Stationarity

Time Series, lesson 5

Which series you can model directly, and how to transform the rest.

You know returns, autocorrelation and the random walk.

Part 1. Make it settle

Below are two simulated prices. Each quarter of the chart shows its own average and spread. Transform the series until every quarter looks alike.

Log, difference, undoLive

A wandering price has a different average in every quarter, so no one average describes it. Its daily changes have the same average and spread throughout. The growing price needed a log first, so that its moves were measured as fractions of the price rather than in money.

Stationary

A series is stationary when its average, its spread and its autocorrelations stay the same whichever stretch you look at. Formally, for weak stationarity: E[xₜ] = μ and Var(xₜ) = σ² for every t, and Cov(xₜ, xₜ₋ₖ) depends only on the lag k.

It matters because everything you estimate from the past, an average, a correlation, a regression, describes the future only if the rules don't change. Fit a model to a series whose average moves and you've fitted one stretch of history.

Your turn

Is a random walk stationary?

Its steps are stationary, but the walk itself isn't: after t steps its variance is tσ². With 1% steps, the spread of where it could be is 10% after 100 days and 20% after 400. No single average or spread describes it.

Show the answer

B. Its steps are stationary, but the walk itself isn't: after t steps its variance is tσ². With 1% steps, the spread of where it could be is 10% after 100 days and 20% after 400. No single average or spread describes it.

Part 2. Unit roots

Keep going with Pro

You've done the first part. Pro opens the rest of this lesson, and every other one.

Still to come

Unit roots
Too much differencing
A worked problem
The same thing, for real

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Next lessonMean reversionSeries that snap back, and how fast: the half-life. Start lesson 6

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05 · Stationarity