All About Spherically Distributed Regression Errors

(This article was first published on Econometrics Beat: Dave Giles' Blog, and kindly contributed to R-bloggers) This post is based on a handout that I use for one of my courses, and it relates to the usual linear regression model,                                  y = Xβ + εIn our list of standard assumptions about the error term in this linear multiple regression model, we include one that incorporates both homoskedasticity and the absence of autocorrelation. That is, the individual values of the errors are assumed to be generated by a random process whose variance (σ2) is constant, and all possible distinct pairs of these values are uncorrelated. This implies that the full error vector, ε, has a scalar covariance matrix, σ2In. We refer to this overall situation as one in which the values of the error term follow a “Spherical Distribution”. Let's take a look at the origin of this terminology.The following discussion is quite general, so you'll realize that it applies to any random variables, not just the error term in our regression model. Further, so that we can look at some diagrams, let’s consider the special [...]