Solved – Unit root test confusion

augmented-dickey-fullerstatastationaritytime seriesunit root

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?

Best Answer

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.

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