Seasonality: Repeating Patterns, How They Are Handled, and What the Adjustment Removes

A pattern is seasonal when it repeats on a fixed calendar period and cancels over one full turn of that period. On the six year record of the Nakshatra unit the January months average 5.00 per cent and the June months average minus 3.00 per cent. Subtracting each month's own factor leaves all twelve calendar months averaging exactly 1.00 per cent.

Six Januaries in a row came in strong on this record and six Junes in a row came in weak. A repeating pattern and a run of luck look identical from the outside, and only arithmetic separates them. Nothing below has to be taken on trust: the whole record is printed out, and the arithmetic that finds the pattern is six additions and one division, done twelve times over.

The Nakshatra unit is an invented traded object written for teaching, and its six year record is one list of 72 monthly changes running from the month ending 31 January 2019 to the month ending 31 December 2024, measured against an opening markThe level a record counts from, set at the moment before its first observation exists. On this one that level was picked rather than measured, and it stands at Rs 100.00/- dated 31 December 2018. of Rs 100.00/- on 31 December 2018. The 72 changes compoundWhat happens when each change is applied to whatever the previous change already left behind, so a run of changes multiplies together instead of adding up. into a price path that ends at Rs 187.4539/-, carried down to the paisaThe hundredth part of a rupee, and the finest step an ordinary amount in rupees gets written down to.. Seasonal adjustment works on the monthly changes rather than on the price path, so every figure below is a percentage and not an amount.

One detail of how the record was put together matters later on. Each month is a steady 1.00 per cent, plus one of twelve repeating calendar values that add to zero across a year, plus an irregular part. The irregular part was chosen so that it adds to zero down each calendar month across the six years. A collected record never behaves like that, and the built in cancellation is the reason the twelve figures below are exact rather than estimated.

On this record the column averages are the pattern rather than an estimate of it, so working them out again off the grid below lands on the same twelve numbers.

What makes a repeating pattern seasonal rather than just a run of good months?

Three things have to hold at once, and only the third of them does any real work. The first is that it repeats on a fixed period, so which observations belong together can be stated in advance: every January with every other January, every Monday with every other Monday. The second is that it comes back, so the same part of the cycle behaves the same way each time round rather than once. The third is that it cancels over one full turn, adding to nothing across a complete twelve months.

The third condition is the one that separates a repeating pattern from a direction, and it is the only one of the three that can be settled with arithmetic alone. Think of a food stall outside one office building. The stall takes more in the week before a big festival every single year. In the fortnight afterwards, with everybody away, it takes less. Across a whole year the extra and the shortfall cancel and the stall is no bigger than it was. The festival swing is a seasonal pattern. Now think of a stall whose takings creep up every year because the office keeps hiring. The creeping rise never cancels, and no amount of grouping by calendar month will make it.

How is the pattern found in a record somebody has handed over?

Sort the observations into twelve piles by the calendar month written on each one, then average each pile. Sorting and averaging is the entire method. On a record of six full years each of the twelve groups holds six observations, so each average is six numbers added together and divided by six, and there are twelve of those to do.

Averaging the six Januaries works because everything in those six numbers that is not about January averages away, leaving behind only what January itself contributes. Each January carries the steady part, the January part and its own irregular part. Add six of them and the steady part is still there, the January part is still there, and the six irregular parts have partly cancelled each other. On this particular record the six irregular parts cancel exactly. The twelve averages below are therefore the pattern itself and not an approximation of it.

The grid leaves out a great deal that other methods need. No record has to be lined up against a shifted copy of itself, and no rolling windowA run of consecutive observations, always the same count of them, which picks up one ahead and drops one behind at every step, so whatever gets worked out inside it gets worked out again. It has notes of its own elsewhere. has to be slid along it. Nothing has to be fitted to anything either. The method needs 72 numbers sorted into twelve piles by the label on each one, and each pile then averaged. Every figure below can be checked with addition alone.

Why is a single memory figure not the test for a repeating pattern?

The shortcut is tempting, and on this record it fails badly. There is a standard figure that measures how closely a record resembles a copy of itself shifted back a fixed number of steps, called autocorrelationOne number, between minus one and one, for how closely a record resembles a copy of itself shifted back a fixed number of steps. It is built and used separately.. If a record repeats every twelve months, the reasoning goes, then shifting it back twelve months should line it up with itself and the figure should be large.

On the six year record that figure, read at a shift of twelve months, is 0.0022. A reading of 0.0022 is as close to nothing as a reading gets. And yet the calendar pattern in this same record is enormous: January's average and June's average are 8.00 per cent apart, and the pattern accounts for a full 28.00 per cent of everything the record does. Both of those statements are true about the same 72 numbers at the same time, so a memory figure cannot serve as a test for a repeating pattern.

