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Lag Notation For Time Series

Learn to read and apply time subscripts and lag operators without confusing the number of steps with the elapsed time.

Beginner 10 min read

What You'll Learn

  • Distinguish a time-indexed process from its observed values.
  • Read \(x_t\) and \(x_{t-k}\) using a specified time index.
  • Apply the lag operator repeatedly and interpret \(L^k x_t\).
  • Identify when a lagged value is unavailable at the start of a series.
  • Preserve missing time periods instead of treating nonadjacent observations as adjacent.

Why Lag Notation Matters

Time-series data are recorded in an order: for example, one observation per month or one per quarter. To describe an observation precisely, we need to identify its position in that sequence. To refer to an earlier observation relative to a current one, we need a compact way to say how many time steps back to look. Time subscripts identify positions, and lag notation specifies backward shifts.

This notation is useful even before fitting a time-series model. It lets us state exactly which observation is being discussed and prevents an easy error: confusing “two time steps earlier” with “two calendar units earlier” when the data are not equally spaced. The notation introduced here is aligned with the SOA Exam SRM (Statistics for Risk Modeling) syllabus.

Definition: Let \(X_t\) denote the value of a time-indexed process at time index \(t\), and let \(x_t\) denote the observed value at that index. The subscript \(t\) identifies the position in the series. A lag of \(k\) steps from index \(t\) refers to the value at index \(t-k\), written \(X_{t-k}\) or, for observed data, \(x_{t-k}\). Here \(k\) is a nonnegative integer.

For example, \(x_8\) is the observation at index 8. The observation two steps before it is \(x_{8-2}=x_6\). The number 2 counts index steps, not necessarily years, months, or days. Its calendar meaning depends on how the index was defined. If one index corresponds to one quarter, a two-step lag is two quarters; if the index corresponds to one trading day, it is two trading days.

The Lag Operator

The lag operator, written \(L\), acts on a time-indexed value by moving its index back one step. Applying it once gives the previous observation; applying it repeatedly gives observations further back. A superscript on \(L\) counts how many times the shift is applied.

Definition: For a time-indexed value \(X_t\), the lag operator is defined by \(LX_t=X_{t-1}\). For a nonnegative integer \(k\), \(L^kX_t=X_{t-k}\), with \(L^0X_t=X_t\). The same notation applies to observed values: \(L^kx_t=x_{t-k}\).
$$ LX_t=X_{t-1}, \qquad L^2X_t=X_{t-2}, \qquad L^kX_t=X_{t-k}. $$

The superscript \(k\) is an operator power: \(L^2\) means apply \(L\) twice. It does not mean square the observation. Thus \(L^2x_t\) is \(x_{t-2}\), whereas \((x_t)^2\) is the square of the value at index \(t\). These expressions describe different things.

Why Repeated Shifts Give \(L^kX_t=X_{t-k}\)

The rule for a \(k\)-step lag follows from the one-step definition. Each application subtracts one from the index. The repeated applications can be written explicitly:

1
Start with one shift.
By definition, \(LX_t=X_{t-1}\).
2
Apply a second shift.
Apply \(L\) to the shifted value: \(L(LX_t)=L(X_{t-1})=X_{t-2}\). Therefore, \(L^2X_t=X_{t-2}\).
3
Continue for \(k\) shifts.
Each additional application reduces the index by one. After \(k\) applications, the index is \(t-k\), so \(L^kX_t=X_{t-k}\).

This rule is a statement about indices and does not require a particular probability distribution, model, or relationship between observations. It tells us which value the notation refers to; it does not tell us whether that value is statistically related to the current one.

Worked Example: Reading a Two-Quarter Lag

A hypothetical insurer records quarterly earned premium, in thousands of dollars, for six consecutive quarters. The observations are \(x_1=82\), \(x_2=88\), \(x_3=91\), \(x_4=86\), \(x_5=95\), and \(x_6=100\). Find \(x_5\), \(Lx_5\), and \(L^2x_5\), and identify the quarter for each value.

Method. Use the subscript as the quarter index and subtract the lag order from the current index.

At index 5, the recorded value is \(x_5=95\), so earned premium in quarter 5 is $95,000.

For a one-step lag, \(Lx_5=x_{5-1}=x_4=86\). This is the value from quarter 4, or $86,000.

For a two-step lag, \(L^2x_5=x_{5-2}=x_3=91\). This is the value from quarter 3, or $91,000.

Check. The indices move from 5 to 4 to 3, one quarter backward per application of \(L\). The values are therefore \(95\), \(86\), and \(91\), respectively, in thousands of dollars. The common mistake is to treat \(L^2x_5\) as \(95^2\); the superscript belongs to the lag operator, not to the observation.

Lag Order, Time Units, and Series Boundaries

A lag order must be interpreted using the index definition. If a series is indexed by consecutive months, then \(x_{t-3}\) refers to the observation three months before \(x_t\). If the index is consecutive business days, it refers to the observation three business-day steps earlier. When the time points are irregular, three index steps still means three positions earlier in the indexed sequence; it may not correspond to a fixed number of calendar days.

Lagged values can also fall outside the observed range. Suppose a series begins at index 1. Then \(Lx_1=x_0\), but \(x_0\) is not among the observations unless the data set provides it. In that data set, the one-step lag at index 1 is unavailable. Similarly, \(L^3x_2=x_{-1}\) is unavailable if the recorded series contains only indices 1 and later. The notation still identifies the requested index; the data simply do not supply its value.

