Is there a preference in the regression performance metric for regression models with the same type of loss minimization? I applied two regression models (ordinary least square (OLS) and linear absolute regression) to the same dataset, where this dataset is split into train and test sets.
Two performance measures are used to check the accuracy of linear regression models:
MSE stands for mean square error.
MAD stands for mean absolute deviation.
I found that the model fit by OLS will have a lower MSE value on unseen data than one fit using linear absolute regression? 
On the other hand, the model that is fitted by linear absolute regression will have a lower MAD value on unseen data than the one that is fitted using OLS.
Therefore, if I use the MSE as a regression performance measure, I will end up saying that the OLS model is the best, and contradictorily, if I use MAD as a regression performance measure, I will say that absolute linear regression is the best?
There is a claim by my colleagues that performance metrics will prefer regression models with the same type of loss minimization. In other words, if I have a comparison study, I can't use MAD alone and say that this absolute linear regression is the best choice because if I report MSE, the OLS model is better.
In my recent question, Dave's answer showed counter-examples on both real and simulated datasets.
My question is as follows:
If their claim is not correct how would you phrase a counter-argument in a few sentences and in a logical manner, not only R codes? If they are correct, why does this happen and how?
 A: Dave's answer has nothing to do with whether there's bias in an in-sample metric vs. an out-of-sample metric when the algorithm optimizes the in-sample metric.  His answer addresses whether minimizing the in-sample metric necessarily also minimizes the (expected) out-of-sample metric (Edit: it doesn't); it says nothing about the relative values of the two.  The bias issue states that if you do minimize an in-sample metric, the corresponding out-of-sample metric can be expected to be worse; it says nothing about whether some other objective function could improve the OOS metric.
A: 
Since the out-of-sample data are different from the in-sample data, all bets are off when it comes to what the out-of-sample metric chooses as its preferred model. In some sense, we are tuning the in-sample loss function as a hyperparameter in order to achieve the best out-of-sample performance on our metric of choice. If the out-of-sample metric prefers a model trained with a different loss function, so be it! That’s why we tune the hyperparameter.

I would present the argument like that. I also would be comfortable giving simulations or empirical data where, for example, out-of-sample square loss prefers a model that was trained with absolute loss than with square loss (such as the examples I gave with Iris and simulations).
