Backshift Operator

Backshift Operator

The backshift operator $B$ (sometimes written $L$ for lag) takes a time series and returns the same series shifted one step into the past, so that
$B{X}_t=X_{t-1},B^k{X}_t=X_{t-k}\text{ for any integer }k.$
It is a linear operator: applied to a weighted sum of two series it returns the same weighted sum of their lagged copies, $B(a{X}_t+b{Y}_t)=a{X}_{t-1}+b{Y}_{t-1}$. The point of introducing $B$ as a symbol rather than just writing $X_{t-1}$ is that powers of $B$ can be combined into polynomials in exactly the same algebraic way as powers of an ordinary variable. So an expression like $\varphi (B)=1-{\varphi }_{1}B-{\varphi }_2{B}^2-\cdots -{\varphi }_p{B}^p$ acting on $X_t$ produces $X_t-{\varphi }_1{X}_{t-1}-\cdots -{\varphi }_p{X}_{t-p}$, and an entire AR or ARMA equation collapses to the compact form $\varphi (B)X_t=\theta (B)W_t$.

The inverse operator $B^{-1}$, defined by $B^{-1}{X}_t=X_{t+1}$, exists in formal calculation but is rarely used in practice: causal time-series models, in which today’s value depends on past observations rather than future ones, work entirely with non-negative powers of $B$. The differencing operator $\nabla =1-B$, which produces $\nabla X_t=X_t-X_{t-1}$, is the most important application of $B$-polynomials and the algebraic ingredient behind the “I” in ARIMA(p, d, q).

Operator vs. Variable $B$ is an operator on time series, not a number, but the algebra of polynomial operators behaves exactly like the algebra of polynomials in a complex variable $z$. So $\varphi (B)=1-{\varphi }_{1}B-\cdots -{\varphi }_p{B}^p$ can be analyzed by replacing $B$ with $z$ and computing the roots of the equation $\varphi (z)=0$. This translation is what makes stationarity (roots of $\varphi (z)$ outside the unit circle) and invertibility (roots of $\theta (z)$ outside the unit circle) checkable by ordinary polynomial-root finding. The operator $B$ inherits the convergence behavior of its $z$-polynomial counterpart only when applied to the right class of time series; failure of the unit-circle conditions corresponds to power series in $B$ that diverge when applied.

Intuition

The backshift operator is bookkeeping. It turns “look at yesterday’s value” into a symbol that can be multiplied, expanded, and factored, and the payoff is that the recursive equation $X_t=\varphi X_{t-1}+W_t$ becomes the algebraic equation $(1-\varphi B)X_t=W_t$, which can be manipulated with the rules of polynomial algebra. Multiplying out $(1-0.6B+0.5B^2)$ feels like multiplying out a quadratic, the locations of its roots in the complex plane translate directly into facts about the time series (stationary or not, oscillating or smoothly decaying, persistent or quickly forgetting), and the same algebra works for products and quotients of polynomials.

This polynomial-vs-operator parallel is the technical reason ARMA model identification can be done by inspecting characteristic polynomials. It is also why “factor out a common $(1-B)$” is a meaningful operation on a time-series model: factoring corresponds to differencing the series, and the rearranged equation $\varphi (B)(1-B{)}^d{X}_t=\theta (B)W_t$ is the ARIMA form that handles non-stationary series by differencing them out before fitting.

Properties

(Linearity) Applied to a weighted sum, $B$ distributes over the sum and pulls out the constants.

For any series $X_t,Y_t$ and any scalars $a,b$, the operator satisfies $B(a{X}_t+b{Y}_t)=a{X}_{t-1}+b{Y}_{t-1}$. Linearity is the property that lets polynomial operators in $B$ be applied term-by-term to weighted sums of series, and is the algebraic reason ARMA’s recursive equations behave the same way as ordinary polynomial equations under multiplication and addition.

(Composition) Powers and products of $B$-polynomials commute and combine like ordinary polynomials.

The two facts are $B^j{B}^k=B^{j+k}$ for integer $j,k$, and $\varphi (B)\theta (B)=\theta (B)\varphi (B)$ for any two polynomials $\varphi ,\theta$. They follow immediately from the definition: shifting first by $j$ and then by $k$ steps gives the same result as shifting by $j+k$ steps in either order, and addition and multiplication commute for the scalar coefficients. Composition is what justifies factorizations like $\varphi (B)(1-B{)}^d{X}_t=\theta (B)W_t$: the differencing factor $(1-B{)}^d$ can be moved across the AR factor $\varphi (B)$ freely.

(Polynomial Roots and Stationarity) Roots of $\varphi (z)$ outside the unit circle in the complex plane are equivalent to stationarity of $\varphi (B)X_t=W_t$.

