Stationary Process

Introduction of time series data

A time series is a sequence of observations in chronological order, for example, daily log returns on a stock.

A common simplifying assumption is that the data are equally spaced with a discrete-time observation index.
This may only hold approximately. For example, daily log returns on a stock may only be available for weekdays, with additional gaps on holidays.
The consecutive observations are commonly regarded as equally spaced, for simplicity.

# Seasonal variation: A repeating pattern within “any fixed period” (not within a year).
# Trend: Systematic change in a time series does not appear to be periodic.
# This time series data’s seasonal oscillation increase over time.

There is a peak time period for travel passengers in July because of the summer vacation.

Stochastic Process

A stochastic process is a sequence of random variables and can be viewed as the “theoretical” or “population” analog of a time series. A time series can be considered a sample from a stochastic process.

“Stochastic” is a synonym for random.
One of the most useful methods for obtaining parsimony in a time series model is to assume some form of distributional invariance over time, or stationarity.

When we observe a time series, the fluctuations appear random, but often with the same type of stochastic behavior from one time period to the next (with the same covariance).
For example, returns on stocks or changes in interest rates can be very different from the previous year, but the mean, standard deviation, and other statistical properties often are similar from one year to the next.

Also, the demand for many consumer products, such as sunscreen, winter coats, and electricity, has random as well as seasonal variation, but each summer is similar to past summers, each winter to past winters, at least over shorter time periods.

Stationary stochastic processes are probability models for time series with time-invariant behavior. A process is said to be strictly stationary if all aspects of its behavior are unchanged by shifts in time.

Weakly Stationary

Often, it will suffice to assume less, namely, weak stationarity. A process is weakly stationary if its mean, variance, and covariance are unchanged by time shifts.

More precisely, {Yt} is a weakly stationary process if

E(Yt)=μ (a finite constant) for all t
Var(Yt)=σ² (a positive finite constant) for all t
Cov(Yt,Y(t-h))=γ(h) for all t and h

The mean and variance do not change with time.
The covariance between two observations depends only on the lag, the time distance |h| between them, not the indices t directly.

The adjective “weakly” in “weakly stationary” refers to the fact that we are only assuming that means, variance, and covariances, not other distributional characteristics such as quantiles, skewness, and kurtosis, are stationary. Weakly stationary is also sometimes referred to as covariance stationary.