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

then

LX_{t}=X_{t-1}

for all

t>1

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

X_{t}=LX_{t+1}

for all

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}

and

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,

\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

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

where $\varphi (L)$ and $\theta (L)$ respectively represent the lag polynomials

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

and

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

means the same thing as

\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 $[\ ]_{+}$ , removes the entries of the polynomial with negative power (future values).

Note that $\varphi \left(1\right)$ denotes the sum of coefficients:

\varphi \left(1\right)=1-\sum _{i=1}^{p}\varphi _{i}

Difference operator

[edit]

In time series analysis, the first difference operator : $\Delta$

{\begin{aligned}\Delta X_{t}&=X_{t}-X_{t-1}\\\Delta X_{t}&=(1-L)X_{t}~.\end{aligned}}

Similarly, the second difference operator works as follows:

{\begin{aligned}\Delta (\Delta X_{t})&=\Delta X_{t}-\Delta X_{t-1}\\\Delta ^{2}X_{t}&=(1-L)\Delta X_{t}\\\Delta ^{2}X_{t}&=(1-L)(1-L)X_{t}\\\Delta ^{2}X_{t}&=(1-L)^{2}X_{t}~.\end{aligned}}

The above approach generalises to the i-th difference operator $\Delta ^{i}X_{t}=(1-L)^{i}X_{t}\ .$

Conditional expectation

[edit]

It is common in stochastic processes to care about the expected value of a variable given a previous information set. Let $\Omega _{t}$ 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[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:

L^{n}E_{t}[X_{t+j}]=E_{t-n}[X_{t+j-n}],

B^{n}E_{t}[X_{t+j}]=E_{t}[X_{t+j-n}].

References

[edit]

Hamilton, James Douglas (1994). Time Series Analysis. Princeton University Press. ISBN 0-691-04289-6.
Verbeek, Marno (2008). A Guide to Modern Econometrics. John Wiley and Sons. ISBN 0-470-51769-7.
Weisstein, Eric. "Wolfram MathWorld". WolframMathworld: Difference Operator. Wolfram Research. Retrieved 10 November 2017.
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. ISBN 978-1-118-67502-1.

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The lag operator, commonly denoted by

L

, is a mathematical construct in time series analysis and econometrics that shifts the values of a time series backward by one time period, such that for a stochastic process

\{ y_t \}

L y_t = y_{t-1}

. This operator, also known as the backshift operator, facilitates the compact representation of dynamic relationships in sequential data. Powers of the lag operator extend this shifting to multiple periods, where

L^k y_t = y_{t-k}

for any non-negative integer

k

, allowing the formation of lag polynomials such as

\phi(L) = \sum_{j=0}^p \phi_j L^j

.^[1] These polynomials are essential for modeling autoregressive processes, where an AR(

p

) model can be expressed as

\phi(L) y_t = \epsilon_t

, with

\epsilon_t

representing white noise innovations.^[2] Similarly, in moving average models and ARMA frameworks, the lag operator enables the inversion of processes and the derivation of infinite-order representations, provided the roots of the characteristic polynomial lie outside the unit circle to ensure stationarity and causality.^[3] Beyond univariate models, the lag operator plays a critical role in multivariate time series econometrics, such as vector autoregressions (VARs) and error correction models, where it simplifies the notation for impulse response functions and distributed lag structures. Its utility extends to forecasting, spectral analysis, and handling nonstationarity, making it indispensable for analyzing economic and financial data exhibiting temporal dependencies.

Fundamentals

Definition

The lag operator, denoted by

L

, is a fundamental mathematical tool in time series analysis that shifts a time series backward by one period. For a discrete-time series

\{X_t\}

, where

t

indexes time, the action of the lag operator is defined as

L X_t = X_{t-1}

, effectively replacing the current value with the previous one.^[1]^[2] This operation represents a one-period backward shift, preserving the structure of the series while delaying its values. The lag operator extends naturally to higher powers for multiple-period shifts. Specifically, for any positive integer

k \geq 1

L^k X_t = X_{t-k}

, indicating a shift backward by

k

periods.^[1]^[2] The inverse of the lag operator, known as the lead operator and denoted

L^{-1}

, shifts the series forward by one period, such that

L^{-1} X_t = X_{t+1}

. More generally, for

k > 0

L^{-k} X_t = X_{t+k}

.^[4] This bidirectional capability allows the operator to model both past dependencies and future expectations in time-indexed data. The lag operator applies to discrete-time stochastic processes, such as random walks or autoregressive series, as well as deterministic sequences like arithmetic progressions.^[5]^[6] For illustration, consider the deterministic sequence

