Stationarity is a central idea in time-series analysis: it describes processes whose statistical behavior does not depend on when we observe them. This lesson introduces the general idea and the strongest common definition, strict-sense stationarity.
A stochastic process is a collection of random variables indexed by time, often written as {Xt}. A process is stationary when its relevant statistical properties remain unchanged as time passes. For example, its average level and variability do not systematically change, and the relationships among observations remain consistent.
All flashcards
What is strict-sense stationarity?
Invariance of every finite-dimensional joint distribution under a common shift of time indices.
What is wide-sense stationarity (WSS)?
A process with a constant mean, finite second moments, and autocovariance depending only on lag.
How do shocks differ in trend-stationary and unit-root processes?
Shocks are transitory in trend-stationary processes and have permanent effects in unit-root processes.
What does joint wide-sense stationarity require?
Each process must be WSS, and their cross-covariance must depend only on the time lag.
What does a Fourier-based surrogate procedure do?
It retains Fourier magnitudes, randomizes phases, enforces conjugate symmetry, and applies an inverse transform.
Quiz questions
Which conditions define wide-sense stationarity for a real-valued process?Answer: A constant mean, finite second moments, and autocovariance depending only on lag.
What additional condition is needed for two individually WSS processes to be jointly WSS?Answer: Their cross-covariance must depend only on the time difference.
If a process is Nth-order stationary, what follows for orders M≤N?Answer: It is also Mth-order stationary.
How do shocks typically behave in a unit-root process compared with a trend-stationary process?Answer: Unit-root shocks have permanent effects; trend-stationary shocks are transitory.
In a Fourier-based surrogate procedure, what is retained while phases are randomized?Answer: The magnitude spectrum, preserving the distribution of power across frequencies.
Based on the Wikipedia article Stationary process, licensed CC BY-SA 4.0. Adapted into a summary, flashcards and quizzes.