# Dickey–Fuller unit-root test

Dickey-Fuller test for GDP
sample size 14
unit-root null hypothesis: a = 1

test with constant
model: (1-L)y = b0 + (a-1)*y(-1) + e
1st-order autocorrelation coeff. for e: 0.060
estimated value of (a - 1): -0.784054
test statistic: tau_c(1) = -2.88716
p-value 0.07195

Dickey-Fuller test for Arrivals
sample size 13
unit-root null hypothesis: a = 1

test with constant
model: (1-L)y = b0 + (a-1)*y(-1) + e
1st-order autocorrelation coeff. for e: -0.247
estimated value of (a - 1): 0.321498
test statistic: tau_c(1) = 4.63155
p-value 1


Can you tell me if P value is 1/0.07 then it's stationary or not?

• Are you just asking how to determine if a p-value is significant? – gung - Reinstate Monica Dec 7 '15 at 14:55
• yes, just I want to determine if they are stationary or non-stationary – Marieta Turabelidze Dec 7 '15 at 15:35
• I am skeptical as to the usefulness of the test for fourteen observations – Christoph Hanck Dec 7 '15 at 19:30

The null hypothesis of the ADF test is the unit root, i.e. the series is nonstationary. To see this, take the simple case of an AR(1) model:

$$y_t = c + \phi y_{t-1} + \epsilon_t \tag{1}$$

and the unit root corresponds to $\phi = 1$, would you agree? So we formulate the hypothesis:

$$H_0 : \phi = 1 \implies \text{unit root}$$

$$H_1 : \phi <1 \implies \text{stationarity}$$

Why is the alternative $\phi <1$ and not the general $\phi \neq 1$? You will have to think about that. It turns out the hypothesis is more easily tested if we subtract $y_{t-1}$ from both sides in 1 getting

$$\Delta y_t = c + \gamma y_{t-1} + \epsilon_t$$

where $\gamma = \phi -1$. This can be estimated by OLS and the unit root corresponds to the usual t-statistic

$$t = \frac{\widehat{\gamma}}{SE(\widehat{\gamma})}$$

the catch being however that under $H_0$ the statistic does not follow the normal but the Dickey-Fuller distribution whence the critical values come from.

So you tell me, if the null hypothesis is of unit root and the p-value is too large what is the conclusion?

I think the question is about interpreting the output of unit root test.

First understand the decision rule for any statistical test:

• If p-value < level of significance (alpha); then null hypothesis is rejected.
• If p-value > level of significance (alpha); then we fail to reject the null hypothesis.

Level of significance (alpha) is chosen by the researcher. Most common value is 0.05

Now understand, Unit Root Test:

• Null Hypothesis is "Series has a unit root"
• Alternative Hypothesis is "Series does not have a unit root"

Finally Stationarity Vs Unit Root

• If a series has a unit root, it is non-stationary

So, if p-value (for unit root test) > 0.05 (at 5% level of sig); then Then the series has a unit root, hence it is non-stationary.