- #1

- 7

- 0

## Main Question or Discussion Point

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:

y

where u

∇y

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

Calculating the first difference based on the equation of the assumed underlying model gives:

∇y

∇y

∇y

Therefore, the first difference ∇y

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?

y

_{t}= a_{0}+ a_{1}t + β y_{t-1}+ u_{t}$where u

_{t}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:∇y

_{t}= α_{0}+α_{1}t+δ y_{t-1}+u_{t}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 ∇y_{t}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:

∇y

_{t}= y_{t}- y_{t-1}= [a_{0}+ a_{1}+ β y_{t-1}+ u_{t}] - [a_{0}+ a_{1}(t-1) + β y_{t-2}+ u_{t-1}]∇y

_{t}= [a_{0}- a_{0}] + [a_{1}t - a_{1}(t-1)] + β[y_{t-1}- y_{t-2}] + [u_{t}- u_{t-1}]∇y

_{t}= a_{1}+ β * ∇y_{t-1}+ u_{t}- u_{t-1}$Therefore, the first difference ∇y

_{t}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?