When we estimate a regression model, the differences between the actual and "predicted" values for the dependent variable (over the sample) are termed the "residuals". Specifically, if the model is of the form:
y = Xβ + ε , (1)
and the OLS estimator of β is b, then the vector of residuals is
e = y - Xb . (2)
Any econometrics student will be totally familiar with this.
The elements of e (the n residuals) are extremely important statistics. We use them, of course, to construct other statistics - e.g., test statistics to be used for testing the validity of the underlying assumptions associated with our regression model. For instance, we want to check, are the errors (the elements of the ε vector) serially independent; are the errors homoskedastic; are they normally distributed; etc.?
What a lot of students don't learn is that these residuals - let's call them "Ordinary Residuals" - are just one type of residuals that are used when analysing the regression model. Lets take a look at this.