Recover the AR(1) coefficient from a half-life

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An analyst estimates that shocks to a mean-reverting AR(1) series decay to half their size in exactly 4 periods. What is the persistence coefficient \phi?

Accepted Answer

The half-life relation is \phi^{h} = 0.5. Solving for \phi given the half-life h = 4: \phi = 0.5^{1/h} = 0.5^{1/4} = 2^{-1/4}. Numerically, \phi = 0.5^{0.25} \approx 0.8409. So a 4-period half-life corresponds to \phi \approx 0.84. A quick sanity check: 0.8409^4 = (0.8409^2)^2 \approx 0.7071^2 \approx 0.5, as required. Takeaway. The half-life formula inverts to \phi = 0.5^{1/h}; a half-life of 4 periods implies a persistence of about 0.84.

Recover the AR(1) coefficient from a half-life

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