Base Effect: When Last Year Distorts This Year's Number

A base effect is a growth rate that is large or small because the period it is measured against was unusual, not because this period was. Every growth rate is built from two levels, and the release prints only the ratio, never the base underneath it. The arithmetic is correct and almost uninformative on its own.

The three figures the panel opens on are the Sankhya volume index, and they are worth having in plain text as well as in the fields. The period being reported stands at 99.00 index points. The period it is measured against stands at 90.00. The gap between them is 9.00 index points, and 9.00 divided by 90.00 is 0.10, so the printed rate is plus 10.00 per cent. The reference period, two periods before that, stands at 100.00. The same 99.00 therefore sits 1.00 index point below where the series began, or minus 1.00 per cent. Both sentences are true of the same period, and only one of them is a headline.

Two neighbouring subjects put this arithmetic to work on a real release. Reading a price release meets the base effect at step two of its own procedure, where the base period is named before the headline is read. An industrial volume index meets it as a comparison month that can be dragged while this month stands still. Both apply what is worked out here. And all of it comes out of a single fact: a rate has a top and a bottom, and a release quotes the answer without quoting the bottom.

Why does a growth rate hide half of itself?

A growth rate is one level divided by another level, minus one. Two numbers go in and one number comes out, and the one that comes out is the only one that gets printed in a headline. Half of every growth rate is the base it was measured against, and nobody quotes the base.

The convention is not a criticism of anyone. A headline has room for one number, and the ratio is the number readers ask for. But it means the printed figure is a fraction with the bottom torn off, and a fraction with the bottom torn off can be made to say almost anything by choosing what the bottom was.

Consider a shop that took Rs 9,900/- yesterday. Is that good going? The question cannot be answered, and not for any want of judgement. The other number is missing. Against Rs 9,000/- the day before, it is up. Against Rs 10,000/- the day before, it is down. The Rs 9,900/- did not change between those two sentences. Only the comparison did, and the comparison is not a fact about yesterday at all.

What happens when only the base moves and nothing else?

Go back to the panel and do the one thing it was built for. Leave the reported period at 99.00 index points and touch nothing but the slider. The slider writes into the comparison period and into nothing else at all. At a base of 90.00 the printed rate reads plus 10.00 per cent. Drag to 100.00 and it reads minus 1.00 per cent. Drag on to 110.00 and it reads minus 10.00 per cent. Drag the other way, down to 82.50, and it reads plus 20.00 per cent. The headline covered 30.00 percentage points, from plus 20.00 to minus 10.00, and the period it claims to describe never moved by a hundredth of a point.

The lower half of the panel draws that as a curve rather than as four separate readings, and the shape of it is worth a minute. A bigger base always gives a smaller printed rate, so the curve slopes downward the whole way. The curve is steepest on the left, where the base is small, and a collapsed comparison period therefore produces the wild rates rather than merely the large ones. The curve crosses zero at exactly one place, where the base equals the reported level, at 99.00. Left of that crossing the release prints a rise and right of it the release prints a fall, and the period being reported on is identical at every point along it.

One caution belongs beside that, stated plainly. None of this says a release is misleading anybody. The arithmetic is correct at every point on the curve, and a compiling body that printed any other number would be wrong. The sweep shows something narrower and more useful: how little of a printed rate is a statement about the period it is printed for.

How can a series grow 10.00 per cent and still sit below where it started?

Here is the case that makes the whole argument click, and it is worth sitting with because it feels wrong the first time. Consider a volume indexA series rebased so that one chosen period equals 100, which allows movement to be read without carrying the awkward absolute size of the thing being measured. The price indices are built separately. for the Republic of Sankhya. The index starts at 100.00 index points. In the next period it falls to 90.00. In the period after that it rises to 99.00.

Now compute the rate for that third period. 99.00 divided by 90.00 is 1.10, so the series grew plus 10.00 per cent. The plus 10.00 per cent is not a rounding, an approximation or a trick. The growth rate is correctly calculated, and any release compiled on that series would print exactly that figure.

