Half life (Dickey-Fuller) If I estimate the regression
$$y_t=\alpha+\rho y_{t-1}+\varepsilon_t$$
the half life of this process for $\rho\neq 1$ is $\text{ln}(0.5)/\text{ln}(\rho)$.
If I instead estimate the Dickey-Fuller regression
$$\Delta y_t=\beta_0+\beta_1 y_{t-1}+\varepsilon_t$$ 
is the half life now simply $\text{ln}(0.5)/\text{ln}(\beta_1+1)?$
 A: I think you are right.
Take the two equations, 
$$y_t = \alpha + \rho y_{t-1} + \varepsilon_t$$
and 
$$\Delta y_t = \beta_0 + \beta_1 y_{t-1} + \varepsilon_t.$$ 
Subtracting $y_{t-1}$ from both sides of the first equation yields the second equation with $\beta_0=\alpha$ and $\beta_1=\rho-1$. And vice versa, adding $y_{t-1}$ to both sides of the second equation yields the first equation with $\alpha=\beta_0$ and $\rho=\beta_1+1$. Thus the two equations are equivalent representations of the same process. 
Then I do not see a problem in substituting $\rho$ with $\beta_1+1$ in the half life formula (as long as the argument fits all the assumptions that underly the half life formula).
So far the coefficient values were taken as given and estimation was not considered. If you need to estimate the coefficients first, the condition $|\rho|<1$ allows consistent estimation of both representations by OLS or by maximum likelihood. I am not sure how fast the estimator will converge if $|\rho|$ is very close to unity, though, and there may be a difference between the two representations in such a case.
