Lag operator

Read the exponent as a distance in time

For a nonnegative integer $k$,

Applying $L$ twice gives $L^2y_t=y_{t-2}$. The exponent counts shifts; it does not square the observation. Nor is $L$ an unknown scalar that multiplies $y_t$. In $aLy_t=ay_{t-1}$, the constant $a$ supplies the multiplication and $L$ supplies the shift. Here and below, the index increases forward in time.

The meaning of a “period” comes from the data grid. With daily trading observations, a lag of one means the previous trading observation, which need not be the previous calendar date. For irregular event data it means the previous event unless the series has first been put on a calendar grid. Declare that choice before interpreting a lag as elapsed time.

Three spreads, several different operations

Take a synthetic quoted spread observed at consecutive daily sampling points:

At the current row $t=2$, $Ly_2=y_1=12$ and $L^2y_2=y_0=10$. Both results retain the spread's basis-point units. The earlier observation is aligned with the current row; it does not become a new observation made today.

The symbol $1$ in an operator expression is the identity: it leaves the series as it is. Thus $1-L$ subtracts the lagged value from the current value:

This is a change in the daily spread change, measured in basis points on this fixed sampling grid. It is neither the square of the change nor the two-observation change. The latter is $(1-L^2)y_2=15-10=5$ basis points. Parentheses decide which calculation you perform.

Why model equations use it

An autoregressive model with constant coefficients can be written either way:

Expanding the polynomial recovers the original equation. The finite algebra works because shifting is linear and successive shifts compose: $L^jL^ky_t=L^{j+k}y_t$. These manipulations use constant coefficients; shifting a time-varying coefficient also shifts its date.

The beginning of the sample still matters

In this sample, $Ly_0$ would require $y_{-1}$, which was not supplied. The operator defines which observation is needed; it cannot manufacture that observation. Preserve the missing value, obtain earlier data, or state an explicit model-based initialisation. Filling it with zero silently changes the calculation.

With only these three consecutive observations, a first difference provides two usable rows, and a second difference provides one. When comparing models with different lag lengths, align their usable samples as well as their dates. A longer lag can otherwise change the estimation sample along with the model.

Sources

• Chris Brooks, Introductory Econometrics for Finance, 2nd edition, Cambridge University Press, 2008, §5.3, pp. 211–212: lag operator, powers and moving-average polynomial notation.
• Rob J. Hyndman and George Athanasopoulos, Forecasting: Principles and Practice, 3rd edition, §§9.1–9.3: differencing, backshift notation and autoregressive models.