\(\begin{equation} \label{eq:5}
X_t^{\prime}=X_t-T_t=X_t-\beta t
\end{equation}\)
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):