X_t = t

, where each term is the time index itself; applying the lag operator yields

L X_t = t - 1

, demonstrating a uniform shift without changing the linear functional form.^[2] The notation

L

is equivalent to the backshift operator

B

, with details on conventions provided elsewhere.^[5]

Notation and Conventions

The lag operator is primarily denoted by the symbol

L

, defined such that for a time series

\{x_t\}

L x_t = x_{t-1}

. This notation facilitates compact representation of lagged values and polynomials in time series analysis. In many econometric contexts, the lag operator is interchangeably referred to as the backshift operator and denoted by

B

, satisfying the same relation

B x_t = x_{t-1}

.^[7]^[5] Although

L

and

B

perform identical shifts, the choice of symbol can reflect disciplinary emphasis;

B

is often favored in econometric literature to highlight the backward-shifting nature of the operation, while

L

underscores the general lagging concept. This convention traces its origins to the Box-Jenkins methodology for ARIMA modeling, introduced in the seminal 1970 text Time Series Analysis: Forecasting and Control, where lag operator notation was employed to simplify the expression of autoregressive and moving average structures.^[8]^[1] Field-specific variations further distinguish the notation: in time series econometrics,

L

(or

B

) remains standard for discrete-time shifts, whereas in digital signal processing, the analogous unit delay is conventionally represented by

z^{-1}

within the z-transform framework, reflecting a frequency-domain perspective on discrete signals.^[9] In practical implementations, statistical software packages numerically realize this operator; for instance, R's lag() function in the stats package shifts a time series backward by a specified number of periods, and MATLAB's lag() method for timetables performs equivalent time shifts on data arrays.^[10]^[11]

Mathematical Properties

Lag Polynomials

In time series analysis, a lag polynomial is defined as a formal power series constructed from the lag operator

L

, written as

\phi(L) = \sum_{i=0}^{\infty} \phi_i L^i

, where the coefficients

\phi_i

are constants and typically

\phi_0 = 1

. This polynomial acts on a time series

\{X_t\}

by producing

\phi(L) X_t = \sum_{i=0}^{\infty} \phi_i X_{t-i}

, effectively weighting current and past values of the series.^[12] Such representations facilitate compact notation for linear combinations of lagged observations.^[1] For finite-order cases, common in autoregressive models of order

p

(AR(

p

)), the lag polynomial takes the form

\phi(L) = 1 - \sum_{i=1}^p \phi_i L^i,

where the leading coefficient is normalized to 1 and higher powers of

L

have zero coefficients.^[12] This structure captures dependencies on up to

p

lags while maintaining the formal series framework.^[2] Lag polynomials exhibit a rich algebraic structure, closed under addition and multiplication, with the latter following standard polynomial rules due to the commutativity of powers of

L

(i.e.,

L^i L^j = L^{i+j}

). For example, multiplying two first-order polynomials yields

(1 - L)(1 - \alpha L) = 1 - (1 + \alpha) L + \alpha L^2,

resulting in another lag polynomial of higher order.^[2] Division, however, often produces an infinite series via polynomial long division; a canonical case is

\frac{1}{1 - L} = \sum_{k=0}^{\infty} L^k,

valid as a formal power series without requiring convergence in the classical sense.^[12] In time series contexts, the interpretation of these infinite expansions ties to process properties, where stationarity requires the coefficients to be absolutely summable (

\sum_{i=0}^{\infty} |\phi_i| < \infty

). This condition ensures the filtered series has finite variance and is equivalent to all roots of the polynomial

\phi(z) = 0

(replacing

L

with complex

z

) lying outside the unit circle in the complex plane.^[1]

Powers and Inverses

The powers of the lag operator

L

extend its basic shifting action to multiple periods. For a positive integer

k

, the

k

-th power is defined as

L^k X_t = X_{t-k}

, which shifts the time series backward by

k

periods, representing a

k

-period lag.^[2] This iterative application allows for compact notation in expressing dependencies on past values in time series models.^[13] The inverse of the lag operator corresponds to forward shifts, or leads. For a positive integer

k

, the negative power is

L^{-k} X_t = X_{t+k}

, advancing the series by

k

periods.^[13] This forward shift operator is useful in contexts requiring anticipation of future values, though it assumes the series is defined for those periods.^[2] Key algebraic properties facilitate manipulation of these powers. The lag operator satisfies

