# Distribution of estimator of AR(1) parameter in presence of unit root

I want to apply the unit root test to the original equation before writing it in the Dickey-Fuller form.

The original equation is: $$Y_t=b Y_{t-1} + u_t.$$ I want to test for unit root directly by testing if $b=1$. I read in Enders's book "Applied Time Series Econometrics" that this is equivalent to the Dickey-Fuller test. However he didn't mention what critical value to compare to.

My $t$-test for $b=1$ hypothesis is $-1$. Can you please help me with the critical value?

Assume, the critical value is $-1.95$; what conclusion about unit root can I draw from my test?

• Is this a question from a course or textbook? If so, please add the [self-study] tag & read its wiki. – gung - Reinstate Monica Oct 3 '16 at 0:33
• It is not from a textbook and it is not for a course I'm taking rn. I'm reviewing my econometrics notes before I tutor a student. – Abed111 Oct 3 '16 at 0:43
• @Abed, was my answer helpful, or do you need some extra elaboration? (Just checking since there was no response from you.) – Richard Hardy Oct 11 '16 at 18:49
• Testing that (standardized) coefficient against the Dickey-Fuller distribution is the Dickey-Fuller test (not "equivalent to"). You can find the critical values in a Dickey-Fuller table. Testing it against the $t$-distribution is wrong, because it does not have a $t$-distribution under the null of a unit root, as mentioned below. – Chris Haug Feb 24 '17 at 15:10

If the critical value were indeed $-1.95$ (which I don't think it is as the estimator distribution is nonstandard under the null of a unit root), a test statistic of $-1$ would not allow rejecting the null hypothesis of $b=1$ at the confidence level corresponding to the critical value -- because the test statistic is less extreme (closer to zero) than the critical value: $-1.95<-1<0$.