Dickey-Fuller test for unit root Why we subtract xt-1 from both sides of AR(1) equation to test if g1 = b1 - 1 is equal to 0, rather than test if b1 is equal to 1 in Dickey-Fuller test process?
 A: It's a matter of preference.
The two statistics are identical, not just in distribution but algebraically. 
First regression
$$
x_t = \rho_1 x_{t-1} + \epsilon_t,
$$
the $t$-statistic for testing $H_0:\rho_1 = 1$ is
$$
t_1= \frac{\hat{\rho_1} - 1}{se(\hat \rho_1)}.
$$
Second regression
$$
\nabla x_t = \rho_2 x_{t-1} + \epsilon_t,
$$ where $\nabla x_t$ is the first difference,
the $t$-statistic for testing $H_0:\rho_2 = 0$ is
$$
t_2= \frac{\hat{\rho_2}}{se(\hat \rho_2)}.
$$
It is clear that $\hat{\rho_1} - 1 = \hat \rho_2$ and (because the two regressions have the same residuals) $se(\hat \rho_1) =  se(\hat \rho_2)$. Therefore 
$$
t_1 = t_2.
$$
In other words, it doesn't matter which regression you run.
The formulation with $\nabla x_t$ is more common because it is consistent with the augmented (ADF) regression with lagged $\nabla x_t$'s. 
This has nothing to do with the fact that (both) regressions are biased in finite sample. (You can perform hypothesis testing. It's the whole point of having a test statistic. The difference here is that the limit distribution under the null cannot be obtained by the Central Limit Theorem as in the stationary case.)
