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I am currently reading the book 'High Frequency Financial Econometrics' by Jacod and Ait-Sahalia and have a problem with the following statement in chapter 2:

'The usual rationale for first-differencing a time series is to ensure that when T [the time horizon] grows the process [X, used for modeling the log of an asset] does not explode: in many models, the process X may be close to a local martingale and thus the discrete date $X_{i\Delta}$ may be close to exhibiting a unit root, whereas the first differences of X will be stationary or at least non-explosive.'

$\Delta$ is the frequency rate at which the process X is sampled with mesh tending to zero.

I am not familiar with the notion of unit root. What do the authors mean by it? Is someone able to give an example, when the process X could explodes? I am also not sure, waht it means to explode.

Thank you in advance!

ArkhamKnight
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  • The concept comes from the statistical time series stream (a setting which is discrete in time). You can view this as the presence of a random walk embedded in the process, so its variance is explosive whereas the variance is constant in the stationary setting. The matter is of utmost importance for statsticians (named econometricians in this field) because the estimation procedure has to be adapted in the presence of unit root (or other motive of absence stationnarity). Best regards – TheBridge Nov 07 '16 at 16:49

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