# Vector Autoregression Algebra, $M_t$, $L$

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,

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That's impossible to understand, without a definition of what $L$ is. But I guess one should have read the previous article that explains that $L$ is a delay operator. ems.bbk.ac.uk/for_students/bsc_FinEcon/fin_economEMEC007U/… – leonbloy Jul 10 '12 at 13:45
Whew - that makes so much more sense. :) Interestingly article says nothing about $L$ being an operator. Note: if you want to make this an answer I'll gladly accept. – BB_ML Jul 10 '12 at 13:49

## 2 Answers

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 $z^n$).

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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.

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