Using Lag Polynomials to Write Autoregressive and ARMA Models

Summary

The document explains how a polynomial in the backward shift operator represents a weighted combination of lagged observations. The operator shifts a time series back by one period, and its powers shift it by additional periods. Replacing the polynomial’s variable with this operator turns each polynomial term into a lag of the series.

It then shows that subtracting a lag polynomial from one gives the familiar autoregressive form: the current observation is expressed using a constant, past observations, and an error term. Extending the expression with a polynomial applied to the errors yields the standard ARMA structure when the polynomials have the stated finite orders. This notation is a compact way to express time series equations; the explanation does not address model estimation, stationarity, invertibility, or how to select polynomial orders.

Key ideas

• The backward shift operator maps a series value to its previous-period value, and its powers produce longer lags.
• Applying a polynomial in the shift operator creates a weighted sum of lagged observations.
• An autoregressive model can be written compactly using one minus a lag polynomial.
• Adding a finite lag polynomial in the error term gives an ARMA representation.
• The notation describes model structure but does not establish stationarity or explain estimation.

Tags

# What does A(B) mean in time series


# What does A(B) mean in time series












So I have been reading some papers regarding time series, mainly from Granger and Engle. I am a bachelor econometrics student, but I have never seen such notation before. For example, A(B)(1-B)x(t) = -az(t-1) + b(t). I know that B is the backward shift operator. Could someone clarify this?

another example would be that time series x(t) = a(B)epsilon(t)

## Answer by Kevin (score 1, accepted)

https://quant.stackexchange.com/a/46728

As you said, $B$ is the lag or backward shift operator such that $BX_t=X_{t-1}$ and $B^pX_t=X_{t-p}$. Let $A$ now be polynomial, say $A(x)=a_1 x + a_2x^2+...+a_px^p$. Then,

\begin{align} A(B) X_t &= \left( a_1 B + a_2B^2+...+a_pB^p\right) X_t \\ &=a_1 X_{t-1} + a_2 X_{t-2} + ... + a_p X_{t-p} \end{align} and \begin{align} \big(1-A\big)(B) X_t &= \big(1-A(B)\big) X_t \\ &= \left( 1- a_1 B - a_2B^2-...-a_pB^p\right) X_t \\ &=X_t-a_1 X_{t-1} - a_2 X_{t-2} - ... - a_p X_{t-p}. \end{align} Thus, if you write $\big(1-A\big)(B) X_t=c+\varepsilon_t$, you get \begin{align} X_t&=c+a_1 X_{t-1} + a_2 X_{t-2} + ... + a_p X_{t-p}+\varepsilon_t \\ &= c + \sum_{i=1}^p a_iX_{t-i}+\varepsilon_t, \end{align} which is simply an AR($p$) model. Thus, polynomials of the backward shift operator allow you to easily write down time series models. For instance, $\big(1-A(B)\big) X_t=c+\big(1+C(B)\big)\varepsilon_t$ is an ARMA($p$,$q$) model (if $C$ is a polynomial of order $q$).

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