What is the difference between RSE in training set and test set? Is the process of calculating Residual Standard Error in Training Set and Test Set same? 
 A: Yes, the process of calculating the Residual Standard Error(RSE) in Training Set and Test Set are the same. The only difference is you use a different data set for each: a TRAIN and a TEST data set, usually corresponding to 70% and 30% of the original data set, respectively. 
There is a third type of RSE, the Generalization RSE. Then, the three types of RSE are:


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*Training RSE: computed over the TRAIN set. The training RSE is biased in the sense that it is an smaller (optimistic) estimator of the actual RSE of the model, and thus, rarely used as a measure of model assessment. 

*Generalization RSE: (theoretically) computed over the complete population from where the sample was taken. Realistically, the distribution of the population is not known, and thus the Generalization RSE cannot be practically computed. 

*Test (or empirical) RSE: computed over the TEST set. If the sample is iid then the RSE of the TEST set is used as a consistent estimator of the Generalization RSE.

A: No, it is not the same. When you calculate the Training Set RSE, you measure deviations of the model you generated, relative to the data set used to generate that model. On the other hand, the Test Set RSE measures deviations of the model you generated with some new data, not used previously to generate this model.
For example, let's say that you create a linear model "Sales (Y) VS. Advertising Expenses (X)". For this you use monthly data from 2010 to 2014 and generate a model from this, let's say it is: Sales = 20 + 0.3*Ad. This model will inevitably have some deviation from your data set (2010-2014) as no model will perfectly fit a given data set. A measure of this deviation is the training set RSE. On the other hand, you may want to test your model with future data. So let's say you have data for the first months of 2015, then you need to calculate the deviation of your model with these new data, that were not used to generate your model. A measure of this deviation is the test set RSE.
