Difference between mean square residual and mean square error In regressions, can someone explain me the difference between mean square residual and mean square error. What is the significance of each?
 A: The answer to this question depends on how you define mean squared error (MSE).
In the context of regression, some define it to be 
$MSE = \sum (y-\hat{y})^2/(n-p)$ 
where p is the number of parameters in the regression (including the intercept).  Note that since residuals are $y-\hat{y}$, this is equivalent to mean squared residuals (MSR). Note that this formula is generally used because it provides an unbiased estimate of the variance of the errors.
It is important to note that in the context of regression, the residuals are not the actual errors $(\epsilon)$, which are random variables.  However, the residuals are estimates of the errors under the assumed model,
$y-\hat{y} = \hat{\epsilon} $
where $y$ is the observed value and $\hat{y}$ is the predicted value.
A: I'll try to explain how I see it from statistical point of view.
I don't think MSE and MSR are the same thing (however most people don't differentiate between those two I guess).
Let's say that you do a simulation of data that can be described using regression model. Let's say that you generate the data "randomly" around parabola curve and you have the exact regression function since you did generate data around that function.
Then the "random" part which "secures" that the data are not (most likely) directly on the parabola curve are actually errors.
However, in reality, you usually don't know the theoretical regression function. Then you can only estimate such regression model. The differences between the observed values and your estimated model are then called residuals. So it can be said that residuals are somehow estimate of theoretical errors.
Hope this helps.
A: There is no difference between the mean square residual and mean square error.
A: In short, Mean squared error (MSE) is the square of RMSE. For linear regression standard equation: Y=a+bX, considering MSE equals to the sum of squared differences between actual sample values of X´ and Y´ that are used to fit the linear model, and divided by number of paird samples (n).
For Mean Squared Residues (MSR), it should start firstly to know Least Squared Method in linear regression. Simply put, this method minimize the sum of squared difference between actual Y and estimated Y (sum of squared residues, SSr), corresponding to ∑(Y-Y´)^2. SSr divided by n-2 equals to MSR. And this is mainly useful for analyzing overall significance of linear regression, but also being an essential component of correlation of determination. By ensuring MSR less, performace of different linear regression models could be compared. 
From my point of understandings, MSR account for dispersions of actual Y and estimated Y derived from linear regresion (thus considering the Ymean), whereas MSE is a direct comparison for prediction errors between predictions and observations.
