Is it correct to say that the two time series are correlated in levels, but not first differences?
$$x_{t}=\phi t+u_{t}\\ y_{t}=\rho t+\epsilon_{t}$$
where $\epsilon_{t}$ and $u_{t}$ are mutually orthogonal white noise processes, and $\phi$, $\rho>0$. The first differences of these time series are: $$\Delta x_{t}=\phi+\Delta u_{t}\\ \Delta y_{t}=\rho+\Delta \epsilon_{t} $$
The level series will clearly be correlated as they both have a deterministic time trend. The first differences, however, do not. Is this reasoning correct? Thank you.