Vector autoregressions: preliminaries
Suppose we are observing two stocks and their respective returns are A vector autoregression for the pair is one way to take into account their interdependence. This theory is undeservedly omitted from the Guide by A. Patton.
Required minimum in matrix algebra
Matrix notation and summation are very simple.
Matrix multiplication is a little more complex. Make sure to read Global idea 2 and the compatibility rule.
The general approach to study matrices is to compare them to numbers. Here you see the first big No: matrices do not commute, that is, in general
The idea behind matrix inversion is pretty simple: we want an analog of the property that holds for numbers.
Some facts about determinants have very complicated proofs and it is best to stay away from them. But a couple of ideas should be clear from the very beginning. Determinants are defined only for square matrices. The relationship of determinants to matrix invertibility explains the role of determinants. If is square, it is invertible if and only if (this is an equivalent of the condition for numbers).
Here is an illustration of how determinants are used. Suppose we need to solve the equation for where and are known. Assuming that we can premultiply the equation by to obtain (Because of lack of commutativity, we need to keep the order of the factors). Using intuitive properties and we obtain the solution: In particular, we see that if then the equation has a unique solution
Let be a square matrix and let be two vectors. are assumed to be known and is unknown. We want to check that solves the equation (Note that for this equation the trick used to solve does not work.) Just plug
(write out a couple of first terms in the sums if summation signs frighten you).
Transposition is a geometrically simple operation. We need only the property
Variance and covariance
Property 1. Variance of a random vector and covariance of two random vectors are defined by
Note that when variance becomes
Property 2. Let be random vectors and suppose are constant matrices. We want an analog of In the next calculation we have to remember that the multiplication order cannot be changed.