L^m L^n = L^{m+n}

for non-negative integers

m

and

n

, reflecting the additive nature of shifts, and

L^0 = I

, where

I

is the identity operator such that

I X_t = X_t

.^[13] These properties ensure that powers commute and can be combined straightforwardly in expressions.^[2] A significant application arises in infinite series expansions for invertible processes. When

|\rho| < 1

, the inverse of a simple lag polynomial yields

\frac{1}{1 - \rho L} = \sum_{k=0}^\infty \rho^k L^k

, an absolutely convergent geometric series that expresses the process as an infinite sum of lagged terms.^[2] For instance, in a first-order autoregressive process defined by

(1 - \rho L) X_t = \epsilon_t

, where

\epsilon_t

is white noise, this expansion gives

X_t = \sum_{k=0}^\infty \rho^k \epsilon_{t-k}

, illustrating the infinite moving average representation under stationarity.^[13]

Difference Operator

The difference operator, denoted as

\Delta

, is defined using the lag operator

L

\Delta = 1 - L

, where

L X_t = X_{t-1}

for a time series

\{X_t\}

.^[14]^[7] This operator produces the first difference of the series:

\Delta X_t = (1 - L) X_t = X_t - X_{t-1}

.^[14]^[8] The first difference is particularly useful for removing linear trends from non-stationary time series, transforming them toward stationarity.^[8] For higher-order differencing, the operator is raised to the power

d

, where

d

represents the order of integration of the series:

\Delta^d X_t = (1 - L)^d X_t

.^[14] This applies the first difference

d

times successively, with the second difference given explicitly as

\Delta^2 X_t = (1 - L)^2 X_t = (1 - 2L + L^2) X_t = X_t - 2X_{t-1} + X_{t-2}

, which eliminates quadratic trends.^[14]^[7] In general, for integer

d

, the

d

th-order difference expands via the binomial theorem as

(1 - L)^d = \sum_{k=0}^d \binom{d}{k} (-1)^k L^k

, yielding a finite linear combination of the series and its lags up to order

d

.^[14]^[8] Applying the first difference to a series with a deterministic linear trend, such as

X_t = \mu t + \epsilon_t

where

\mu

is the constant slope and

\{\epsilon_t\}

is stationary noise, results in

\Delta X_t = \mu + \Delta \epsilon_t

, yielding a constant mean and removing the trend component.^[8] To address seasonal patterns with period

s

, the seasonal difference operator is defined as

\Delta_s = 1 - L^s

, producing

\Delta_s X_t = (1 - L^s) X_t = X_t - X_{t-s}

.^[15]^[7] This operator isolates changes across the same seasonal point in consecutive cycles, such as differencing monthly data at lag 12 to remove annual seasonality.^[15]^[7]

Applications

Autoregressive and Moving Average Models

The lag operator provides a compact notation for expressing autoregressive (AR) models, which capture the linear dependence of a time series on its own past values plus a white noise error term. An AR process of order

p

, denoted AR(

p

), is defined as

\phi(L) X_t = \varepsilon_t

, where

\phi(L) = 1 - \sum_{i=1}^p \phi_i L^i

is the autoregressive lag polynomial,

L

is the lag operator such that

L X_t = X_{t-1}

, and

\{\varepsilon_t\}

is a white noise process with mean zero and constant variance

\sigma^2

. For the process to be stationary, all roots of the characteristic equation

\phi(z) = 0

must lie outside the unit circle in the complex plane.^[16] A moving average (MA) model of order

q

, denoted MA(

q

), represents the time series as a linear combination of current and past white noise errors. It is specified as

X_t = \theta(L) \varepsilon_t

, where

\theta(L) = 1 + \sum_{j=1}^q \theta_j L^j

is the moving average lag polynomial. For invertibility, which ensures the model can be expressed as an infinite-order autoregression useful for estimation and forecasting, all roots of

\theta(z) = 0

must also lie outside the unit circle.^[16] The autoregressive moving average (ARMA) model combines these structures to model more complex serial dependencies, defined for orders

p

and

q

\phi(L) X_t = \theta(L) \varepsilon_t

. This general form inherits the stationarity condition from the AR component (roots of

\phi(z) = 0

outside the unit circle) and the invertibility condition from the MA component (roots of