The condition is that every solution $z$ of the polynomial equation $\varphi (z)=1-{\varphi }_{1}z-\cdots -{\varphi }_p{z}^p=0$ must satisfy $|z|>1$. The “outside the unit circle” convention is a consequence of writing the polynomial in $B$ rather than in the inverse $B^{-1}$: roots outside the unit circle in $z$ correspond to a convergent infinite-MA representation, which is what makes the implied process stationary and causal. Different texts use different conventions (some use $z=B^{-1}$, in which case the condition flips to roots inside the unit circle). This vault follows the Shumway-Stoffer convention of polynomials in $B$ with roots outside.

(Invertibility) Roots of $\theta (z)$ outside the unit circle are equivalent to invertibility of $X_t=\theta (B)W_t$.

Invertibility means that the polynomial operator $\theta (B)$ can be inverted as a power series, expressing the white noise as a convergent function of past observations: $W_t=\theta (B{)}^{-1}{X}_t$. Without this property, two different MA models, one with parameter $\theta$ and one with $1/\theta$, produce the same observable distribution, so the parameters cannot be recovered from data. The reasoning behind both the stationarity and the invertibility root conditions is developed in 2.5.5 Causality and Invertibility.

(Difference Operator) Powers of $\nabla =1-B$ remove polynomial trends.

Expanding the binomial ${\nabla }^d=(1-B{)}^d$ gives the explicit formula ${\nabla }^d{X}_t={\sum }_{j=0}^d\left(\genfrac{}{}{0ex}{}{d}{j}\right)(-1{)}^j{X}_{t-j}$. First differencing ($d=1$) reduces a series with a linear trend to a stationary series whose mean is the trend’s slope; second differencing ($d=2$) reduces a quadratic-trend series to a stationary one whose mean is twice the second derivative; and in general, $d$-th differencing collapses a degree-$d$ polynomial trend down to a constant. The same ${\nabla }^d$ is the algebraic ingredient that turns ARMA into ARIMA: a series whose mean drifts polynomially in time can be modeled by ARMA after differencing $d$ times.

(Seasonal Differencing) The operator ${\nabla }_s=1-B^s$ removes seasonality of period $s$.

For monthly data with annual seasonality, applying ${\nabla }_{12}$ produces ${\nabla }_{12}{X}_t=X_t-X_{t-12}$, the change from the same month one year ago, which subtracts off any deterministic annual cycle. Combined with regular differencing as $\nabla {\nabla }_{12}$, the operator handles series with both a trend and a yearly cycle in one pass. This combination is the ingredient behind seasonal-ARIMA models like the famous “airline model” ARIMA$(0,1,1)\times (0,1,1{)}_{12}$.

Examples

AR(1) in operator form

Express $X_t=\varphi X_{t-1}+W_t$ in operator form and identify the characteristic polynomial.

Solution $X_t-\varphi X_{t-1}=W_t$, which in operator form reads $(1-\varphi B)X_t=W_t$. The autoregressive polynomial is $\varphi (B)=1-\varphi B$, and its $z$-counterpart $\varphi (z)=1-\varphi z$ has the single root $z=1/\varphi$. Stationarity demands $|1/\varphi |>1$, equivalent to $|\varphi |<1$, recovering the standard AR(1) stationarity condition by polynomial-root inspection.

Moving the lagged term to the left gives

AR(2) characteristic polynomial

Write $X_t=0.6X_{t-1}-0.5X_{t-2}+W_t$ in operator form, locate the roots, and decide stationarity.

Solution $(1-0.6B+0.5B^2)X_t=W_t$. The $z$-polynomial $1-0.6z+0.5z^2=0$ has roots $z=(0.6\pm \sqrt{0.36-2})/1=0.6\pm 1.28i$ from the quadratic formula, with modulus $|z|=\sqrt{0.36+1.64}=\sqrt{2}\approx 1.41$. Both roots lie strictly outside the unit circle, so the process is stationary. The roots being complex (rather than real) tells us in addition that the autocovariance oscillates as it decays: the series rings at a characteristic frequency before settling back to its mean, with the period given by $2\pi$ divided by the angular argument of the root.

Operator form:

Seasonal ARIMA in compact operator form

An airline-passenger model with monthly seasonality is often written as ARIMA$(0,1,1)\times (0,1,1{)}_{12}$, which in operator form reads $\nabla {\nabla }_{12}{X}_t=(1-{\theta }_{1}B)(1-{\Theta }_1{B}^{12})W_t$.

What is the operator polynomial doing? $W_t,W_{t-1},W_{t-12},W_{t-13}$: $W_t-{\theta }_1{W}_{t-1}-{\Theta }_1{W}_{t-12}+{\theta }_1{\Theta }_1{W}_{t-13}.$ The compact operator form contains all the structure in two factored polynomials; expanding it would produce a sparse but lengthy MA representation. The whole point of the backshift operator is that "compact" is also "computable": every standard software package (R forecast, Python statsmodels) parses these polynomial factors directly, builds the corresponding sparse coefficient vector internally, and runs the fit.

The left side applies non-seasonal differencing (which removes a trend) and seasonal differencing of period 12 (which removes the annual cycle), in either order; the two operators commute. The right side multiplies a non-seasonal MA(1) operator by a seasonal MA(1) operator, and expanding the product gives four terms involving the shocks