A paper I saw used the Pearson correlation together with MSE to measure the performance of a machine learning model. After doing some research, I have seen that using Pearson correlations to evaluate a model can work if the exact values do not need to be precisely approximated but rather guessing correctly whether the output should be low/high or in between.

So my question is whether using both MSE as a metric for precise values and correlation as a metric for the overall “shape” of the output data gives any advantage over using only one or the other.

Also for correlation metrics. I assume that a good correlation value does not necessarily mean that the model is accurate, but does a bad correlation value mean that the model is definitely performing poorly?

I would also be grateful if anyone could tell me if there are any papers or books that discuss the use of correlation as a performance metric for regression models, as I could only find the standard metrics such as MSE etc in the papers or books.