Rolling Windows: Recomputing a Figure as Time Moves On

A rolling window recomputes a figure over the last fixed number of observations, then moves one step and does it again. A twelve month window over the Nakshatra unit's 72 months gives 61 readings, the first at December 2019, and its rolling spread runs from 2.0548 per cent to 6.9462 per cent. When the record changes, the first reading free of the old stretch arrives eleven months later.

Almost every figure anyone meets about a record over time is a rolling figure. The twelve month average, the three year volatility, the rolling correlation, the moving average drawn across a chart: all of them are the same manoeuvre. A length is fixed, something is computed over that many observations, the window slides along by one, and the computation runs again. The output is not a number. The output is a whole new series, one reading per position, and that new series has habits of its own that have nothing to do with the record it was computed from.

The habits of that new series are the subject here. A rolling figure arrives late by an amount that can be worked out exactly. A rolling figure moves when the record underneath it has not moved at all. And it is dated in a way that quietly changes what it is describing. None of that is a flaw in the arithmetic. The arithmetic behaves that way by construction. Knowing how it behaves is the difference between reading one of these lines and being led by it.

Everything below is worked on one invented record. The Nakshatra unit is a traded something with a price and nothing else decided about it. Its six year record holds 72 dated monthly changes. The earliest belongs to the month that closed on 31 January 2019 and the latest to the month that closed on 31 December 2024, and all of them run against an opening markA starting price fixed one period ahead of a record's earliest observation. No measurement is taken on that day at all. The figure is there purely so that observation number one has a level to be compared against. of Rs 100.00/- set on the final day of 2018. Taken together those 72 months carry a mean of 1.00 per cent and a spread of 5.00 per cent. The 72 months were written down rather than gathered, and they stand for no security and no market.

What is taken as already settled?

Three things, each built elsewhere and used here without being rebuilt.

• The mean and the standard deviation. Those are the two figures being recomputed here. The spread throughout is always the standard deviation on the n denominatorDividing the summed squared distances by the number of observations rather than by that number less one. It is the choice that fits describing a complete set of figures rather than estimating from a sample of them. Covered separately., because a window of twelve months is a complete set of twelve months being described, not a sample of some larger thing.
• The moving average. That is one particular thing computed over a rolling window, and how the three kinds of moving average differ, and how far behind the price each of them sits, is covered separately. The window itself is the subject, not the averages that can be put inside it.
• The record's two stretches. Over its first three years the six year record's spread is 3.00 per cent. Over its last three it is 6.4031 per cent. Both stretches have a mean of exactly 1.00 per cent. That the change exists, and what one spread figure for the whole record costs, is covered separately. That change is taken as given, and a window is watched crossing it.

What is a rolling window, and what does recomputing over one produce?

Here is the whole manoeuvre in four steps. Fix a length. Take the last that many observations. Compute something over them. Move one step forward and do all of it again. Those four steps are the whole of it, and the manoeuvre is genuinely no more complicated.

The part that is not obvious is what comes out the other end. Recomputing over a rolling window does not produce a number, it produces a new series, with one reading for each position the window can occupy, and that new series has to be dated. The convention is to date each reading to the last month of the window that produced it, and once that convention is accepted, every difficulty below follows from it.

Think about a food stall outside one office building. A single week is too jumpy to tell the owner anything, so every Friday she writes down what the last twelve weeks took. The number she writes down on Friday is labelled with Friday's date. But eleven of the twelve weeks inside it are already over, and some of them are nearly three months over. The label says today. The contents are mostly history. Nobody is being dishonest and the arithmetic is correct; it is just that a figure dated to today can be almost entirely about a period that has finished.

The Nakshatra unit's record behaves the same way. The reading dated December 2022 in the picture above was computed from January 2022 through December 2022, so eleven of its twelve inputs are from months that were already in the past when it was published. A rolling figure is always a report about a stretch, wearing the date of the stretch's last day.

How many readings does a twelve month window give on 72 months?

Sixty one, and the first of them falls at December 2019. The count is worth doing slowly. The arithmetic is easy and the mistake is common.

January 2019 cannot carry a reading. Neither can February, and so on through November. A twelve month window needs twelve months, and for January 2019 there are no earlier months in the record at all. The first month that has eleven months in front of it inside the record is the twelfth month, December 2019. From there every remaining month carries a reading, right through to December 2024.

The count is the record length, less the window length, plus one: 72 less 12 plus 1 gives 61. The plus one catches people out, so check it on something small. On a record of five observations with a window of three, the readings fall at the third, fourth and fifth observations. Three readings, and 5 less 3 plus 1 is 3. The rule holds.

The consequence is worth taking seriously before a length is chosen. A long window on a short record leaves almost nothing to look at. A thirty six month window on this same record yields 37 readings instead of 61. A sixty month window yields 13. Seventy two yields exactly one. One reading is not a series at all. It is the whole record computed once. And the frequencyHow often a record records something: daily, weekly, monthly, yearly. Changing it changes how many observations a given stretch of calendar time contains, and a good deal else besides. Covered separately. of the record decides how much calendar time a window of twelve actually covers. Twelve on this monthly record is a year. Twelve on a daily record is under a fortnight.

