Heteroskedasticity: When Volatility Itself Varies

Heteroskedasticity means the spread of a record is not the same all the way through it. The six year record of the Nakshatra unit averages exactly 1.00 per cent a month across its first three years and exactly 1.00 per cent across its last three. The spreads over those two stretches are 3.00 per cent and 6.4031 per cent. The single figure of 5.00 per cent describes neither of them.

Somebody hands over 72 months of numbers and asks how much the thing moves about. One spread figure comes out, it gets written down, and without anyone meaning it a question has been answered that this particular record cannot answer with one number. Not because the arithmetic went wrong. The arithmetic is perfect. The record has two personalities, and only one of them was allowed to be described.

A word first about the thing being recorded. The Nakshatra unit, an invented one written to be taught with, trades nowhere. Its six year record holds 72 monthly changes, one for every month from the month ending 31 January 2019 through to the month ending 31 December 2024, all of them counted against an opening markThe starting level a record is measured against. The mark is decided rather than observed, and here it was written in as Rs 100.00/- on the last day of 2018, before any month of the record existed. of Rs 100.00/- set on 31 December 2018. String the changes together and they build a price path quoted to the nearest paisaOne hundredth of a rupee. An amount in rupees is not normally quoted any finer than that.. The path itself is never needed. Every quantity below is a percentage, and all of them can be rebuilt out of the 72 numbers on their own.

Two facts about how the record was put together matter for what follows. Every month in it is three things added up: a fixed 1.00 per cent, a calendar value drawn from a repeating set of twelve that cancels over any full year, and an irregular part. Both facts belong to that third part. The irregular part was arranged to cancel across each half of the record as well, so the two halves land on exactly 1.00 per cent rather than merely near it. Its size trebles partway along, and that trebling is the change in spread the record was built to show. Every observation arrives at the same frequencyThe spacing of a record: whether it carries one observation a day, one a week or one a month. Reading the same movement at a different spacing is covered separately., once a month, so none of them outweighs another.

Where these numbers come from: every figure below is recomputed from the 72 monthly changes themselves rather than lifted out of a table, and the changes are set out in full so that any of it can be redone by hand. No maintained series stands behind the record, and no authority is named in the third column of the table at the foot.

What does the word heteroskedasticity actually claim?

Far less than the length of it suggests. Heteroskedasticity is one of the rare cases where all of the difficulty sits in the name and none of it sits in the idea. Split the word: hetero means different, skedasis means scatter. Different scatter. The claim is that the spread of the record is not the same in one stretch of it as it is in another, and it says nothing whatsoever about the average, nothing about the direction, and nothing about what caused the change.

A tea stall stands outside one office building. Across an ordinary month it takes roughly the same amount every day, give or take a little. Then the festival fortnight arrives and the takings start behaving strangely: one day is triple the usual because somebody is buying for a whole floor, the next day is nearly nothing because half the building has gone home. Added up across the year, the stall took what it always takes. Its average day did not move. The scatter moved: a single day can now land much further from that average. Asked how much a typical day varies, and held to one number for the whole year, the owner would be wrong twice over: far too jumpy a description of the quiet months, and far too calm a description of the fortnight.

The tea stall shows the entire shape of the matter, and the record below does the same thing in numbers. The opposite condition has its own long name, homoskedasticity, meaning the same scatter everywhere. Homoskedasticity is worth knowing only because so much of what gets done with a spread figure quietly assumes it, and nobody ever says so out loud.

How is a changing spread spotted with no machinery at all?

The record is cut in two and the spread of each half worked out. Cutting and comparing is the whole test at its cheapest. The test needs nothing beyond the ordinary arithmetic: the same squared distances, the same division, the same square root, simply run twice instead of once. There are more careful procedures for deciding whether a difference found this way is large enough to take seriously, and not one of them is needed to notice the thing in the first place.

Where should the cut fall? Where there is a reason for it: a date when something plausibly changed about the conditions the record was collected under. A record long enough to worry about is usually long enough to halve, so with no reason at all the middle will do. The reason this check gets skipped is not that it is hard, it is that almost everybody who checks anything checks the average, and a check on the average passes this record without a murmur.

