Stochastic Time Series Concepts

A time series is just a random variable indexed by time. The whole game is figuring out which pieces of the index pattern are signal you can model and which pieces are noise that ruins forecasts.

A stochastic process is a sequence of random variables $\{Y_t\}$ indexed by time $t=1,2,\dots ,T$. Each $Y_t$ has its own distribution, and crucially the $Y_t$ are usually not independent. That dependence across time is what makes time series different from cross-sectional regression.

KEY: In cross-sectional data, independence across observations is assumed. In time series, dependence across observations is the entire point.

The additive decomposition is $Y_t=T_t+S_t+C_t+{\epsilon }_t$. The multiplicative version is $Y_t=T_t\times S_t\times C_t\times {\epsilon }_t$, preferred when seasonal amplitude grows with the level.

A white noise process $\{{\epsilon }_t\}$ has three defining properties: mean $E[{\epsilon }_t]=0$, constant variance $\text{Var}({\epsilon }_t)={\sigma }^2$, and zero autocorrelation $\text{Cov}({\epsilon }_t,{\epsilon }_s)=0$ for $t\ne s$. It is the atomic noise input that drives larger models.

Common mistakes

• Calling a series stationary just because the mean looks flat. Constant mean is necessary but not sufficient. Variance must be constant and autocovariance must depend only on lag. A series with flat mean but fan-shaped variance is still non-stationary.
• Confusing strict and weak stationarity. Strict requires the full joint distribution to be shift-invariant. Exam questions about "stationary" with no qualifier always mean weak stationarity.
• Treating white noise as Gaussian by default. White noise needs mean 0, constant variance, and zero autocorrelation. Normality is an extra assumption layered on top.

Bottom line

• A stochastic time series $\{Y_t\}$ decomposes into trend, seasonality, cyclical, and irregular (noise) components. Most modeling work is isolating each piece.
• Weak (covariance) stationarity requires constant mean, constant variance, and autocovariance that depends only on lag $k$, not on absolute time $t$. This is the workhorse exam definition.
• White noise has mean 0, constant variance ${\sigma }^2$, and zero autocorrelation at every nonzero lag. Normality is an extra assumption, not required.
• A random walk $Y_t=Y_{t-1}+{\epsilon }_t$ is NOT stationary. Its variance $t{\sigma }^2$ grows with time; first-differencing makes it stationary.

Exam shortcut

If the prompt mentions constant mean AND constant variance AND autocovariance depending only on lag, the answer is weak stationarity. If even one of those three fails, the series is non-stationary. When in doubt, scan the question for "variance grows with t" or "mean trends upward" as instant disqualifiers. For random walks, memorize the trio: mean $Y_0$ (or $Y_0+\delta t$ with drift), variance $t{\sigma }^2$, ACF near 1 across many lags.

The full lesson (about 2,600 words, 17 min read) adds 2 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• 3a

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