Lag Function

We now define the lag function

We further assume that

for any constant c and any variables x and z. We also use the following notation for any variable z and non-negative integer n.

We can express the AR(p) process

using the lag function as

or even

where 1 is the identity function and we use the notation (f+g)x to mean f(x) + g(x) for any functions f and g. This can also be expressed as

By Property 1 of Autoregressive Process Basic Concepts, this can also be expressed as

or

Note that

is a p^th degree polynomial which is equivalent to the characteristic polynomial of the AR(p) process, as described in Characteristic Equation for Autoregressive Processes. This polynomial can be factored (by the Fundamental Theorem of Algebra) as follows

where the values r₁, r₂, …, r_p are the characteristic roots of the AR(p) process.

Based on the vector φ = [φ₁, …, φ_p] of coefficients, we can define the operator φ(L)

and so an autoregression process can be expressed as

Observation: The lag function is also called the (back) shift operator and so sometimes the symbol B is used in place of L.