VAP/Multivariate Time Series

[ActAus]

Hi all,

I'm not sure if someone has asked questions regarding expressing a univariate time series into a multivariate vector form (turning a AP(p) into a VAP(1)), or if there is a general rule about it.

My question specifically is how we know how many rows of the matrix to write when turning a AP(2) into a VAP(1). For example, the example the Notes (bottom of p45 of Ch13 under the heading "Multivariate time series"). In this example, there are 3 rows (Xt, Xt-1, Xt-2)T in the matrix. However, in an assignment question Q4.13 part (iii) Xn = 0.7Xn-1 - 0.1Xn-2 + en, the solution shows there are only 2 rows (Xn, Xn-1)T in the matrix. They are both AR(2), yet one VAP(1) has 3 rows and the other 2 rows. Is there a general rule of how many rows of matrix or both expressions are valid??

My second question is how does the matrix under the heading "Multivariate time series" (again bottom of p45 of Ch13) show that it is a vector in the form of Xt = AXt-1 + Bet. More specifically, where we do get the "Xt-1" part from (the matrix above)?

Thanks.

 

[John Lee]

Yeah. I see the issue. The summary (Ch13 p45) is an ARMA not an AR and that is the difference between the 2 questions. The approach for the AR one in the assignment is what I would do.

For the ARMA I can see the logic (so we have corresponding terms in the X and e vectors on the RHS) but you could also write it as:

\(\pmatrix{X_t \\ X_{t-1}} = \pmatrix{\alpha_1 & \alpha_2 \\ 1 & 0} \pmatrix{X_{t-1} \\ X_{t-2}} + \pmatrix{1 & \beta \\ 0 & 0} \pmatrix{e_t \\ e_{t-1}} \)