\(\begin{equation} \label{eq:5} X_t^{\prime}=X_t-T_t=X_t-\beta t \end{equation}\)
After calculating r1 for the detrended series X't, the lag-one autoregressive was removed from the X't:
\(\begin{equation} \label{eq:6} Y_t^{\prime}=X_t^{\prime}-r_1 X_{t-1} \end{equation}\)
Tt the identified trend and Y't the residual mixed (Yt):