A Coefficient Is Not a Percentage: What '333%' Means in a Study of College Competition

The question

When a study logs an outcome variable and then reports a regression coefficient, what does that coefficient mean when a reader converts it into a percentage? More precisely: a study estimates how market competition affects the social benefits and social costs of higher education, reports instrumental-variable coefficients of −3.33 and −3.10 on logged outcomes, and states that moving from monopoly to near-perfect competition "raises total social benefits by approximately 333% against a 310% rise in total costs." Is that sentence correct?

The question has two layers, and they must be kept apart. The first layer is arithmetic: given a coefficient of −3.33 on a logged outcome and a one-unit change in the regressor, what percentage change in the underlying outcome does the model imply, and does any standard convention produce 333? The second layer is causal: does the study identify the effect of competition on quality at all, or does the estimate reflect something else? The arithmetic is checkable from the published table alone. The causal claim is not, and no amount of arithmetic fixes it.

This brief does both, but it separates them deliberately. The arithmetic is where a reader can be misled by a single word. The causal claim is where the study's scientific value lives or dies.

Why it matters

Elasticities and semi-elasticities are how most quantitative social science communicates magnitudes. Policy is argued in percentages: a 333% gain against a 310% gain sounds like a transformation; a 7% relative advantage sounds like a rounding error. When the same coefficient can honestly be written as 333%, as 2,694% or as a ratio of 1.07, the choice of convention is doing a lot of the persuasion.

That matters for anyone reading applied economics, but it matters especially for readers of policy research on education, where market structure — how many providers compete, whether they are publicly or privately run, how concentrated the market is — is a live question in many countries. Brazil's private higher-education sector is one of the largest in the world and has consolidated heavily over the past two decades [1,2], so a credible estimate of what competition does to quality would be valuable. A number reported with the wrong order of magnitude is worse than no number, because it anchors the debate.

The general lesson is portable: log transforms are the default in labor, education and development economics, the coefficient's percentage interpretation is only approximately valid, and the approximation degrades exactly where effects are largest and most interesting.

The study

The paper at issue is a working paper by Israel Gottschalk, "Does Competition Undermine Higher Education Quality? Evidence from Brazil" [3]. It studies subject-city markets in Brazilian private higher education, measures competition with a Herfindahl-Hirschman Index (HHI), and instruments the HHI with its value eight years earlier. The outcome family is built from INEP's publicly distributed higher-education data: final-year course scores (ENADE), entry scores (ENEM) used to control for incoming ability, median fees, and programme-quality indicators. The repository accompanying the paper contains a single Stata do-file, an ownership crosswalk, and the paper PDF [4]. The author is at the Centre for Competition Policy, University of East Anglia; the README describes the paper as "in preparation / under review."

Table 7 of the working paper reports the social cost-benefit calculation [3]. The four columns are the ordinary least squares (OLS) and IV estimates for benefit and for cost, all on logged market-level outcomes, all with N = 4,988. The IV coefficients are −3.33 for benefit and −3.10 for cost, both significant at conventional levels. The sentence that follows reads:

"The IV coefficients imply that moving from monopoly (HHI = 1) to near-perfect competition (HHI ≈ 0) raises total social benefits by approximately 333% against a 310% rise in total costs: competitive markets deliver more social benefit from higher education per unit of cost than concentrated ones." [3]

The thesis version reports the same coefficients with standard errors (0.346 and 0.333) and repeats the claim as "the social benefits would increase by 333% while costs would increase by 310%" [5]. No version of the sentence states the convention being used.

Reading a log coefficient: three conventions, three answers

The outcome here is logged. The do-file that produces Table 7 defines the outcome with gen benefit=log(benefit1) and gen cost=log(cost1), where benefit1 is a score-weighted enrollment measure and cost1 is a weighted fee measure [4]. So the coefficient β in the regression

log(y) = β · HHI + controls + error

relates a change in HHI to a proportional change in y. The change in HHI from 1 (monopoly) to about 0 (near-perfect competition) is a change of ΔHHI ≈ −1, a large change by the standards of the approximation.

