I have a series for Portuguese GDP from 1995 to 2015 (quarterly data).

The plot of GDP over time shows clearly a trend. I ran a Dickey Fuller and the series turned out to be stationary by using the option drift (p_value is close to zero).

How shall I interpret this result? There is clearly a trend so the series is (at least)not mean stationary. Running the Dickey Fuller,by adding the trend option, the series results non-stationary. When I notice a trend behaviour shall I always include a trend in my regression in order to make my analysis valid? Including more lags the situation doesn't improve. Can you help me with this puzzling outcome? Shall I consider the series non stationary and take the first difference to solve it?(I include stata outputs below)

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Normally you would expect $\operatorname{log}(GDP)$ to be integrated of order one and $\Delta \operatorname{log}(GDP)$ (the growth rate of $GDP$) to be stationary. The long-run average growth rate will be a couple of percent (rather than zero) per year so it is reasonable to include a drift term when testing for a unit root in $\operatorname{log}(GDP)$. This is the same as to say that $\operatorname{log}(GDP)$ has a linear time trend, or that $GDP$ is growing exponentially.

Meanwhile, if you allow for a linear trend in the augmented Dickey-Fuller test specification, that means you allow for a linear trend in the growth rate of $GDP$, or a quadratic trend in $\operatorname{log}(GDP)$. That would be difficult to interpret from the subject-matter perspective. Would you expect the growth rate to have a linear trend? I doubt that.

I doubt it makes sense testing for a unit root in the level of $GDP$, as the level is growing or diminishing exponentially, if we follow the ideas in the first paragraph. So if you conduct the test on $\operatorname{log}(GDP)$, perhaps your results will match the expected findings.

If they do not, then you have to make a choice: (1) believe the scheme above and blame the short sample period, a structural change or yet something else for a "weird" test result, or (2) ditch the scheme (say that it is it that is "weird") and follow the test result.

  • $\begingroup$ Is it possible that the initial ADF test rejected a unit root because the Portuguese GDP data were characterized by a deterministic, rather than stochastic trend? $\endgroup$ – Amaziah Oct 20 '17 at 17:58
  • $\begingroup$ @Amaziah, Yes, if the ADF test specification allowed for the deterministic trend. If the deterministic trend was not allowed for, the null hypothesis would have not been rejected in presence of a deterministic trend (because a deterministic trend is even stronger than a stochastic trend). $\endgroup$ – Richard Hardy Oct 20 '17 at 18:25

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