And now compute where the series actually is. 99.00 against a starting level of 100.00 is one index point lower. In per cent that is minus 1.00. The series grew 10.00 per cent and its level is 1.00 per cent below where it began, and both of those statements are completely true at the same time. One of them will be in the headline and the other will not.

Nothing has gone wrong here. The two sentences answer different questions. The 10.00 per cent answers how this period compares with the one immediately before it, and that one was a bad period. The minus 1.00 per cent answers where the series stands. A reader holding only the first sentence does not have a partial view of the level; they have no view of it whatsoever.

What does that look like at the scale of one household?

Take the fruit stall outside a hospital gate. In an ordinary week it takes Rs 10,000/- a day. Then the road outside is dug up for a month and takings drop to Rs 9,000/- a day. The road reopens and takings come back to Rs 9,900/- a day.

The stallholder tells his supplier that business is up 10.00 per cent. He is right: Rs 9,900/- against Rs 9,000/- is exactly plus 10.00 per cent. He is also Rs 100/- a day worse off than before the road was dug. Over a month of trading that is Rs 3,000/- he is not seeing. The rate he quoted is arithmetically perfect and it describes the road works rather than his stall. Both numbers live in the same three figures, and only one of them will come up in conversation.

Why do a fall and a rise of the same per cent not cancel?

The asymmetry is the engine underneath the whole thing, and once it is seen in index points rather than in per cent it stops being surprising. A percentage is not a quantity. A percentage is a quantity divided by whatever it was measured against, so the same percentage applied to two different bases produces two different quantities.

Work it through on Sankhya. The fall in period 1 is 10.00 per cent of 100.00. In index points that is 10.00. The rise in period 2 is 10.00 per cent of 90.00. In index points that is only 9.00. Same percentage, different base, and so a different number of points: 10.00 points out and only 9.00 points back in. The series is left 1.00 point short. The fall is taken off a bigger number than the rise is added to, so down then up by the same per cent always lands lower.

The shortfall has a shape worth remembering. On a starting level of 100.00, an equal down-then-up move of size d leaves a shortfall of 100 multiplied by d squared. At 10.00 per cent that is 1.00 point. At 20.00 per cent it is 4.00 points. At 30.00 per cent it is 9.00 points. The gap does not grow in step with the move. The gap grows with the square of it, and large equal moves therefore leave surprisingly large holes. The order makes no difference either: up then down by 10.00 per cent also lands at 99.00.

The same asymmetry turns up elsewhere. Exchange rates carry the same asymmetry. The two percentages are measured against different bases, so a currency weakening by some per cent and then strengthening by the same per cent does not return to its old quote. The asymmetry is the same arithmetic wearing different clothes.

What does the whole four period path look like, levels and rates together?

One more period and the effect not only appears but disappears again. The disappearance is the part that makes it identifiable. The Sankhya index runs 100.00, then 90.00, then 99.00, then 100.98 index points. The three rates that come out of those four levels are minus 10.00 per cent, plus 10.00 per cent and plus 2.00 per cent.

A rate on its own is the exact fault at issue, so the table reads by the row and not by the column. The 10.00 per cent period is the one where the level is still below the start, and the 2.00 per cent period is the one where the level finally passes it. The rate column on its own ranks those two periods the wrong way round, and by a wide margin.

Vintage of every number here: an illustrative first-print series for the Republic of Sankhya, carried unchanged through every figure and through both panels. The levels were chosen for clean arithmetic rather than realism, so every step can be recomputed by hand. Each level divided by the one above it gives the rate in the same row, and each rate applied to the level above it gives the level, so the two columns check each other in both directions.

Now run the same three rates against a base that behaved ordinarily, with period 1 holding at 100.00 instead of falling. The rates are identical in both columns. The levels are not, and the distance between them is the size of the misreading.

Two series, the same printed rate of plus 10.00 per cent in period 2, and 11.00 index points between where they actually stand. A reader working from the rate column alone cannot tell them apart.

