Lag operator

In time series analysis, the lag operator (L) or backshift operator (B) operates on an element of a time series to produce the previous element. For example, given some time series

X

=

{

X

1

,

X

2

,

…

}

{\displaystyle X=\{X_{1},X_{2},\dots \}}

then

L

X

t

=

X

t

−

1

{\displaystyle LX_{t}=X_{t-1}}

for all

t

>

1

{\displaystyle t>1}

or similarly in terms of the backshift operator B:

B

X

t

=

X

t

−

1

{\displaystyle BX_{t}=X_{t-1}}

for all

t

>

1

{\displaystyle t>1}

. Equivalently, this definition can be represented as

X

t

=

L

X

t

+

1

{\displaystyle X_{t}=LX_{t+1}}

for all

t

≥

1

{\displaystyle t\geq 1}

The lag operator (as well as backshift operator) can be raised to arbitrary integer powers so that

L

−

1

X

t

=

X

t

+

1

{\displaystyle L^{-1}X_{t}=X_{t+1}}

and

L

k

X

t

=

X

t

−

k

.

{\displaystyle L^{k}X_{t}=X_{t-k}.}

Lag polynomials

Polynomials of the lag operator can be used, and this is a common notation for ARMA (autoregressive moving average) models. For example,

ε

t

=

X

t

−

∑

i

=

1

p

φ

i

X

t

−

i

=

(

1

−

∑

i

=

1

p

φ

i

L

i

)

X

t

{\displaystyle \varepsilon _{t}=X_{t}-\sum _{i=1}^{p}\varphi _{i}X_{t-i}=\left(1-\sum _{i=1}^{p}\varphi _{i}L^{i}\right)X_{t}}

specifies an AR(p) model.

A polynomial of lag operators is called a lag polynomial so that, for example, the ARMA model can be concisely specified as

φ

(

L

)

X

t

=

θ

(

L

)

ε

t

{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}}

where

φ

(

L

)

{\displaystyle \varphi (L)}

and

θ

(

L

)

{\displaystyle \theta (L)}

respectively represent the lag polynomials

φ

(

L

)

=

1

−

∑

i

=

1

p

φ

i

L

i

{\displaystyle \varphi (L)=1-\sum _{i=1}^{p}\varphi _{i}L^{i}}

and

θ

(

L

)

=

1

+

∑

i

=

1

q

θ

i

L

i

.

{\displaystyle \theta (L)=1+\sum _{i=1}^{q}\theta _{i}L^{i}.\,}

Polynomials of lag operators follow similar rules of multiplication and division as do numbers and polynomials of variables. For example,

X

t

=

θ

(

L

)

φ

(

L

)

ε

t

,

{\displaystyle X_{t}={\frac {\theta (L)}{\varphi (L)}}\varepsilon _{t},}

means the same thing as

φ

(

L

)

X

t

=

θ

(

L

)

ε

t

.

{\displaystyle \varphi (L)X_{t}=\theta (L)\varepsilon _{t}.}

As with polynomials of variables, a polynomial in the lag operator can be divided by another one using polynomial long division. In general dividing one such polynomial by another, when each has a finite order (highest exponent), results in an infinite-order polynomial.