Regression of a time series difference Suppose $x(t)$ and $y(t)$ are two time series. I regress $y(t)$ against $x(t)$, and obtain
$$y(t)=ax(t)+b+z(t) $$
for some regression constants $a, b$ and residue $z(t)$. Define $\Delta u(t):=u(t+1)-u(t)$ for a given time series $u(t)$. Now regress $\Delta y(t)$ against $\Delta x(t)$. It seems the following should hold
$$\Delta y(t) \sim a\Delta x(t)$$
However, it seems the actual regression coefficient turns out to be $0$.
Can someone give a model to account for this phenomenon? Is there where cointegration should be used?

That's the main difference between math and statistics in time series. If you assume that the actual data generating process (DGP) of $y(t)$ is your equation, then your reasoning applies. But if it's not, then many things may be happening. Basically you are trying to combine two time series that are supposed to have two independent DGPs, which come with noise and/or trend. Those elements are distorting your model in differences, and the noise of that model is greater than the one in levels. You should indeed use cointegration models, for example de Error Correction Model (ECM) or the ARDL-ECM Model, to verify if the variables are in fact related or not.
