Chap 14 section 5 multivariate time series -Vector notation

[Bill SD]

Hi,
I really appreciate the appendix explaining Eigen values.
But Found it difficult to understand the core reading and provided examples in Chapter 14 section 5. I have two specific questions:

Chap 14 section 5 "multivariate time series" (Pg 33 in 2019 version) states: "The components of X_t will be denoted X_t(1),...X_t(m)"

Question 1) Is this a one-step ahead forecast or simply the same as Xt-1, Xt-2,...Xt-m?

The Chapter 14 Summary sheet says that an AR(2) time series can be shown as X_t=AX_t-1+et

Question 2) How exactly does the vector notation (eg. X_t) work- How would one know that 'X_t' notation means a vector only including the terms Xt and Xt-1; while the notation X_t-1 refers to a vector only including the terms Xt-1 & Xt-2?

 

[Andrew Martin]

Hello

Question 1)

Here we have m different variables. Xt(1) and Xt(2) are random variables representing the values taken by two different processes at time t. They are not meant to represent values of the same process at different times. Note however that although the processes are different, the values of one process may be related to the values of another, see Example 2 in the notes for an example.

If we have just a few different variables then we can represent them easily with different letters. For example, in the case where we have 3 different processes, then we can represent these as, say Xt, Yt and Zt. Using the notation in the notes, this is Xt^(1), Xt^(2) and Xt^(3)

Question 2)

This is not clear without explanation. Noting that an AR(2) can be written as a VAR(1) requires defining the random vector denoted by Xt underscore as (Xt, Xt-1). For an AR(3), we would need to define Xt underscore as (Xt, Xt-1, Xt-2). In terms of what is shown in the summary, it should be clear to what the vectors and matrix relate, as the matrix equation is above it.

Hope this helps!

Andy