Lag operator

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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={X1,X2,}

then

LXt=Xt1 for all t>1

or similarly in terms of the backshift operator B: BXt=Xt1 for all t>1. Equivalently, this definition can be represented as

Xt=LXt+1 for all t1

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

L1Xt=Xt+1

and

LkXt=Xtk.

Lag polynomials

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

εt=Xti=1pφiXti=(1i=1pφiLi)Xt

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)Xt=θ(L)εt

where φ(L) and θ(L) respectively represent the lag polynomials

φ(L)=1i=1pφiLi

and

θ(L)=1+i=1qθiLi.

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

Xt=θ(L)φ(L)εt,

means the same thing as

φ(L)Xt=θ(L)εt.

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.

An annihilator operator, denoted [ ]+, removes the entries of the polynomial with negative power (future values).

Note that φ(1) denotes the sum of coefficients:

φ(1)=1i=1pφi

Difference operator

Main page: Finite difference

In time series analysis, the first difference operator  :Δ

ΔXt=XtXt1ΔXt=(1L)Xt.

Similarly, the second difference operator works as follows:

Δ(ΔXt)=ΔXtΔXt1Δ2Xt=(1L)ΔXtΔ2Xt=(1L)(1L)XtΔ2Xt=(1L)2Xt.

The above approach generalises to the i-th difference operator ΔiXt=(1L)iXt .

Conditional expectation

It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let Ωt be all information that is common knowledge at time t (this is often subscripted below the expectation operator); then the expected value of the realisation of X, j time-steps in the future, can be written equivalently as:

E[Xt+j|Ωt]=Et[Xt+j].

With these time-dependent conditional expectations, there is the need to distinguish between the backshift operator (B) that only adjusts the date of the forecasted variable and the Lag operator (L) that adjusts equally the date of the forecasted variable and the information set:

LnEt[Xt+j]=Etn[Xt+jn],
BnEt[Xt+j]=Et[Xt+jn].

See also

References