# Unit root test confusion

For my time-series regression, I am regressing the difference in variable x on a difference in variable y. Before proceeding, I want to check for stationarity of my variables. Regressing d.x (difference in x) over L.d.x (lag of the difference in x), I get a coefficient of $-0.9995261$ and a 95% interval: $[-1.000654, -0.9983979]$. As far as I remember, the abs value of the coefficient (and the values in the interval) must be less than 1 for us to be sure of the stationarity, which is not the case here. So I conducted a Dickey-Fuller test (all of this in Stata) and got the following results:

Interpolated Dickey-Fuller
Test Statistic  |  1% Critical Value |  5% Critical Value | 10% Critical Value

Z(t)       -3474.557   |         -3.430     |     -2.860         |    -2.570

MacKinnon approximate p-value for Z(t) = 0.0000


I am confused because they very strongly suggest that there is no unit root, whereas I expected a close call. Is this because I have ~5000 observations? Or am I wrong, and the condition for the unit root is a coefficient in excess of 1, not its absolute value in excess of one?

A (simple) unit root process is an AR(1) process such that $x_t = \rho x_{t-1} + \nu_t$, where $\rho = 1$. Subtract $x_{t-1}$ from both sides, giving $\Delta x_t = (\rho - 1) x_{t-1} + \nu_t$. The Dickey-Fuller test asks whether $\gamma \equiv \rho - 1 = 0$. See that it is testing a different hypothesis than the one implied by the confidence interval that you offer. Hence, your test and the DF test results are not comparable.
• The regression that you are using is the wrong one to assess stationarity. The standard errors are incorrect, leading to the incorrect interval. If you want to test the stationarity of $x_t$, run a Dickey-Fuller test on $x_t$. If you want to test the stationarity of $\Delta x_t$, run a Dickey-Fuller test on that (replace $x_t$ in my exposition above with $\Delta x_t$). The DF test takes into consideration that the standard errors are incorrect under the null hypothesis of a unit root and provides the appropriate critical values to do your testing. Regular confidence intervals are wrong. – Charlie Jan 29 '13 at 13:32