I don´t think this is correct here if we consider MSE to be the sqaure of RMSE. For instance, you have a series of sampled data on predictions and observations, now you try to do a linear regresion: Observation (O)= a + b X Prediction (P). In this case, the MSE is the sum of squared difference between O and P and divided by sample size N.
But if you want to measure how linear regression performs, you need to calculate Mean Squared Residue (MSR). In the same case, it would be firstly calculating Residual Sum of Squares (RSS) that corresponds to sum of squared differences between actual observation values and predicted observations derived from the linear regression.Then, it is followed for RSS divided by N-2 to get MSR.
Simply put, in the example, MSE can not be estimated using RSS/N since RSS component is no longer the same for the component used to calculate MSE.