Classical Time Series — Course Guide

Preparation — Stationarity in 30 Minutes

A compact diagnostic primer on stable dependence, trends, unit roots, seasonality, and breaks.

Preparation — Stationarity in 30 Minutes

The idea in one sentence

A series is weakly stationary when its mean and variance are constant and the covariance between two observations depends on their lag, not their calendar date.

\mathbb E(X_t)=\mu,\qquad \operatorname{Var}(X_t)=\gamma(0),\qquad \operatorname{Cov}(X_t,X_{t-h})=\gamma(h).

Stationary does not mean flat or independent. It means that dependence operates under a stable rule, so different parts of one path can inform the same population quantities.

Three processes, three diagnoses

Process	What a shock does	Stationary?
$X_t=\varepsilon_t$	disappears immediately	yes, if the noise has stable moments
$X_t=0.8X_{t-1}+\varepsilon_t$	decays as $0.8^h$	yes
$X_t=X_{t-1}+\varepsilon_t$	remains permanently	no: variance grows with $t$

For the AR(1), repeated substitution gives

X_t=\varepsilon_t+0.8\varepsilon_{t-1}+0.8^2\varepsilon_{t-2}+\cdots.

If $\operatorname{Var}(\varepsilon_t)=4$ , then

\operatorname{Var}(X_t)=\frac{4}{1-0.8^2}=11.11.

For the random walk, $X_t=\sum_{j=1}^{t}\varepsilon_j$ , so $\operatorname{Var}(X_t)=4t$ . The calendar time now changes the distribution.

Do not collapse five different problems into “non-stationary”

Pattern	Small example	First response
deterministic trend	demand rises by 2 MWh each year	trend regressor or detrending
stochastic trend/unit root	shocks permanently shift the level	first difference, if justified
deterministic seasonality	every January is about 10 units higher	month indicators or seasonal terms
seasonal unit root	this January resembles last January through accumulated shocks	seasonal difference
structural break	a heat pump raises demand after installation	intervention term, regime model, or shorter window
changing variance	seasonal amplitude grows with level	log/Box–Cox transform or variance model

Different operations answer different diagnoses. Applying log, first difference, and seasonal difference automatically can destroy useful structure.

A useful counterexample: over-differencing

Suppose $X_t=\varepsilon_t$ is already white noise. Differencing gives

\Delta X_t=\varepsilon_t-\varepsilon_{t-1},

an MA(1) process with coefficient $-1$ and lag-1 autocorrelation $-1/2$ . The transformation has manufactured dependence and made the model non-invertible. “More stationary-looking” is not always better.

Trend-stationary or difference-stationary?

Two upward paths can look alike:

\text{trend-stationary: }X_t=\alpha+\beta t+u_t,

\text{difference-stationary: }X_t=X_{t-1}+c+\varepsilon_t.

After a shock, the first returns toward its deterministic trend; the second carries the shock forward. Detrending is appropriate for the first and differencing for the second. A short sample may not distinguish them cleanly, so report uncertainty rather than declaring a diagnosis from one test.

ADF and KPSS ask opposite questions

Test	Null hypothesis	A small p-value suggests
ADF	a unit root is present	evidence against a unit root
KPSS	the series is level- or trend-stationary	evidence against stationarity

Read them with plots, subject knowledge, and break dates. A break can make a stationary regime look like a unit root; low power can make a genuinely persistent process inconclusive.

Do not write “the ADF proves stationarity.” Write: “At the stated deterministic terms and lag choice, the test rejects/does not reject a unit root; plots, KPSS evidence, and break sensitivity lead us to…”

Four-case interpretation

ADF	KPSS	Reasonable next step
rejects	does not reject	stationary modeling is plausible
does not reject	rejects	differencing or a unit-root model is plausible
rejects	rejects	inspect breaks, nonlinearity, variance, and test specification
does not reject	does not reject	sample may be weakly informative; compare robust alternatives

Thirty-minute diagnostic routine

Name the unit and frequency. A “lag of 12” means nothing without monthly sampling.
Plot levels and mark interventions. A break is not an autocorrelation pattern.
Compare seasonal positions. Ask whether January behaves like January.
Inspect variance against level. Log only when a proportional interpretation is sensible.
Compare detrending and differencing. Explain which shock behavior each assumes.
Use ADF/KPSS as supporting evidence. State deterministic terms and lag choice.
Replot and inspect the transformed series. Watch for strong negative lag-1 correlation from over-differencing.
Validate both choices out of sample. Transformation is part of the model, not a preprocessing fact.

Exit check

For each statement, say true or false and give one sentence:

A stationary series must be independent. False—its lag covariance may be non-zero.
A random walk with zero drift has constant mean, so it is stationary. False—its variance grows.
Seasonal differencing and month indicators express the same assumption. False—one removes a seasonal stochastic trend; the other models stable seasonal means.
Differencing can harm a stationary series. True—it can inject an MA unit root.
Passing one unit-root test settles the model. False—specification, breaks, power, and the forecast task still matter.

Continue to Stochastic Processes and Stationarity for the population language behind these decisions.

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Classical Time Series — Course Guide

A matrix-first course in covariance-stationary processes, ARMA equations, linear prediction, likelihood, state space, and spectra.

Module 1 — Processes, Dependence, and Stationarity

Represent a stochastic process through finite random vectors, Toeplitz covariance matrices, stationarity, and Wold innovations.