\theta(z) = 0

outside the unit circle). A simple example is the AR(1) model,

X_t = \mu (1 - \phi) + \phi X_{t-1} + \varepsilon_t

, which rewrites as

(1 - \phi L) (X_t - \mu) = \varepsilon_t

; here, stationarity requires

|\phi| < 1

. For parameter estimation in ARMA models, the invertible form allows expressing

X_t = [\theta(L) / \phi(L)] \varepsilon_t

, representing the series as an infinite autoregression in past errors, facilitating maximum likelihood methods.^[16]^[17]

Conditional Expectations

In time series analysis, conditional expectations are defined with respect to an information set

\Omega_t

, which contains all observable data up to time

t

. The conditional expectation of a future value

X_{t+j}

given this information is denoted

E_t[X_{t+j}] = E[X_{t+j} \mid \Omega_t]

, representing the best forecast based on past and present observations.^[13] A key implication involves the law of iterated expectations, which states that for

k > 0

E_t[E_{t+k}[X_s]] = E_t[X_s]

. This tower property ensures that expectations formed with nested information sets converge to the outer conditioning, facilitating recursive computations in dynamic models. In rational expectations models, where agents form forecasts optimally using all available information, lag operators simplify the derivation of multi-step forecasts. By representing expectation shifts compactly, such as through polynomials in

L

, these models express long-horizon predictions as iterative applications of one-step rules, reducing computational complexity in economic simulations.^[13]

Forecasting and Stationarity

In time series analysis, stationarity is a fundamental property ensuring that the statistical characteristics of a process, such as its mean and variance, remain constant over time. For autoregressive processes represented using the lag operator

L

, where

L X_t = X_{t-1}

, stationarity holds if the roots of the associated lag polynomial

\phi(L)

lie outside the unit circle in the complex plane. This condition guarantees that the process does not exhibit explosive behavior or persistent trends, allowing for reliable inference and modeling.^[18] Similarly, for moving average components, invertibility—a related concept—requires roots outside the unit circle to express the process as an infinite autoregression.^[19] To handle non-stationary series, the autoregressive integrated moving average (ARIMA) model extends the ARMA framework by incorporating differencing. The ARIMA(p,d,q) model is expressed as

\phi(L) (1 - L)^d X_t = \theta(L) \epsilon_t

, where

\phi(L)

is the autoregressive lag polynomial of order p,

\theta(L)

is the moving average lag polynomial of order q,

d

is the degree of differencing to achieve stationarity, and

\epsilon_t

is white noise. This formulation, introduced by Box and Jenkins, applies the difference operator

(1 - L)^d

to transform an integrated series into a stationary one before fitting the ARMA structure.^[20] Forecasting with the lag operator involves computing conditional expectations based on the inverted model representation. The h-step-ahead forecast is defined as

\hat{Y}_{t+h|t} = E_t [X_{t+h}]

, where the expectation is taken with respect to information available at time t. For a simple AR(1) model

X_t = \mu (1 - \phi) + \phi X_{t-1} + \epsilon_t

with

|\phi| < 1

, the forecast simplifies to

\hat{Y}_{t+h|t} = \phi^h (X_t - \mu) + \mu

, reflecting the geometric decay of the influence of the current observation as the horizon increases. This approach extends to higher-order ARIMA models by iteratively substituting forecasts for future values and setting future errors to zero.^[21] Unit root testing assesses stationarity by checking for roots on the unit circle, with the lag operator facilitating model specification. In the Dickey-Fuller test, the hypothesis of a unit root is examined through an augmented regression

\Delta X_t = \alpha X_{t-1} + \sum_{i=1}^k \beta_i \Delta X_{t-i} + \epsilon_t

, where

\Delta = 1 - L

and the lags control for serial correlation. Rejection of the null hypothesis (

\alpha = [0](/page/0)

) indicates stationarity, enabling appropriate differencing in ARIMA modeling.^[22]

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Lag operator

Fundamentals

Definition

Notation and Conventions

Mathematical Properties

Lag Polynomials

Powers and Inverses

Difference Operator

Applications

Autoregressive and Moving Average Models

Conditional Expectations

Forecasting and Stationarity

References

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Lag operator

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Lag operator

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Difference operator

Conditional expectation

See also

References

Lag operator

Fundamentals

Definition

Notation and Conventions

Mathematical Properties

Lag Polynomials

Powers and Inverses

Difference Operator

Applications

Autoregressive and Moving Average Models

Conditional Expectations

Forecasting and Stationarity

References

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