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I recently read about the Dickey-Fuller test. Firstly about the transformation from $$ y_t=\rho y_{t-1}+\epsilon_t $$

to: $$ y_t-y_{t-1}=(\rho-1) y_{t-1}+\epsilon_t $$

I assumed it is to get the statistic of $\delta=\rho -1$ which is a pivotal parameter not dependant on some unknown statistic? or am i mistaken?

Secondly, as I understood the statistical test made on delta is one-sided. why do we omit negative values whose absolute value of rho is larger than 1?

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First, the distribution is not easier, but we can take the default t-statistic reported by regression packages.

Second, we "omit" positive values of $\delta$ where $\rho$ is larger than one because these correspond to explosive processes which are typically viewed as implausible in applications. See Explosive processes, non-stationarity and unit roots, how to distinguish?, though.

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  • $\begingroup$ As to my knowledge in statistics we try to asses pivot parameters because they don't depend on the unknown quantity in question.I actually meant that delta is a pivotal parameter while rho is not? Or am i completely mistaken ? $\endgroup$
    – Tomer Gigi
    Commented Jul 10 at 15:16
  • $\begingroup$ A pivot is a quantity whose distribution does not depend on the parameter(s), e.g. $(X-\mu)/\sigma$ for a normal random variable would be a pivot. Both $\delta$ and $\rho$ are parameters, so that indeed has nothing to do with pivotalness, I would say. $\endgroup$ Commented Jul 10 at 16:07
  • $\begingroup$ Ok ,thank you very much ! Ill try to look up the original paper to understand this transformation $\endgroup$
    – Tomer Gigi
    Commented Jul 11 at 5:21

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