Unit roots and the ADF test

Is there a unit root?

In the autoregression $x_t=\varphi x_{t-1}+\varepsilon_t$ the coefficient $\varphi$ decides the behaviour. With $|\varphi|<1$ the series is stationary and pulled back to its mean. With $\varphi=1$ the series has a unit root: it is a random walk, shocks never fade and the best forecast is the last value. The question matters for trading, because a mean-reversion strategy needs the first case and a trend strategy can live with the second. The Dickey-Fuller test rewrites the model as $\Delta x_t=\gamma x_{t-1}+\varepsilon_t$ with $\gamma=\varphi-1$ and tests the null $\gamma=0$ (unit root) against $\gamma<0$ (stationary).

\Delta x_t=\alpha+\beta t+\gamma x_{t-1}+\sum_{i=1}^{p}\delta_i\,\Delta x_{t-i}+\varepsilon_t,\qquad \mathrm{DF}=\frac{\hat\gamma}{\mathrm{se}(\hat\gamma)},\qquad t_{1/2}=\frac{\ln(1/2)}{\ln\varphi}

(1)

The augmented version (ADF) adds lagged differences $\Delta x_{t-i}$ so that the residuals are free of autocorrelation. The statistic is the usual $t$ -ratio of $\hat\gamma$ , but under the null it does not follow Student's $t$ distribution. It follows the Dickey-Fuller distribution, whose critical values are more negative: about $-2.86$ at 5% with a constant and about $-3.41$ with a constant and a trend. Reject the unit root only when the statistic is below the critical value. Failing to reject is weak evidence: the test has little power when $\varphi$ is close to one or the sample is short, so a stationary but persistent series is often mistaken for a random walk. For a stationary AR(1) the half-life of a shock is $\ln(1/2)/\ln\varphi$ .

Reading an ADF resultA regression of

\Delta x_t

x_{t-1}

gives

\hat\gamma=-0.08

with standard error

0.03

, so the statistic is

-0.08/0.03\approx-2.67

. Against the 5% critical value

-2.86

this is not low enough: do not reject the unit root. The implied

\hat\varphi=0.92

would have a half-life of

\ln(0.5)/\ln(0.92)\approx8.3

days if the series were really stationary, so a statistic of

-3.4

instead would clear the bar and the series could be treated as mean-reverting.

ADF null: a unit root. Reject only when the statistic is below the Dickey-Fuller critical value.
Low power: a persistent stationary series often fails to reject, so absence of rejection is not proof.
Half-life of a shock in a stationary AR(1): $\ln(1/2)/\ln\varphi$ .

Lab · what's graded

The Dickey-Fuller coefficient15 pts
A half-life15 pts
The test statistic15 pts
Reject or not15 pts
Match the vocabulary15 pts
Running a unit-root test10 pts
Not rejecting is not proving15 pts

Is there a unit root?

\Delta x_t=\alpha+\beta t+\gamma x_{t-1}+\sum_{i=1}^{p}\delta_i\,\Delta x_{t-i}+\varepsilon_t,\qquad \mathrm{DF}=\frac{\hat\gamma}{\mathrm{se}(\hat\gamma)},\qquad t_{1/2}=\frac{\ln(1/2)}{\ln\varphi}

(1)

Reading an ADF resultA regression of

\Delta x_t

x_{t-1}

gives

\hat\gamma=-0.08

with standard error

0.03

, so the statistic is

-0.08/0.03\approx-2.67

. Against the 5% critical value

-2.86

this is not low enough: do not reject the unit root. The implied

\hat\varphi=0.92

would have a half-life of

\ln(0.5)/\ln(0.92)\approx8.3

days if the series were really stationary, so a statistic of

-3.4

instead would clear the bar and the series could be treated as mean-reverting.

ADF null: a unit root. Reject only when the statistic is below the Dickey-Fuller critical value.
Low power: a persistent stationary series often fails to reject, so absence of rejection is not proof.
Half-life of a shock in a stationary AR(1): $\ln(1/2)/\ln\varphi$ .

Lab · what's graded

The Dickey-Fuller coefficient15 pts
A half-life15 pts
The test statistic15 pts
Reject or not15 pts
Match the vocabulary15 pts
Running a unit-root test10 pts
Not rejecting is not proving15 pts

Is there a unit root?

Lab · what's graded

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Unit roots and the ADF test

Is there a unit root?

Lab · what's graded

Log in to run your code.