I have a time series of climate data that I'm testing for stationarity. Based on previous research, I expect the model underlying the data to have an intercept term, a positive linear time trend, and some normally distributed error term. In other words, I expect the underlying model to look something like this: yt = a0 + a1t + β yt-1 + ut $ where ut is normally distributed. Since I'm assuming the underlying model has both an intercept and a linear time trend, I tested for a unit root with equation #3 of the simple Dickey-Fuller test, as shown: ∇yt = α0+α1t+δ yt-1+ut This test returns a critical value that would lead me to reject the null hypothesis and conclude that the underlying model is non-stationary. However, I question if I'm applying this correctly, since even though the underlying model is assumed to have an intercept and a time trend, this does not imply that the first difference ∇yt will as well. Quite the opposite, in fact, if my math is correct. Calculating the first difference based on the equation of the assumed underlying model gives: ∇yt = yt - yt-1 = [a0 + a1 + β yt-1 + ut] - [a0 + a1(t-1) + β yt-2 + ut-1] ∇yt = [a0 - a0] + [a1t - a1(t-1)] + β[yt-1 - yt-2] + [ut - ut-1] ∇yt = a1 + β * ∇yt-1 + ut - ut-1$ Therefore, the first difference ∇yt appears to only have an intercept, not a time trend. Because the underlying model has an intercept and a time trend, should I use the Dickey-Fuller test that includes an intercept and time trend when it tests for a unit root, or should I use the Dickey-Fuller test that only includes an intercept because the first difference of the original time series only has an intercept?