A missing observation within the series is different from a time point that does not exist in the index definition. If an indexed month is missing from the records, do not silently renumber later observations. Keep the time index attached to its intended month. Otherwise, a requested lag may point to the wrong calendar period.

Worked Example: A Lag at the Start of the Series

A hypothetical monthly policy-retention index is recorded from January through December. January is index 1, February is index 2, and so on. The recorded values are \(x_1=84\), \(x_2=85\), \(x_3=83\), \(x_4=87\), \(x_5=88\), \(x_6=86\), \(x_7=89\), \(x_8=90\), \(x_9=88\), \(x_{10}=91\), \(x_{11}=92\), and \(x_{12}=93\). Find \(L^3x_{11}\) and \(L^3x_2\), and state whether each is available.

Method. For each expression, subtract 3 from the current index.

For November, \(t=11\). Therefore,

$$ L^3x_{11}=x_{11-3}=x_8=90. $$

Index 8 is August, so the three-step lag of the November value is the August value, 90.

For February, \(t=2\). Therefore,

$$ L^3x_2=x_{2-3}=x_{-1}. $$

The recorded series begins at index 1, so \(x_{-1}\) is not observed. This lagged value is unavailable; it is not valid to substitute January’s value or to treat the result as zero.

Check. November shifted back three monthly steps is August. February shifted back three steps falls before January, outside the recorded range. The common mistake is to report a value for an unavailable lag without an assumption or additional data.

Keep the Time Index Attached to the Observation

When data contain gaps, a lag refers to the intended index, not merely the previous row in a shortened table. For instance, if weeks 1, 2, 4, and 5 are recorded, then the row immediately before week 5 may be week 4, but \(x_{5-2}\) is \(x_3\), the value for week 3. If week 3 is missing, the two-step lag at week 5 is unavailable. It is not the value for week 2.

This distinction is particularly important when a data file omits dates with no recorded activity. If the analytic index is calendar weeks, a week with no activity may still need to be represented according to the study’s definition, rather than removed and replaced by a consecutive row number. First establish what one index step means; then apply lag notation.

Worked Example: A Missing Weekly Observation

A hypothetical claim-count series is indexed by calendar week. Recorded counts are week 1: 4 claims, week 2: 7 claims, week 4: 9 claims, and week 5: 6 claims. Week 3 has no recorded value. Determine \(x_4\), \(Lx_5\), and \(L^2x_5\).

Method. Use the calendar-week index, even though one week is absent from the recorded rows.

The value at index 4 is \(x_4=9\), the count for week 4. The one-step lag at week 5 is

$$ Lx_5=x_{5-1}=x_4=9. $$

The two-step lag at week 5 is

$$ L^2x_5=x_{5-2}=x_3. $$

Week 3 has no recorded value, so \(x_3\) and consequently \(L^2x_5\) are unavailable in these data.

Check. Week 4 is one calendar step before week 5; week 3 is two steps before it. The common mistake is to count backward by rows and use week 2’s value of 7 as the two-step lag. That would be incorrect because week 3 is a missing indexed observation between weeks 2 and 4.

Common Notation Traps

  • Confusing a lag with a value. \(L^2x_t\) means \(x_{t-2}\), not the square \(x_t^2\).
  • Subtracting the wrong amount. A lag of \(k\) steps changes index \(t\) to \(t-k\); it does not change it to \(t+k\).
  • Ignoring the time scale. A two-step lag is not automatically two years or two calendar days. Read the index definition.
  • Compressing a series with missing dates. Do not treat the previous recorded row as the previous time step unless the row order truly represents consecutive steps in the defined index.
  • Inventing a boundary value. If the requested earlier index is not observed, identify the lag as unavailable rather than assuming it is zero.
Key takeaway: To interpret a lag, identify the current index \(t\), subtract the lag order \(k\), and look up index \(t-k\). The operator \(L^k\) shifts by \(k\) steps; it does not square a value or guarantee that the requested observation exists.

Check Your Understanding

For each problem, apply the stated time index and report whether the requested lag is available.

  1. A quarterly expense series has \(x_1=12\), \(x_2=15\), \(x_3=14\), \(x_4=18\), and \(x_5=20\), in thousands of dollars. Find \(Lx_5\) and \(L^3x_5\). (Answer: \(Lx_5=x_4=18\); \(L^3x_5=x_2=15\), in thousands of dollars.)
  2. A daily series is indexed by consecutive business days, with \(x_7=103\) and \(x_4=98\). What is \(L^3x_7\), and what does the lag order count? (Answer: \(L^3x_7=x_4=98\); it counts three business-day index steps.)
  3. A monthly series is recorded only for indices 1 through 8. Is \(L^2x_2\) available? Show the requested index. (Answer: \(L^2x_2=x_0\), which is unavailable in the recorded series.)
  4. A weekly table contains values for weeks 10, 11, 13, and 14, with week 12 missing. What is \(L^2x_{14}\), and can its value be read from the available observations? (Answer: \(L^2x_{14}=x_{12}\); it is unavailable because week 12 is missing.)
  5. For an indexed series with \(x_6=11\), interpret \(L^0x_6\) and \(L^2x_6\). (Answer: \(L^0x_6=x_6=11\); \(L^2x_6=x_4\), whose value is not specified.)

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