The reason is worth stating plainly. The near zero reading is a property of this record rather than a rule of arithmetic. The reading at twelve months lands close to zero because the second half of the record carries a far heavier irregular part than the first, and that noise swamps the twelve month agreement in a single summary figure. On a different record the twelve month reading might well come out large and agree with the calendar averages. The size of the figure is not the general lesson. The general lesson is that the calendar averages answer the question actually asked, month by month. A shifted correlation answers a different question and returns one number for the whole record.

Naming the convention whenever such a figure is quoted

There is a second reason to be careful with these readings. More than one defensible convention exists for computing them, and they disagree. Take the record after its calendar pattern has been removed and read the one month figure two ways. The series convention takes deviations from a single average, the record's, and puts every one of the record's squared deviations underneath as the divisor. The series convention gives minus 0.0162. The paired convention treats the 71 overlapping pairs as two columns and correlates them, each column centred on its own average, and gives minus 0.0166.

The two conventions part company further as the shift gets longer, so a figure quoted without its convention is a figure nobody else can reproduce. At a shift of twelve months on that same adjusted record the series convention reads minus 0.2785 and the paired convention reads minus 0.3465. On the record before adjustment the two read 0.0022 and 0.0000 at twelve months. Both still say the same thing, that there is nothing much to see. But the numbers are not the same numbers, and only one of them is right for whichever convention somebody else is using.

What are the six year record's twelve calendar month averages?

Here they are, in order: January 5.00 per cent, February 3.00 per cent, March 1.00 per cent, April minus 1.00 per cent, May minus 2.00 per cent, June minus 3.00 per cent, July minus 2.00 per cent, August minus 1.00 per cent, September 1.00 per cent, October 3.00 per cent, November 4.00 per cent and December 4.00 per cent.

The shape is a single fall and a single recovery: strongest in January, sliding down to a trough in June, then climbing back to 4.00 per cent by November and holding there through December. The twelve months trace one turn of a cycle, and the record does it six times. Whether such a shape is common in anything real is a separate question; what matters is that when a shape like this exists, this is how it is extracted.

Two of the twelve deserve a separate sentence. March averages exactly 1.00 per cent and September averages exactly 1.00 per cent, and 1.00 per cent is also what the whole record averages. The match is arithmetic rather than a coincidence: those two calendar values are exactly zero in this record, so March and September carry no calendar effect at all and sit on the record's own average by construction. Any month whose calendar value is zero will do the same thing on any record.

What is a seasonal factor, and why do the twelve of them add to zero?

A seasonal factor is the calendar average with the record's own average taken out of it. Subtracting 1.00 per cent from each of the twelve gives 4.00, 2.00, 0.00, minus 2.00, minus 3.00, minus 4.00, minus 3.00, minus 2.00, 0.00, 2.00, 3.00 and 3.00. Each one answers a narrow question: what does landing in this calendar month do to a figure, over and above what landing anywhere in this record already does?

The twelve add to exactly zero, and that is not luck. Here is the reason, and it takes one line. Every calendar month on this record carries the same number of observations, six, so the average of the twelve calendar averages is the record's own average. Subtracting that same average from each of the twelve therefore removes exactly as much as the twelve contained. Five of the factors sit above the line and add to 14.00 per cent, five sit below and add to minus 14.00 per cent, and two are zero.

One condition sits inside that reason, and it is the first thing to give way once a record stops being tidy. Six full years gives six observations in every column. A record that stops in the middle of a year does not: it might hold seven Januaries and five Julys. The moment the columns hold different counts, the twelve averages no longer average to the record's own average, and the twelve factors no longer add to zero. The factors will come out close, and close is not the same as zero, and a set of factors that does not add to zero will quietly move the average of whatever it is subtracted from.

What is Seasonally Adjusted Data, and what has actually been taken out of it?

The operation is a subtraction and nothing more. Take every observation, look up the factor for the calendar month it landed in, and subtract that factor from it. Every January observation loses 4.00. June's factor is minus 4.00, and subtracting a negative adds, so every June observation gains 4.00. March and September have factors of zero, so they are left exactly as they were. Sixty of the 72 observations move, twelve do not, and the largest single move anywhere in the record is 4.00 per cent.

An adjusted figure is an original figure with a number subtracted from it, and that number came from other years. A seasonally adjusted January reading for 2021 has had 4.00 per cent taken off it, and that 4.00 per cent was worked out from the Januaries of 2019, 2020, 2021, 2022, 2023 and 2024 together. The adjusted figure is a hybrid: one month's observation, corrected by six years of Januaries. The hybrid is not wrong, and it is not what was observed.

