Is log-log model considered to be nonlinear? I am currently revising a paper, in which I tested an empirical model in the following form:
,
where EP is indicator of environmental performance, FDI - foreign direct investment which is the main independent variable of interest, Ci controlled variables which were also log-transformed and Vj - dummy variables.
I am confused with one of the reviewers' comments. Namely, the reviewer stated that the relationship between the main independent variable and the dependent variable could be nonlinear and that, therefore, nonlinear model should be estimated and reported. Here is the exact quote of the comment:

Many literature papers suggest a nonlinear effect of foreign direct investment on air pollution. Authors did not use nonlinear models to examine the effect of foreign direct investment on air pollution. I suggest that linear and nonlinear regressions should be considered in empirical models in order to compare the estimated results. 

To the best of my understanding, the main log-log model (previously mentioned) is derived from:

and such model already describes nonlinear relationship between the variables (or am I wrong?).
Could anyone give me advice on how I should approach this. My idea was to present level-level and log-log model estimations side by side as a response to the comment, but I am not sure if it is appropriate solution. Perhaps, there is some standard procedure I should be using for estimating possible nonlinear relationships?
The same reviewer also requested the explanation for using the log-log model, and I precisely used it because I believed that the relationship between variables could be non-linear (in addition to: the assumption of constant elasticity between the variables which was theory-driven; the variables using different measurement scales, skewness of the dependent variable values; and the fact that log-log provided a better fit for the data compared to level-level model). I wanted to ask if the arguments in this paragraph could be used as justification for using log-log model, and, finally, is there a need to justify the use of log-log model at all, if I present estimations for both log-log and level-level models?
I apologize for the long question and thank you in advance!
 A: I think by "non-linear" the reviewer refers to a modeling approach which is agnostic to the functional form relating the exposure to its response. Smoothers, GAMs, splines, or even some forms of model selection are ways of fitting "non-linear models".
I would respond possibly a couple of ways, depending on what your analysis is supposed to do.


*

*Thank you for your suggestion.

*We inspected the trend of the fit with scatterplots, residual tests, and other forms of diagnostics and found that the model fit was good/not good. We considered departures from the specified model form by fitting higher order terms and found...

*Since the objective of this analysis was inference, we focused on modeling an association with a direct interpretation. We used robust error estimates to ensure 95% CIs and p-values are consistent even in the presence of model misspecification.

*We fit a more general model using splines and summarized the predicted trends using a series of model smoothers which we are including for your review. We would/would not be amenable to including these figures as an online supplemenet.
