# Does this series have a unit root?

I am trying to figure out whether my time series is stationary or not. In order to do so I run four different tests: the three test from the Dickey-Fuller test (standard, drift and trend) given by the library urca and function ur.df and one corresponding to the Augemented Dickey Fuller test adf.test (library "tseries"). Here is how my time series looks like:

And here are the outcome from the tests:

I) UNIT-ROOT TEST (NO DRIFT, NO TREND)

x = ur.df(AC.HousePrices, type = "none", selectlags = "AIC")
summary(x)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
z.lag.1    -0.12044    0.04723   -2.55  0.01195 *
z.diff.lag -0.24261    0.08544   -2.84  0.00526 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03546 on 128 degrees of freedom
Multiple R-squared:  0.135, Adjusted R-squared:  0.1214
F-statistic: 9.985 on 2 and 128 DF,  p-value: 9.338e-05

Value of test-statistic is: -2.5499

Critical values for test statistics:
1pct  5pct 10pct
tau1 -2.58 -1.95 -1.62


II) UNIT-ROOT WITH DRIFT

x = ur.df(AC.HousePrices, type = "drift", selectlags = "AIC")
summary(x)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.005721   0.003748   1.526  0.12940
z.lag.1     -0.169456   0.056915  -2.977  0.00348 **
z.diff.lag  -0.217301   0.086602  -2.509  0.01336 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03528 on 127 degrees of freedom
Multiple R-squared:  0.1504,    Adjusted R-squared:  0.137
F-statistic: 11.24 on 2 and 127 DF,  p-value: 3.198e-05

Value of test-statistic is: -2.9774 4.4497

Critical values for test statistics:
1pct  5pct 10pct
tau2 -3.46 -2.88 -2.57
phi1  6.52  4.63  3.81


III) UNIT-ROOT WITH DRIFT AND DETERMINISTIC TREND

x = ur.df(AC.HousePrices, type = "trend", selectlags = "AIC")
summary(x)

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  1.491e-02  7.893e-03   1.889  0.06116 .
z.lag.1     -2.034e-01  6.227e-02  -3.266  0.00141 **
tt          -1.193e-04  9.022e-05  -1.322  0.18857
z.diff.lag  -2.001e-01  8.732e-02  -2.291  0.02361 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.03518 on 126 degrees of freedom
Multiple R-squared:  0.162, Adjusted R-squared:  0.1421
F-statistic: 8.121 on 3 and 126 DF,  p-value: 5.489e-05

Value of test-statistic is: -3.2656 3.5665 5.3323

Critical values for test statistics:
1pct  5pct 10pct
tau3 -3.99 -3.43 -3.13
phi2  6.22  4.75  4.07
phi3  8.43  6.49  5.47


IV) AUGMENTED DICKEY-FULLER TEST

 adf.test(AC.HousePrices)

data:  AC.HousePrices
Dickey-Fuller = -2.1267, Lag order = 5, p-value = 0.5238
alternative hypothesis: stationary


The way I see it.

• We can reject the null hypothesis in the first two tests and accept the alternative hypothesis since the absolute value of the test statistic is greater than the critical values. This will indicate that the series is stationary. Moreover, these results are in a 95 percent confidence interval.
• Nevertheless, the third test seems to indicate that at the 95 percent confidence interval we cannot reject the null hypothesis, which will indicate that the series is NOT stationary.
• On the other hand, the augmented Dickey-Fuller test seems to indicate that the series is nonstationary (large p-value that does not allow rejecting the null hypothesis).

The question then seems to be if the series has a drift and a trend. If it does, then the series is not stationary, but if it does not, then it is. Si there any way, to determine whether or not it has these two components?

• Recall that the null hypothesis of the ADF test is presence of unit root (nonstationarity)! – Richard Hardy Jul 11 '16 at 10:42
• Hi Richard, true. I re-edited the question. I also realized that adf.test checks for unit-root using drift and trend. Do you know of a way to see if the time series has a trend and drift? Cheers – Economist_Ayahuasca Jul 11 '16 at 11:12
• Try some basic time series textbook. Perhaps Zivot & Wang "Modeling Financial Time Series with S-PLUS", they have a good exposition of unit root tests. – Richard Hardy Jul 11 '16 at 11:18