The blindness of an average check is the part worth carrying away. On the record below, the two halves have averages that agree to the last decimal place. A reviewer comparing the first three years with the last three on any measure of the centre finds nothing at all, files the record as well behaved and moves on. The scatter changed, and the scatter was never in the column being looked at. Recomputing a spread over a rolling windowA figure worked out again and again over a fixed length stretch that slides forward one observation at a time, so the record produces a moving reading instead of a single one. Covered separately. that slides along the record is the other way of seeing this and is covered separately; the split test is the one for five minutes and a suspicion.

What does the six year record look like when it is cut in half?

A cut at December 2021 gives 36 months on each side. Equal halves make the tidiest cut available, and that is why it was chosen. Here is what comes out of each pile, and then out of both piles put back together.

The two averages are identical to the last decimal place, and that is what makes this record such a clean case: nothing about the centre moved and everything about the scatter did. The later stretch is 2.1344 times as wide as the earlier one. The gap is not a subtle difference needing a test to see, but the difference between a record whose months land within a few points of the average and one whose months regularly land ten points away.

One thing the cut did not disturb is worth saying plainly. The record carries a repeating calendar shape, the same twelve values in both halves, so it contributes equally to each and cannot be what makes the two spreads differ. Removing the calendar shape by seasonal adjustmentSubtracting from each observation the average amount its own position in the cycle contributes, so that a repeating calendar shape is taken out of a record. Covered separately. would shrink both halves and would leave the gap between them standing. The irregular part changed between the two stretches, and nothing else did.

Why does one figure for the whole record hide both stretches?

Because of where it lands, and the arithmetic of that is worth doing slowly. A variance is the average squared distance from the average. Each of the first 36 months sits, on average, 9.00 squared points from the centre, so those months contribute 324 in total. Each of the last 36 sits 41.00 squared points away, contributing 1476. The two together make 1800 squared points across 72 months. Divided by 72 that gives a variance of 25.00. The square root of that is 5.00 per cent.

Because the two halves share exactly the same average, the whole record's variance is the plain average of the two half variances, so the single figure is not a compromise between the two stretches, it is the midpoint of them, and it is equally wrong about both. Sixteen points of variance above the first stretch, sixteen points below the second. The single figure sits precisely nowhere.

The midpoint happens to land on the round number 5.00 per cent here, and that is a property of how this record was built rather than anything to expect elsewhere. Which quantity does the averaging matters as well. Variances average; spreads do not. The plain average of 3.00 and 6.4031 is 4.7016. The square root bends the scale on the way through, so that average was never going to come out at 5.00. Averaging two spread figures directly is the step to stop at.

What does using the wrong spread figure actually cost?

Count it in months. Take the average and add and subtract 1.959964 spreads, and a 95 per cent band comes out. Built out of the whole record spread, the band runs from minus 8.7998 to 10.7998 per cent. The band is arithmetically correct on the whole record. Now lay it over each stretch in turn and count what escapes.

A 95 per cent band on 36 observations should leave about 1.80 of them outside. With each stretch given its own band, that is roughly what comes out: 1 outside in the first, 3 in the second. With one band used for everything, the first stretch produces 0 months outside out of 36. The second produces 5: Oct 2022, Nov 2023, May 2024, Jul 2024, Dec 2024.

The single band is wrong in both directions at once. The average of the two errors looks exactly like no error at all. Across all 72 months it leaves 5 outside against the 3.60 expected, and nobody would blink at that. The widths tell the same story: 19.5996 points wide against the first stretch's own 11.7598, so it is 1.6667 times wider than that stretch needs and only 0.7809 of the width the second stretch needs. Too loose in one place, too tight in the other, unremarkable on average.

Does a changing spread mean the centre moved as well?

No, and this record was built so that the point cannot be missed. A record whose spread trebles feels like a record where something happened, and on this one nothing happened to the average at all. The average reads 1.00 per cent across the first three years, 1.00 per cent across the last three, and 1.00 per cent across the whole thing.

Think about a household living on one salary that arrives on the same date every month. For three years the gap between what comes in and what goes out sits within a few hundred rupees either way. Then somebody in the house starts taking freelance work: some months bring in a great deal, some months bring in nothing, and across a year the household ends up with what it always ended up with. Nobody got richer. The household can no longer say in advance what any single month will look like. A wider spread claims that much about a record and no more.

A changing spread is not a change in direction, not a change in level, and not evidence that anything has gone wrong with the thing being recorded. A changing spread is a statement about how far the observations scatter, and it stops there. Nor is a changing spread a matter of memory. Whether this month says anything at all about next month is measured by autocorrelationA measure of how far a record resembles itself shifted along by a fixed number of observations, which is how memory in a record gets put into a number. Covered separately., a separate reading covered on its own. Whether the average of a record shifts over time is a third question again, and this record answers it with a flat no.

