I want to understand the importance of optimizing the correct loss function. Say that I am building a linear regression model $p$ for predicting some values $y_1,\ldots,y_n$.
I choose to fit my linear model such that it minimizes the mean squared errors. Now I send my model off to a statistical prediction competition, where they instead of using MSE as an error metric uses mean absolute error (so no squares). How will this affect my models predictive power? In general, what can be said about the importance of optimizing the "correct" loss function?
Edit: If one should put the question in a more specific context, then that context would be predictive modelling competitions like those featured on Kaggle.com. I want to understand the importance of choosing models and loss functions which correspond to the evaluation metric for the competition. One reason for this is this comment by the winner of a Kaggle competition.