An Introduction to Econometric Theory

6More on Distributions

6.1 Random Vectors

The first task of this chapter is to extend the analysis of one or two random variables to an arbitrary number. The tools of matrix algebra are of course brought into play. Let

be an ‐vector whose elements are random variables. With a different set of conventions for matrices, note that the capital‐letter convention for random variables has to be foregone here. Without enough distinct symbols to allow notational conventions to apply universally, there is no alternative to just keeping in mind how a particular object has been defined.

The new idea is the generalization of mean and variance to the vector case. The expected value of is, straightforwardly enough, the vector whose elements are the expected values of the elements of :

However, generalizing the idea of the variance requires more than a vector of variances, because the random elements don't exist in isolation. Covariances also need to be represented. The key idea is the expected outer product ...