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 enter image description here

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: enter image description here

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:

enter image description here

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?

  • $\begingroup$ How is the progress? What do you think about my answer? $\endgroup$ Commented Mar 1, 2018 at 6:19
  • $\begingroup$ I understand it. But can you provide me the details mathematical background about the drift, trend specification? $\endgroup$ Commented Mar 3, 2018 at 4:43

1 Answer 1


I will give a somewhat informal answer.

To me the data looks stationary around a linear trend or perhaps an exponential trend with a small exponent. Even if there is a random walk component in it, it seems more or less negligible compared to the linear trend and the stationary variations around it. But the first test tells you otherwise; I wonder if this is because of lack of power or the unaccounted exponential trend or what.

Regarding the second test, the model allows for a quadratic time trend in the data (equivalent to a linear time trend in first-differenced data). It might capture the slight exponent trend well within the sample. But do you have grounds to expect a quadratic time trend in your data? If not, you have a substantive argument (instead of a statistical one) for not choosing this specification.

You can also try modelling, say, 70% of your data with different model specifications and extrapolating for the remaining 30%. By comparing the pseudo out-of-sample forecasts with the observed data you could get an idea of which model suits the data better. (My guess is an exponential trend with stationary variations around it should do quite fine.)

Side note: I have some doubts about whether ARIMA is an effective forecasting technique for yearly data. Sure, if you have no other variables and the only information available to you is the series itself, then ARIMA might be the best you can do. But this is not that often the case. E.g. if the series represents some aggregate food production by a country, I would guess some governmental institutions or banks might have (publicly available) forecasts superior to those of ARIMA. Or you could include more variables in the model, perhaps some leading indicators, and see if that produces better forecasts.


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