Note that $\log(x_1^2)=2\cdot\log(x_1)$. Consequently, $\text{cor}(\log(x_1),\log(x_1^2))=1$. The log of the square adds no value when the log is already there.


Is using ACF and PACF plots to check whether residuals follow and AR(1) process a valid methodology? I suppose it is valid unless there is something special about this model which I do not know. The plots suggests AR(0), AR(1) or AR(5) depending on how sensitive to marginally significant ACs and PACs you want to be. (An AR(12) could also be considered but I ...

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