Should this regression be tested for stationarity? This is a fairly basic question I am struggling with. I am using the oil prices I have currently to forecast the number of cars bought 8 months from now using the regression below. Would this regression be considered a time-series and thus require a test of stationarity? I believe this is a time-series band thus stationarity needs to be tested but I'm getting feedback that this is more of a simple linear regression and I'm unable to argue my case. 
\begin{eqnarray*}
\\carsbought_{t+8} = 7.1+ 16.42*oilprice_{t}
\end{eqnarray*}
Thanks
 A: Yes, the data is a sequence of observations in time, so a time series is probably the way to go. The first thing I would check is the presence/absence of autocorrelation both on each individual series and between the two (I am pretty sure there are, as the oil price today is very likely to influence oil price in a few days)
If those autocorrelations are there, you can then check for stationarity, differenciate the series as many times as you need before achieving stationarity, and then fit a 2-component vectorial (cars bought, oil price) time series model (VAR is often used here)
This would be the "orthodox" approach. However, sometimes a "wrong" model can make it just as well. I would try validation techniques for linear regression models (QQ-plots, autocorrelation on residuals tests and so on...). Most likely, you will see that the residuals (which should be just noise) show some recogniseable pattern
A: You have a good possibility of both dependent and independent series non-stationary, in this case regression results are junk in many cases. For instance, if you have unit roots in series and no cointegration then the whole construct is useless. Therefore, yes, you have to look at both series and understand whether they're stationary or not. What's important is the sample time frame. Financial and economic time series sometimes can be stationary or not depending on the time frame. A typical AR(1) or ornstein uhlenbek type of process can have properties of unit root in short range and stationary properties over long periods.
For instance, here's the oil price index in two different time frames. One is clearly nonstationary the other one may or not look stationary.


