Vector Autoregression Algebra, $M_t$, $L$

Question

In the paper here

http://www.ems.bbk.ac.uk/for_students/bsc_FinEcon/fin_economEMEC007U/VAR.pdf

It shows VAR(p) model as

$$ W_t = A_1W_{t-1} + A_2W_{t-2} + ... + A_pW_{t-p} + \epsilon_t $$

But then it makes a simplification and says the formula above equals to

$$ (I - A_1L - A_2L^2 - ... - A_pL^p)W_t = \epsilon _t $$

How does the author make this switch? Are all $W_t$ vectors somehow combined to give $L$? But then why is taking 2nd, 3rd, etc powers come into play?

Thanks,

Answer 1

A previous article in the same site-series, devoted to (scalar) AR-MA processes, explains that $L$ is delay operator: http://www.ems.bbk.ac.uk/for_students/bsc_FinEcon/fin_economEMEC007U/arma.pdf

Further, when one applies Z-transform (or, basically equivalent, Generating Functions), the $n-$delay operator maps to $z^{-n}$ (or $z^n$).

Answer 2

In much of the time series literature the notation is B instead of L as it is also called the backshift operator which to the first power maps the vector Xt into the vector Xt-1. Knowing this the change you saw in the formula was just a notational change using the backshift operator. However the polynomial formed in the characteristic polynomial and for univariate time series its roots determine stationarity or nonstationarity.