Let's think about the standard linear regression model that we encounter in our introductory econometrics courses:
y = Xβ + ε . (1)
By writing the model in this form, we've already made two assumptions about the stochastic relationship between the dependent variable, y, and the regressors (the columns of the X matrix). First, the relationship is a parametric one - hence the presence of the coefficient vector, β; and second, the relationship is a linear one. That's to say, the model is linear in these parameters. If it wasn't, we wouldn't be able to write the model in the form given in equation (1).
However, the model isn't fully specified until we lay out any assumptions that are being made about the regressors and the random error term, ε. Now, let's consider the full set of (rather stringent) assumptions that we usually begin with:
- The X matrix has full column rank, say "k".
- The columns of X are either non-random (strictly, "fixed in repeated samples"); or if they are random, they are independent of the values of ε.
- The mean of the distribution that generates the (unobserved) elements of the ε vector is zero.
- The variance of the distribution that generates each elements of ε is the same (say, σ2). That is, the errors are "homoskedastic".
- The unobserved values of the ε vector are pair-wise uncorrelated. That is, they don't exhibit any "autocorrelation".
- The random errors are generated according to a Normal process.
The last of these assumptions can be relaxed in quite a general way without affecting any of the usual results that we typically establish by using it. (See here.)
You'll remember we can combine assumptions 3 to 5, and express them in the form:
ε ~ [0 , σ2In] , (2)
and we usually describe the statement, (2), by saying that the errors of the model are "spherically distributed".
Students have often asked me where this last this piece of language comes from.
A while back I put together a handout that discusses the notion of "spherically distributed errors", and I'm using it right now with my introductory graduate econometrics course. Rather than replicate the information in that handout here, you can download the pdf file.
If you use it with your own classes, an acknowledgment would be appreciated.
© 2012, David E. Giles