What does the six year record's rolling spread actually do?

The rolling spread climbs, roughly, and wobbles the whole way. At December 2019 it reads 2.0548 per cent, the lowest of all 61 readings. At December 2021 it reads 3.4359 per cent. At December 2022, 5.7319 per cent. At December 2024 it reads 6.9462 per cent, the highest of all 61.

Now put those beside what the record's spread actually is. The record's spread takes two values and only two: 3.00 per cent for the first thirty six months, 6.4031 per cent for the last thirty six. Not one of the 61 rolling readings lands on either of those two numbers. Thirteen of them fall below 3.00 per cent, four rise above 6.4031 per cent, and the other forty four land somewhere between the two, including a long run through the middle where the record's spread is not anywhere near the middle at all.

Why does a correct calculation miss the correct answer sixty one times out of sixty one? Because twelve months is a small number of months. A spread computed from twelve figures is not the spread of the process that produced them; it is what those particular twelve happened to do. Roll the window on by one, drop the oldest month, add a new one, and the answer moves whether or not anything about the record changed. The wobble in that line is the window's own noise, not the record's news.

Notice also the shape of the climb. The climb is not a step. The record steps, at January 2022, from one spread to the other in a single month. The rolling line ramps upward across most of 2022 instead.

Why does the rolling figure arrive late when the record changes?

Because a window that straddles a change point is computing over both stretches at once, and it goes on doing that until the change point has passed clean out of the back of the window.

Follow it month by month. The six year record's spread changes at January 2022. The reading dated December 2021 covers January to December 2021, all of it inside the calm stretch. The reading dated January 2022 covers February 2021 to January 2022: eleven calm months and one wide one. February 2022 gives ten and two. The mixture keeps shifting, one month at a time, and the reading keeps drifting upward as it does. The reading dated December 2022 is the first one that covers January to December 2022, twelve months all from the new stretch, with no trace of the old one left in it.

The delay is the window length less one, eleven months here. That is arithmetic, not an observation about this particular record. A three month window would be clear of the change in two months. A twenty four month window would take twenty three. A thirty six month window on this record would not be clear of it until December 2024, the last month there is. Change the record and the number does not change; change the window and it does.

One sentence is worth carrying away. A rolling window is a lag in disguise. It looks like a smooth line describing now, and it is a report about a period that is mostly over, arriving a known number of months after the fact. How far behind a smoothed line sits, and why, is worked through in detail separately. The lateness is not a side effect. Lateness is what a window is.

Why does a rolling mean move when nothing underneath it moved?

The next point is the most useful of the lot, and most treatments of rolling figures stop before they reach it.

Take the six year record's average monthly change. Over the first three years it is exactly 1.00 per cent. Over the last three years it is exactly 1.00 per cent. Over all six it is exactly 1.00 per cent. The centre of this record does not move. Not a little; not at all. The record was built with a flat centre on purpose, so that anything the rolling mean does can only have come from the window.

Now compute the twelve month rolling mean over it. At December 2019 it reads 0.3333 per cent. At December 2021, 1.1667 per cent. At December 2024, minus 1.5000 per cent. Across all 61 readings the rolling mean ranges from minus 2.9167 per cent at November 2024 up to 3.0833 per cent at October 2022, a swing of exactly 6.0000 percentage points.

Not one point of that six point swing is a change in the record. It is twelve months of scatter added up and divided by twelve, and that is all it is. In the second stretch the individual months have a spread of 6.4031 per cent, so a twelve month average of them carries scatter of its own of roughly 1.8484 per cent, that being 6.4031 taken over the square root of twelve. Two of those either side of 1.00 per cent covers roughly minus 2.70 per cent to 4.70 per cent, and every one of the observed readings is inside or barely outside that. The line is doing what a twelve month average of noisy months does.

A household version makes this concrete. A small shop's takings vary a lot week to week for reasons nobody controls: weather, a festival, a wedding down the road, a road being dug up. Take a twelve week average of them and it will rise and fall in long smooth waves. The waves look enormously meaningful. The waves are twelve weeks of weather, added up and divided by twelve. The shop is doing exactly what it was doing before.

Two things make this trap worse than it sounds. First, a rolling line looks convincing in a way a scatter of raw months does not: smooth things read as trends. Second, consecutive rolling readings share eleven of their twelve inputs, so they cannot help moving in gentle runs rather than jumping about. A rolling series manufactures the appearance of persistence out of nothing but overlap. The overlap is also why a memory measurement taken on a rolling series is not measuring the record, a point handled where autocorrelationA measure of how much a reading resembles the reading a fixed number of steps earlier in the same record. More than one recipe carries this name and they do not all give the same answer, so a quoted figure is only readable if the recipe is named beside it. Covered separately. is built.

What do the two rolling series read on the same four dates?