Three conventions are in common use for moving from β to a percentage:

Convention A: ordinary proportional change. The exact proportional change is exp(−β·ΔHHI) − 1. With ΔHHI = −1 this is exp(β) − 1, so a coefficient of 3.33 (sign aside) means the outcome multiplies by exp(3.33) ≈ 27.9, i.e. grows by about 2,694%.

Convention B: natural-log (Naperian) percentage. β × 100, valid to first order for small β. This gives 333%, which is the number the paper states.

Convention C: initial-point percentage change. exp(β) − 1 again in the large-change case, but often written differently when the change in the regressor is itself fractional; for a one-unit change in a variable bounded in [0,1], this is the same as Convention A.

The three conventions are not equally wrong. As β → 0 they agree: exp(β) − 1 ≈ β + β²/2 + …, so the difference between 333% and 2,694% is entirely the second-order term, which becomes dominant once |β| exceeds roughly 1. There is no error when β = 0.05 (5% either way). The error is a large-coefficient problem.

There is a fourth reading worth naming, because it is the one a careful reader might defend: the sentence's "approximately" may be doing the work of flagging that the number is a log-scale approximation. "Approximately 333%" is honest as a log-percentage statement. It is misleading only if a reader takes 333% as a statement about the size of the real change. Which a policy reader will.

Under any convention, what is checkable from the paper alone:

The table's point is that the direction is robust under the natural-log reading but the magnitude is not: the ratio of benefit gain to cost gain is 1.07 under Naperian percentages and 1.26 under exponentials. Both exceed 1, so "benefits outpace costs" survives. But the paper's statement that the ratio is 333/310 ≈ 1.07 is a statement about the log scale, not about the money.

The causal question the arithmetic cannot settle

Everything above is arithmetic. It establishes that the sentence's percentage is a log-scale percentage, and it establishes the size of the gap between conventions. It does not establish that the causal estimate is what the paper says it is.

The study's identification rests on instrumenting current market concentration with concentration eight years earlier. A valid instrument must affect quality only through current competition. Concentration eight years ago plausibly affects today's quality through many channels: it shaped which institutions grew, which markets attracted entrants, and which providers exited. Any of those channels is a separate pathway, and the study does not (as reported in the text read here) supply an exclusion test that separates the competition pathway from the persistence pathway. The first stage is reported as strong (F ≈ 174.65 in Table 2), which speaks to instrument strength, not to exclusion.

That is a standard concern with any persistence-based instrument, and it is not unique to this paper. It is worth stating here because the headline sentence invites a reader to treat a large coefficient as a large causal fact.

The structural features of the cost-benefit measure matter too. The benefit measure multiplies a course score by enrollment; the cost measure is aggregate fees. Both are logged and then regressed on the same HHI. The sentence that compares 333% against 310% is therefore comparing two coefficients on two different outcome scales, then describing the comparison as a ratio of benefit to cost. Whether that comparison is meaningful depends on whether log-benefit and log-cost are on comparable units — a question the paper does not address in the text read here.

A further detail occurs in the do-file that produces Table 7: lines 2252–2253 replace missing log values with zero, replace benefit=0 if benefit==. and replace cost=0 if cost==.. Because benefit is defined as log(benefit1), a zero value corresponds to a market whose measured benefit is 1 rather than a missing market. Whether such markets are in the Table 7 sample (N = 4,988) is unknown from the do-file alone, and the paper does not comment on the imputation. This is exactly the kind of implementation detail that a replication would resolve; it is noted here because it bears on whether the reported coefficients are estimated on the sample the table implies.

What the study does establish, and what it does not

The study's contribution is the combination of a market-level competition measure built from a careful ownership crosswalk (treating commonly owned providers as one competitor), an instrument, and a cost-benefit framing. That combination is rarer than it should be: most higher-education competition work studies enrollment or price, not the joint social accounting.

