# Augmented Dickey-Fuller test: which specification for my data?

I've a yearly time series variable. My aim is to forecast the variable using ARIMA methods. For this purpose I need to know the order of integration of the series. The data set is provided here.

After plotting the series, I get the following

So, clearly the data series has an upward time trend.

Then I have applied the Augmented Dickey Fuller test with urca package using the following codes

summary(ur.df(Production, type = "drift",lags = 13, selectlags = "AIC"))


The results are as under:

From this above result, it is clear that the data series is non-stationary (as we fail to reject the $H_0$) ($\tau$ is not significant). Also $\phi_1$ is significant which implies that there is a drift. So, the model seems to be a random walk model with drift.

Secondly, I use the ADF test with 'drift and time trend' using the following code:

summary(ur.df(Production, type = "trend",lags = 13, selectlags = "AIC"))


The following results are obtained:

Now the $\tau_3$ is statistically significant at 5% level implying that the data is stationary. However, according to the values of $\phi_2$ and $\phi_3$ (both are significant at 5% level) which implies that there is drift as well as time trend.

I'm confused at this point. Which ADF model to use? and How to interpret the model results?

• How is the progress? What do you think about my answer? – Richard Hardy Mar 1 '18 at 6:19
• I understand it. But can you provide me the details mathematical background about the drift, trend specification? – Bidyut Ghosh Mar 3 '18 at 4:43