mean reversion speed

[Meritas]

Hi,

Given the autoregressive model,
Xt = a + b*Xt-1 + e

the mean reverting level (unconditional mean) is calculated as a / (1-b).

What about the spead of mean reversion? i.e, if data daily, how to calculate the nb of days it takes for the series to mean revert to the unconditional mean?

Thanks a lot!

[trubador]

This is a little bit tricky. Mean reverting value can only be reached in the limit (as the forecast horizon approaches to infinity). Therefore, "half-life" is the most common way of measuring the speed of mean reversion. This is to calculate the number of periods needed for the forecast to attain half of its original value. And the formula is: t = ln(0.5)/ln(abs(b))

[Meritas]

thank you, trubador!

in order to use the half-life formula, is it necessary to run the AR(1) regression with logs of the variable?

[trubador]

No, it is not. That is a general formula for AR(1) model.

[utah777]

This is a little bit tricky. Mean reverting value can only be reached in the limit (as the forecast horizon approaches to infinity). Therefore, "half-life" is the most common way of measuring the speed of mean reversion. This is to calculate the number of periods needed for the forecast to attain half of its original value. And the formula is: t = ln(0.5)/ln(abs(b))

Trubador, don't you think the mean reversion half time should be ln(0.5)/b ???

[trubador]

Does not matter what I think, that is the formula...

[utah777]

Sorry, you're right, for autoregressive AR(1) process presented in first post the formula is ln(0.5)/ln(b).

Mine is also correct I guess, but for Ornstein–Uhlenbeck representation.