One record, two rolling series, 61 readings each, computed the same way over the same twelve month blocks. Reading them side by side is the fastest way to see that they report two different kinds of thing.

The two columns behave completely differently, and the record explains why. The rolling spread climbs because the record's spread genuinely changed, from 3.00 per cent to 6.4031 per cent at January 2022, so the climb is late and wobbly but it is reporting something real. The rolling mean swings because the record's mean did not change at all, so every point of its six point swing is the window's own noise. The same calculation, on the same record, over the same blocks of twelve months, and one of them is news while the other is not.

The difference is why the four items further below are worth writing beside any rolling figure. Nothing about the shape of a rolling line settles which of these two cases is in front of the reader. Only the record settles that, and only if somebody goes and asks it.

How is a window length chosen?

By deciding which of two errors is the better one to make. Both cannot be avoided.

A short window reacts quickly. A short window is also jumpy, and a jumpy line reports changes that are not there. Compute the rolling spread on this record over a three month window and, across the calm first stretch alone, where the spread is a flat 3.00 per cent throughout, the readings run from 0.4714 per cent to 3.3993 per cent. The high reading is more than seven times the low one, on a stretch where nothing whatsoever changed.

A long window is steady. A long window is also late, and a late line misses changes that are there. Compute the same rolling spread over a thirty six month window and it is beautifully behaved. Its lowest reading is exactly 3.0000 per cent and its highest is exactly 6.4031 per cent, precisely the record's two true values. But it only gives 37 readings, its first does not arrive until December 2021, and it does not become clear of the January 2022 change until December 2024, the last month in the record.

There is no length that is both quick and steady, and choosing a window is choosing which of the two errors is tolerable. A shorter window if a real change would be expensive to miss and a false alarm is cheap to check. A longer window if the reverse. The choice and its reason belong in writing, before anything is computed.

And one honest addition that costs nothing to state. A window length chosen after looking at what each length produces is not a choice, it is a result. Where nine lengths are tried and the one whose line tells the tidiest story is kept, the line is now partly a picture of the searching. Searching does not make the line useless. It makes the search something to declare. Whoever reads the line should know how many lengths were tried.

What is the difference between a rolling window and an expanding one?

A rolling window keeps its length. Every time it moves forward it takes one new observation in at the front and drops the oldest one out of the back. An expanding window keeps everything: it starts at the beginning of the record and grows by one observation each step, so by the end it holds the lot.

On this record the two part company. Both are computing the spread. At December 2019 twelve months is everything there is, so the two agree exactly, at 2.0548 per cent. By December 2021 the rolling figure reads 3.4359 per cent and the expanding one reads exactly 3.0000 per cent, the true spread of everything so far. Then the record changes, and the two go different ways. At December 2024 the rolling figure reads 6.9462 per cent and the expanding one reads exactly 5.0000 per cent. Five per cent is the spread of the whole six years, and not the spread of anything the record is currently doing.

Each new observation is a smaller and smaller share of what an expanding figure holds, so the figure gets steadier and steadier and less and less able to notice anything new. Over the last twelve months of this record the rolling spread moved 1.2435 points while the expanding spread moved 0.6583. By the end, the expanding figure is understating the current stretch by 1.4031 percentage points and will keep understating it for years, quietly, without ever being wrong enough to look wrong.

Neither is right in general. An expanding figure is the honest one where the question is everything known so far and there is no reason to think the record has changed its behaviour. A rolling figure is the honest one where there is such a reason. The same word, the same units and the same chart shape can come out of either, so saying which of the two produced the number is never optional.

What does an analyst state beside a rolling figure?

Four things, and none of them takes more than a line. The four turn the arithmetic into working practice, and the list is the same whether the reader is a credit analyst looking at a borrower's rolling revenue, a research team publishing a rolling volatility, or a household treasurer tracking a twelve month average of the electricity bill.

1. The window length. Without it the figure cannot be reproduced, compared with anything, or interpreted. A rolling spread of 6.9462 per cent is a different statement at twelve months than at thirty six.
2. Which end of the window it is dated to. Almost always the last observation. Every reading above is dated that way. Occasionally the middle, which is a perfectly ordinary way to describe history and a serious problem if the result is then treated as though it had been available on that date. That specific dating error is covered separately.
3. How many readings the record produced. Sixty one is a series. Thirteen is a handful. One is not a rolling figure at all. Stating the count tells a reader immediately how much room there was for the line to have a shape.
4. Whether the movement is bigger than the window's own noise. This is the one that does real work. Before a move in a rolling figure is called a change, the distance that figure travels on a stretch where nothing changes has to be worked out and compared. On this record it is one division: the second stretch's spread of 6.4031 per cent divided by the square root of twelve gives 1.8484 per cent, and a rolling mean that has wandered less than a couple of those has said nothing yet.

The fourth check costs one calculation and would have stopped the failure below. It is not a formal test but a rough size comparison, and it settles whether a movement is even in the range where a question is worth asking.

What a rolling window covers, and what it does not