# 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?

• Is it because the regression $x_t = b_0 + b_1 \times x_{t-1}$ is biased when $b_1=1$ and we cannot perform hypothesis testing, because estimation of $b_1$ and its error is corrupted?
– mbt
Commented Feb 19, 2020 at 8:32
• Hi: I think it's because the test for zero comes straight out of the regression. The tests are identical because testing H0:: B1 = 1 is the same as testing H_0 : B1 - 1 = 0. Note that your subscripts should have time index. They are coefficients. Commented Feb 19, 2020 at 14:39
• Thanks. What do you think about my explanation why $H_0: b_1 = 1$ is corrupted. If $H_0$ is true, regression results are not reliable.
– mbt
Commented Feb 24, 2020 at 13:41
• yes, if you don't difference the time series, then the regression isn't valid ( under the null ) so that's why you estimate the differenced model. Estimating the differenced model allows you to test $b_1 - 1 = 0$ which is the same as testing $b_1 = 1$. So, yes you are correct. That's why yo do that. Also, disregard my original comment. It's not wrong but it doesn't address your question and my apologies for noise. Commented Feb 25, 2020 at 14:42

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.)