What the read evidence supports:

• Observed: the paper reports IV coefficients of −3.33 and −3.10 on logged benefit and cost, N = 4,988, and a first-stage F of about 174.65 [3,5].
• Verifiable arithmetic: those coefficients reproduce 333 and 310 under the natural-log percentage convention, and about 2,694 and 2,120 under the exponential convention.
• Not established in the text read: which convention the authors intended, whether the exclusion restriction holds, and whether the reported sample corresponds to the estimation sample after the missing-value imputation.

Evidence quality

The arithmetic claim is as strong as arithmetic gets: it is checkable by anyone with the published table, and it is reproduced here. Its confidence is effectively certain.

The causal claim is much weaker as evidence. The paper is a working paper, not a peer-reviewed article; no published replication, comment, erratum or correction was located in the searches run for this brief [6]. The instrument's exclusion restriction is asserted rather than tested in the material read. The cost-benefit measure's units are not reconciled. And the reported figures include at least one implementation detail (the zero-replacement of missing logs) that could change the sample.

A genuine correction: the paper's Table 2 first-stage F of 174.65 was read at source; an earlier note in this brief's provenance gave the first-stage F as "about 58," which does not appear in the paper's Table 2 and is recorded here as superseded [7].

Competing explanations for the reported sentence

Three explanations are consistent with the sentence being written as it is:

1. Convention, correctly applied but unclearly labelled. The authors used log percentages throughout and simply did not flag the approximation. Most likely. Under this explanation the sentence is imprecise rather than wrong: 333% is a defensible log-percentage statement.
2. Convention confusion. The authors intended ordinary percentages and miscomputed. Less likely, because 333 and 310 are exactly the log percentages of 3.33 and 3.10 — a coincidence is improbable if the number were meant to be an ordinary percentage.
3. A different construction. The coefficients in the sentence are not the logged-outcome coefficients at all, but a percentage constructed from an unlogged or differently scaled model. The do-file shows the logged model is the one that produces benefit.tex and thus Table 7, so this explanation is unlikely, but it cannot be excluded without the online appendix, which was not located.

The first explanation is the most consistent with the material. It also happens to be the one that makes the sentence technically defensible and practically misleading — the worst combination for a reader.

Limitations and counterevidence

• The published article. The working paper and the PhD thesis are the versions read here. A journal version, if one exists, was not located; the repository says "paper in preparation / under review" and lists no journal [4]. If a journal version rewrote the sentence to state the convention, the reporting question would be resolved. It has not been checked against a journal text, because none was found.
• The online appendix. The working paper's Table 7 note says standard errors are clustered at municipal and subject level and reported in an online appendix; the appendix was not located. It could carry the calculation that maps the coefficient to 333%.
• The target may be version-sensitive. The repository's newest commit is dated 14 September 2026 and the PDF's hash was recomputed in this run and matches the value recorded in the program's charter [7], so the text read is current as of this run.
• The arithmetic does not touch the causal claim. A reader persuaded by this brief should not conclude that competition does or does not raise higher-education quality. It concludes only that one sentence's percentage is a log-scale percentage.
• Selection into the sample. Programs with missing fee or score data are handled differently across the fee and benefit constructions (the fee construction substitutes a fallback value at lines 2233–2235; the benefit construction does not), which could make the benefit and cost regressions run on different effective samples. The paper does not discuss it. This is a candidate for a replication, not a finding.

The best current answer

The sentence is a log-scale sentence. Coefficients of 3.33 and 3.10 on logged outcomes are exactly 333% and 310% as natural-log percentages, and about 2,694% and 2,120% as ordinary proportional changes. The paper's direction — benefits outpace costs — survives both readings, but the size a reader takes away depends on which convention is being used, and the paper does not say. That is a reporting problem of consequence for a policy audience, and it is the kind of problem a replication can settle.

Confidence: ~97% that the coefficients 3.33 and 3.10, read as log percentages regardless of intent, are the origin of the 333% and 310% in the sentence. The two numbers match exactly, and the sentence appears in both the working paper and the thesis with the same numbers.

Confidence: ~85% that if the paper's authors were asked, they would say they meant log percentages. The exact match and the standard convention in applied microeconomics both point that way.

