29-04-2021



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

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  • General Backshift Operator For an autoregressive model of orders p, The equation can be re-written using the backshift operator B to represent all of the related past lag points on the LHS, while.
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Ski Doo 850 Summit, Freeride Clutch Kit 2020 (3.0') 965 ramps. uncountable (linguistics) the changing of a tense when reporting what somebody said, for example when reporting the words ‘What are you doing?’ as ‘He asked me what I was doing.’ See backshift in the Oxford Advanced American Dictionary.

X={X1,X2,}{displaystyle X={X_{1},X_{2},dots },}

Backshift Rule

then

LXt=Xt1{displaystyle ,LX_{t}=X_{t-1}} for all t>1{displaystyle ;t>1,}

or similarly in terms of the backshift operator B: BXt=Xt1{displaystyle ,BX_{t}=X_{t-1}} for all t>1{displaystyle ;t>1,}. Equivalently, this definition can be represented as

Xt=LXt+1{displaystyle ,X_{t}=LX_{t+1}} for all t1{displaystyle ;tgeq 1,}

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

L1Xt=Xt+1{displaystyle ,L^{-1}X_{t}=X_{t+1},}

and

LkXt=Xtk.{displaystyle ,L^{k}X_{t}=X_{t-k}.,}

Lag polynomials[edit]

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{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)Xt=θ(L)εt{displaystyle varphi (L)X_{t}=theta (L)varepsilon _{t},}

Define Shifting

where φ(L){displaystyle varphi (L)} and θ(L){displaystyle theta (L)} respectively represent the lag polynomials

φ(L)=1i=1pφiLi{displaystyle varphi (L)=1-sum _{i=1}^{p}varphi _{i}L^{i},}

and

θ(L)=1+i=1qθiLi.{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,

Xt=θ(L)φ(L)εt,{displaystyle X_{t}={frac {theta (L)}{varphi (L)}}varepsilon _{t},}

means the same thing as

φ(L)Xt=θ(L)εt.{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.

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

Note that φ(1){displaystyle varphi left(1right)} denotes the sum of coefficients:

Operator
φ(1)=1i=1pφi{displaystyle varphi left(1right)=1-sum _{i=1}^{p}varphi _{i}}

Difference operator[edit]

In time series analysis, the first difference operator :{displaystyle nabla }

Backshift

Backshift Hours

Xt=XtXt1Xt=(1L)Xt.{displaystyle {begin{array}{lcr}nabla X_{t}&=X_{t}-X_{t-1}nabla X_{t}&=(1-L)X_{t}~.end{array}}}

Similarly, the second difference operator works as follows:

(Xt)=XtXt12Xt=(1L)Xt2Xt=(1L)(1L)Xt2Xt=(1L)2Xt.{displaystyle {begin{aligned}nabla (nabla X_{t})&=nabla X_{t}-nabla X_{t-1}nabla ^{2}X_{t}&=(1-L)nabla X_{t}nabla ^{2}X_{t}&=(1-L)(1-L)X_{t}nabla ^{2}X_{t}&=(1-L)^{2}X_{t}~.end{aligned}}}

The above approach generalises to the i-th difference operatoriXt=(1L)iXt.{displaystyle nabla ^{i}X_{t}=(1-L)^{i}X_{t} .}

Conditional expectation[edit]

It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let Ωt{displaystyle Omega _{t}} 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].{displaystyle E[X_{t+j}|Omega _{t}]=E_{t}[X_{t+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],{displaystyle L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}],}
BnEt[Xt+j]=Et[Xt+jn].{displaystyle B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}],.}

See also[edit]

References[edit]

  • Hamilton, James Douglas (1994). Time Series Analysis. Princeton University Press. ISBN0-691-04289-6.
  • Verbeek, Marno (2008). A Guide to Modern Econometrics. John Wiley and Sons. ISBN0-470-51769-7.
  • Weisstein, Eric. 'Wolfram MathWorld'. WolframMathworld: Difference Operator. Wolfram Research. Retrieved 10 November 2017.CS1 maint: discouraged parameter (link)
  • Box, George E. P.; Jenkins, Gwilym M.; Reinsel, Gregory C.; Ljung, Greta M. (2016). Time Series Analysis: Forecasting and Control (5th ed.). New Jersey: Wiley. ISBN978-1-118-67502-1.
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Backshift

English[edit]

Etymology[edit]

Backshifted

back +‎ shift

Noun[edit]

Examples

The sentence:

Fred says 'I play football'.

becomes the sentence:

Fred said that he played football.

backshift (countable and uncountable, pluralbackshifts)

  1. (grammar) The changing of the tense of a verb from present to past in reported speech.
  2. (statistics) A function on a stochastic process whose value on any random variable is the preceding random variable in the process.
Backshift

Verb[edit]

backshift (third-person singular simple presentbackshifts, present participlebackshifting, simple past and past participlebackshifted)

  1. (grammar,transitive) To change the tense of a verb from present to past in reported speech.
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