have a mean of zero.' While many stationary series can be adjusted to have a zero mean, stationarity does not require the mean to be exactly zero at all times; it only requires the mean to be constant over time. So this is too strict and not the defining property.
Option 2: 'Its statistical properties (like mean and variance) are constant over time.' This captures the essence of stationarity: constant mean, variance, and autocovariance structure over time. This is the core requirement that makes a time series stationary.
Option 3: 'It cannot have any seasonal component.' A stationary series can exhibit seasonal patterns only if those patterns do not cause the mean or variance to change over time. In many practical contexts, deterministic seasonality would break stationarity, but a process with seasonal behavior that preserves constant statistical properties could still be stationary in a broader sense. Therefore, saying it cannot have any seasonal component is too restrictive and not universally true.
Option 4: 'It must be a straight, horizontal line.' A horizontal line would imply a constant value with zero variance, which is a degenerate case of a stationary process but not the general characteristic. Stationarity allows nonzero, constant variance and fluctuations around the mean.
In summary, the defining feature among the options is that the statistical properties being constant over time (mean, variance, and autocovariance structure) best captures stationarity, whereas the other statements impose unnecessary or false restrictions on what a stationary time series must look like.

What is the primary characteristic of a stationary time series?

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