How is a base effect separated from a genuine improvement?

A large rate is sometimes exactly what it looks like, so something better than suspicion is needed. Three tests, and they get progressively harder to fool.

The first test is to read the level instead of the rate. A level does not care what happened in any other period, so no other period can distort it. On Sankhya, period 2 is 99.00 index points against 100.00 at the start, and that one line settles the question. Reading the level is the fastest test, and it is available whenever the compiling body publishes the level series.

The second test is to compare against a period from before the unusual one, so the unusual period sits inside the comparison rather than at the edge of it. Sankhya from period 0 to period 2 is 100.00 to 99.00. The two periods together come to minus 1.00 per cent, or about minus 0.50 per cent each once allowance is made for compoundingApplying a rate to a total that already includes the effect of earlier rates, so growth multiplies rather than adds. Compounding is why two periods of plus 5.00 per cent come to plus 10.25 per cent and not plus 10.00.. Stretching the window this way dilutes the odd period instead of letting it sit in the denominator on its own.

The third test is the one that actually identifies a base effect rather than merely doubting a rate, and it is a test of patience. Wait one period and watch the rate with nothing new happening. A base effect unwinds on its own the moment the unusual period leaves the comparison, and an improvement does not. The third test is therefore the signature, and the other two are only checks. On Sankhya the rate goes from plus 10.00 per cent to plus 2.00 per cent between period 2 and period 3, and nothing had to happen for it to fall: the base simply changed from 90.00 to 99.00. Had the series genuinely shifted to a faster path, the rate would have stayed up.

When does a base effect stop mattering?

The moment the unusual period stops being the comparison period. The whole answer is that one sentence, and notice what is not in it: nothing about the economy, nothing about policy, nothing about anybody doing anything. The comparison window moves forward one period at a time whether or not anything else changes, so a base effect ends on a schedule set by arithmetic.

The schedule has a genuinely useful consequence. The date on which a base effect leaves a series can be stated in advance. Very few numbers allow anything of the kind. Counting the periods until the odd one drops out of the comparison window gives the point at which the printed rate changes character, without any knowledge at all of what the series will do. Almost nothing else in reading a release permits that much precision about the future, and it costs nothing but counting.

The corollary is a discipline. When the odd period is visible in the denominator, the rate about to be read was largely decided some time ago, and treating it as news about the current period is a category error.

So what does a growth rate not say?

Three things, and they are the three things people most often take from it. A growth rate does not give the level. A growth rate does not say whether the level has made up ground lost earlier. And a rate can be large purely because of the period underneath it, so it does not establish that anything about the way the series behaves has changed.

The cleanest way to hold this is a distinction. A rate is a comparison and a level is a fact, and the two answer questions that do not overlap. A rate needs two periods and moves when either of them moves, so a rate can be rewritten by something that happened a year ago. A level needs one period and is untouched by every other period in the series, so no base effect can reach it. Neither one is the better number; they are answers to different questions, and the trouble starts only when one is used to answer the other.

Who keeps the level series open beside the rate, and why?

Watch what an equity research analyst covering a cyclical business actually does with a release. The rate goes into the note because a note is expected to carry a rate, but the working file has the level series next to it, and the level column is the one that gets looked at when a decision hangs on the answer. The reason is narrow and practical: the level is immune to the base effect entirely, so putting both series side by side dissolves every argument about the base before it starts.

The same habit shows up wherever the stakes are real. A credit officer sizing a working capital limit for a components supplier wants to know whether the volumes that will service the loan are back to where they were. The question is about a level, and no rate can answer it. A policy economist tracking a sector wants the level path so that a period of arithmetic unwinding is not mistaken for a change in direction. A household deciding whether the shop can support a second earner needs to know what the takings are, not what they did against a bad month.

The practical form of it is a habit rather than a technique: a growth rate written down carries the two levels beside it on the same line. The habit takes a few seconds and makes a base effect impossible to miss. The moment the base is visible, the rate stops being able to mislead anybody.