Does the gap survive if the record is cut somewhere else?

The question below is worth answering before the dial. Guessing first and being wrong teaches more here than reading on and nodding. The panel puts the cut anywhere from month 12 to month 60 and recomputes both sides as it travels.

Two things happen as the dial moves, and they are worth separating. The ratio of the two spreads never falls below 1.5411 and never climbs above 2.6230: at every one of the 49 cuts the dial allows, the later stretch comes out wider than the earlier one. The gap between the two averages behaves differently. The gap sits at exactly nothing at the halfway cut, stays under a single point at 37 of those 49 cuts, and opens up only when the cut is pushed so far along that one side is down to about a year of months.

The wobble at the far cuts is not the centre moving. A short stretch is being asked to report an average out of very few observations. How far an average built on twelve numbers can be trusted is exactly what a standard errorA figure saying how far an average computed from a limited set of observations is likely to sit from the average of everything it was drawn out of. A standard error shrinks as the count of observations grows. Covered separately. is for, and that is settled elsewhere. Set side by side as the cut moves, one of the two readings holds its ordering at every setting, and the other loosens as its stretch gets short.

What can honestly be done once it has been found?

Four things, and none of them asks for a method beyond those already set out.

Report both spreads instead of one. Two numbers with a date between them tell a reader more than one number does, and they take the same amount of room. Say which stretch any figure came from, every time. A spread with no stretch attached to it is the sentence that caused all the trouble in the first place. Compute any band or interval inside a stretch rather than across the record. The figure then describes months that actually resemble each other. And when somebody insists on one number, hand it over with the other two underneath it, so the reader who needs the detail can find it and the reader who does not is never made to care.

Nobody can say from this record alone when the change happened without first choosing where to look, and choosing where to look after seeing the data is a different problem, covered separately. The cut at December 2021 is a decision, not a discovery. The cut divides the record into equal halves and happens to sit where the record was built to change, and neither of those two facts was found by the arithmetic above. Write that down honestly and the reader can judge the cut for themselves.

Which three signs should prompt a search for a cut?

Three signs turn the idea into a habit, and they matter most to people who read records for a living rather than build them. A lender sizing a limit against a household's monthly surplus, an analyst writing the one line that describes a series to somebody who will never see it, an investor deciding how much room to leave around an estimate: all three are handed a single spread figure, and all three are entitled to ask which stretch it came from.

The first sign is a spread figure that feels too wide for what the recent part of the record shows. The eye is comparing the recent months against the number, and the number is describing months nobody is looking at. The feeling is worth trusting for long enough to run the split.

The second sign is a band that almost nothing falls outside of, and it is at once the most useful of the three and the least used. A 95 per cent band catching every single observation is not evidence of a well behaved record. A correct spread figure would have left roughly one observation in twenty outside, so a band that catches everything is evidence of a wrong one. On the first three years of this record, the whole record band catches all 36, and the furthest month from the average, Jan 2021 at 7.00 per cent, still leaves 3.7998 percentage points of clear air before the upper edge. Room like that is not comfort. Room like that is a symptom.

The third sign is the weakest and the broadest: any record long enough that conditions could plausibly have changed inside it. Six years of anything qualifies. So does a run of daily observations through a period when the thing being recorded changed how it operated. The check costs two minutes, so the bar for running it should be low.

One caution about all that tidiness: it does not travel. The Nakshatra record was assembled so that both halves land on the same average exactly rather than nearly, and a record that was collected instead of built will not oblige. The method survives the move without a scratch. The exactness does not, and every exact figure above belongs to this record alone.

Is anything here resting on somebody else's word?

Nothing whatsoever, and an empty answer is worth explaining rather than letting it look like something went missing. Splitting 72 numbers at a date, totting up the squared distances on either side and taking two square roots is arithmetic. Arithmetic issues from nobody, applies in every jurisdiction at once and is never revised. No regulator, exchange or data publisher could agree with 6.4031 per cent in a way that makes it a better answer for the second stretch than the addition above already makes it. An official name set against a made up number would lend it a weight it has not earned.

The Nakshatra unit and its six year record are invented.
Educational material. Not advice on any investment, tax, budget or market position.