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2

Any data transformation to assure stationarity eliminates part of the signal in many cases the signal is not completely eliminated so you can still perform the required analyses but in some others as may the your case the signal is erased and the results seem to indicate your variables have lost predictible power although its predictive power may have been ...

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Suppose we estimate the regression model $$\triangle y_{t}=\alpha + \beta y_{t-1}+\varepsilon_{t}$$ This is actually quite similar to the Dickey-Fuller test. If $\beta=0$, then the process has a unit root. Let's proceed assuming that $\beta<0$, i.e. that the process is stationary. The first equation is also similar to the continuous time ...

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What you are saying might be correct for discrete time processes. In continuous time the process $$dX_t = X_t^2 dW_t,\quad X_0 > 0$$ is stationary but not mean reverting.

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