Confidence: ~30% that the reported 333% and 310%, if intended as statements about real changes in benefits and costs, are correct as ordinary percentages. This is essentially ruled out by the arithmetic.

Confidence: ~55% that the study's causal claim — competition raises higher-education quality and social benefit per unit cost — would survive a registered replication over the same public inputs. The instrument's exclusion restriction and the cost-benefit construction are both untested in the material read. This is a genuine uncertainty, not a doubt manufactured from arithmetic.

Reversal conditions

The confidence in the arithmetic reading would fall if the authors supplied an appendix showing a different construction of the percentage, for example one based on an unlogged model or on a level-on-level regression. It would rise if a second, independent reading of the same table reproduced 333 and 310 only under the log convention. The causal confidence would rise with a registered replication that recovers the sample, tests the exclusion restriction, and reproduces Table 7 within a pre-specified tolerance; it would fall if that replication shows the reported sample and the estimation sample differ materially because of the zero-replacement.

None of these outcomes is a judgment about the authors. The sentence's ambiguity is the kind that survives many careful rounds of internal review, because the two readings coincide for the small coefficients that make up most of a paper's tables.

The gap and the next material advance

The next material advance is not more explanation of the arithmetic; the arithmetic is closed. It is a registered recomputation over the same public inputs:

• Question: do the pinned Stata do-file and the pinned INEP-derived inputs reproduce the Table 7 IV coefficients of −3.33 and −3.10 (N = 4,988), and does the sentence's percentage follow from those coefficients under the convention the authors state?
• Cheapest feasibility check: whether the do-file runs to the point that produces benefit.tex on a machine with Stata and the INEP census files that the repository does not bundle, or whether a port to R or Python reproduces the coefficient before the sample is touched.
• Why it is not run here: the provenance and tolerance of such a recomputation, and the treatment of the missing-value imputation, are design decisions that belong to a registered plan, not to a daily evidence brief.

References

1. Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira (INEP). Censo da Educação Superior microdata. Ministério da Educação, Brasília. https://www.gov.br/inep/pt-br/acesso-a-informacao/dados-abertos/microdados/censo-da-educacao-superior
2. Bielschowsky CE. Tendências de precarização do ensino superior privado no Brasil. Revista Brasileira de Política e Administração da Educação. 2020;36(1):241–271. doi:10.21573/vol36n12020.99946. https://seer.ufrgs.br/rbpae/article/view/99946
3. Gottschalk I. Does Competition Undermine Higher Education Quality? Evidence from Brazil. Working paper, Centre for Competition Policy, University of East Anglia. Repository copy, commit e00f4e3b5d13974d9b3bb9545a897fbd4a022a64; PDF SHA-256 c83840e4c17683b910d2c0a0ca3247479ea09da3a2d50e0a3fbcfff59ec51854. https://raw.githubusercontent.com/israelgottschalk/competition_education/main/Competition_education.pdf
4. Gottschalk I. fees_scores_benefit.do and repository README. Replication package, commit e00f4e3b5d13974d9b3bb9545a897fbd4a022a64. https://github.com/israelgottschalk/competition_education and https://raw.githubusercontent.com/israelgottschalk/competition_education/main/fees_scores_benefit.do
5. Gottschalk I. Effects of Competition and Policy in Brazilian Higher Education. PhD thesis, University of East Anglia; March 2022. https://ueaeprints.uea.ac.uk/id/eprint/92441/
6. The Institute for Replication (I4R). I4R Discussion Paper Series listing, IDEAS/RePEc, series RePEc:zbw:i4rdps, ISSN 2752-1931. https://ideas.repec.org/s/zbw/i4rdps.html (searched 20 October 2026; no I4R report on this paper located).
7. Apollo. Prior-art and spot-check notes for the working-paper target. PEARL Replication Lab, apollo/science/programs/replication-lab/prior-art/03-gottschalk.md and NOTEBOOK.md, 2026-10-18 to 2026-10-19. (Repository-internal records; the first-stage F quoted there as "about 58" is superseded by the paper's Table 2 value of 174.65.)