Log a squared variable or square a logged variable I'm estimating an OLS log-log model like so:
\begin{equation}
ln(y) = \alpha + log(x_1) + \varepsilon
\end{equation}
and now want to account for non-linearities in the response of $ln(y)$ to $log(x1)$.
Do I now estimate:
\begin{equation}
ln(y) = \alpha + log(x_1) + log(x_1)^2 + \varepsilon
\end{equation}
or:
\begin{equation}
ln(y) = \alpha + log(x_1) + log(x_1^2) + \varepsilon
\end{equation}
These obviously give me different results - my question is, which criteria do I use to make this decision?
In my case, $x_1$ is GDP and $y$ is carbon emissions.

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.