Now the results, all of which can be read off the grid above. Every one of the twelve calendar months of the adjusted record averages exactly 1.00 per cent. The record's own average is unchanged at 1.00 per cent. Its variance falls from 25.00 to 18.00 and its spread falls from 5.00 per cent to 4.2426 per cent, the square root of 18.00.

The calendar pattern accounts for exactly 7.00 of the record's variance of 25.00, or 28.00 per cent of it, and the remaining 18.00 is the irregular part that no grouping can reach. Notice that the fall in the spread is much less dramatic than the fall in the variance: the variance drops by 28.00 per cent while the spread drops by only 15.1472 per cent. A spread is the square root of a variance, and a square root flattens a fall. Quoting one and thinking of the other is an easy way to overstate what the adjustment achieved.

Trend vs Seasonality: which single test separates a direction from a repeating pattern?

Add it up over one full turn of the cycle and see whether it comes back to nothing. The running total over one full turn is the whole test, and everything else anybody says about the two follows from it.

Run it on this record. The calendar pattern's running total climbs to 6.00 per cent by March, falls to minus 8.00 per cent by August, and arrives back at exactly 0.00 in December. Twelve months later it traces the identical shape. The record's steady 1.00 per cent a month has a running total of 1.00, then 2.00, then 3.00, and it reaches 12.00 per cent by December and keeps going. One of them cancels over a full cycle and comes back; the other cancels over no span at all and never comes back, and that is the difference.

The consequence for the adjustment is immediate. One of them sums to zero over a year and the other does not, so taking the calendar pattern out cannot touch the direction. Twelve numbers that add to zero, subtracted from twelve months, take nothing away from the year. The grid shows exactly that: every year's own average is identical before and after the adjustment, and the adjusted record still climbs at 1.00 per cent a month. Pulling a direction out from under the noise is a harder job than this one, and it is covered separately.

What does the adjustment not do?

Four things, and the last one is the one that catches people.

The adjustment does not change the average, either for the whole record or for any calendar month in it. Every one of the twelve months came in at 1.00 per cent afterwards, the record's own average, and the record's average was 1.00 per cent before as well. Nor does it remove the direction, for the reason set out just above. Nor does it settle this record's changing spread: before the adjustment the first three years have a spread of 3.00 per cent against the last three years' 6.4031 per cent, and afterwards those two read 1.4142 per cent and 5.8310 per cent. The pattern that was removed was sitting equally in both halves, so the gap between the two halves actually widens in relative terms, from the second stretch being 2.1344 times as wide as the first to being 4.1231 times as wide. Whether that changing spread matters and what to do about it is covered separately.

And it does not make the record predictable. Anybody who has just watched a large pattern be lifted out of a record will assume that whatever is left must be tidy. On this record the exact opposite holds. The remainder is 18.00 of variance with no repeating shape in it at all. The adjusted record's month to month reading, on the series convention named earlier, is minus 0.0162. A reading that small is about as close to no memory as a record gets. The pattern was the only orderly thing in there, and removing it did not reveal an order underneath. Removing the pattern removed the order.

What has to be stated beside an adjusted figure?

Three things, every time, and none of them is optional.

State that the figure is adjusted. State what the factor was for that particular month. And state how many turns of the cycle the factor was computed from: six, on this record. The third one is the one that gets left off, and it is the one that decides how much the factor is worth: a factor built from two turns and a factor built from twenty are the same kind of number and not remotely the same quality of number, and once the subtraction has happened the difference is invisible. Six Januaries averaged together is a thin base, and the adjusted figure shows no sign of that thinness unless somebody wrote it down.

The label matters wherever a monthly figure gets compared against something. Somebody deciding whether to advance money to a small business wants to know whether a weak month is the business slipping or the calendar repeating itself as it does every year. A household looking at an electricity bill that is high every summer is doing the same arithmetic informally, and getting it right without ever computing a factor. Anybody handed a column of monthly numbers and asked what changed is one unlabelled column away from answering the wrong question. The habit that protects all three of them is the same: label the column before anybody reads it, and never mix the two kinds in one comparison.

Is anything here resting on somebody else's word?

No, and the empty answer is worth spelling out rather than leaving as a gap. Sorting 72 numbers into twelve piles by the label on each one, averaging each pile and subtracting the result is arithmetic. Arithmetic has no issuing body, no jurisdiction, no revision date and nothing that could be confirmed at a source. There is no regulator, exchange or data publisher whose agreement would make 5.00 per cent a better answer for January than the six additions above already make it, and there is no published series that this record stands in for. Putting an official name in the third column below would lend an invented record 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.