Mean reversion

Time Series, lesson 6

Series that snap back, and how fast: the half-life.

You know the form xₜ = φxₜ₋₁ + εₜ from "Stationarity", and that φ below 1 means shocks fade.

Part 1. Push it and let go

Below is a simulated spread hovering around its average. Push it out and watch it come back. The slider sets how many days it takes to close half the gap.

Push and let goLive

Each day the gap closes by the same fraction, so the way back is a curve, not a straight line: fast at first, then slower, never quite reaching zero. The time to close half the gap is the same wherever it starts.

Mean reversion and half-life

A series pulled back toward a long-run average is mean-reverting. Its half-life is the number of days it takes, on average, to close half of any gap from that average.

Your turn

A spread closes 20% of its gap to the average each day. Roughly what's its half-life?

Each day 80% of the gap survives, so after h days 0.8ʰ of it is left. Set 0.8ʰ = 0.5: h = ln 0.5 / ln 0.8 = 3.1 days. The quick rule, 0.69 / 0.2 = 3.5, is close but a little long.

Show the answer

D. Each day 80% of the gap survives, so after h days 0.8ʰ of it is left. Set 0.8ʰ = 0.5: h = ln 0.5 / ln 0.8 = 3.1 days. The quick rule, 0.69 / 0.2 = 3.5, is close but a little long.

Part 2. The formula, and fitting it

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Still to come

The formula, and fitting it
A worked problem
The same thing, for real

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Next lessonMomentum and trendSeries that keep going, and how to tell the difference. Start lesson 7

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06 · Mean reversion