Stationarity Testing on different time series data

Question

For linear regression modeling, I have macroeconomic data that goes from 1985-2016 which i will use as my independent variable. My dependent variable data ranges from 2002-2016. My question is for stationarity testing do I run those tests on macroeconomic data from 1985 onwards or from 2002 onwards since that is what i would be using to model. The reason as to why I am asking this question is because what if a series is stationary from 1985 to 2016 but is non-stationary from 2002-2016.

Answer 1

Say you have your regressand $y$, the matrix of regressors $X$ and a regression model

$$ y = X\beta + \varepsilon. $$

Say you have a relevant sample (of length $T$) of $y$ and $X$. Say you also have an initial sample (of size $\tilde{T}$) of $X$. Taken the initial sample and the relevant sample together yields an extended sample (of size $\tilde{T}+T$) of $X$.

You have to ensure that the data generating process (DGP) of the regressand as well as the DGPs of the columns of the regressor matrix are stationary in the period of the relevant sample. There are no requirements on the DGPs of the initial sample as it is not being modelled.

What may happen is that you cannot reject a unit root (or some other form of nonstationarity) for the relevant sample, but can reject it for an extended sample. Then you have to decide whether the behaviour of the series has changed between the initial sample and the relevant sample.

• If you believe there was a structural change so that the DGP that generated the relevant sample was not quite the same as the DGP that generated the initial sample, then you could just ignore the test results for the extended sample.
• If, on the other hand, you believe that the DGP has stayed the same, then with a longer sample you may have more power in the tests. You could say that the results from the extended sample should be trusted more than the ones from the relevant sample.

So it all depends on what you assume about the DGP in the relevant vs. the initial sample.