# Dickey-Fuller test statistical significance

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?

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.

• 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 ? Commented Jul 10 at 15:16
• 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. Commented Jul 10 at 16:07
• Ok ,thank you very much ! Ill try to look up the original paper to understand this transformation Commented Jul 11 at 5:21