Another ways to interpret regression coefficient (i.e. γ) is that γ is a measure of how long it take for a price to mean revert. So, Half-life measures number of days that asset prices takes to revert back to half of its initial deviation from the mean. We can interpret γ this way by transform the discrete time series into the differential form such that the change in price (∆yt) become infinitesimal quantities. Moreover, if we also ignore the drift term (βt) and the lag difference (∆yt−1, . . . , ∆yt−k) then it becomes recognisable in stochastic calculus as the Ornstein-Uhlenbeck formula for mean-reverting process.
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