Econometrics · Regression
How to Interpret Log Coefficients in Regression
Your output says 0.087. Is that 8.7%, 0.087%, or 8.7 percentage points? Here’s the answer, and here’s why it’s never explained properly.
It’s half eleven. The assignment is due at nine. Your regression has run, Stata has printed a table, and sitting there in the coefficient column is the number 0.087.
Your dependent variable is log(wage). Your regressor is years of education.
So what do you write?
You know it’s something to do with percentages. You have a vague memory of your lecturer saying “multiply by 100.” You also have a vague memory of a footnote saying that’s only an approximation, and you cannot remember when the approximation stops being fine.
This post fixes that permanently. There are exactly four cases. Each has one interpretation. Once you’ve got them, you’ll never look this up again — and you’ll be able to read any empirical economics paper ever written, because virtually all of them are written in logs.
Let’s do the rules first, then the reasons.
The four rules, on one card
Everything depends on which side of the equation the log is on. That’s it. That’s the whole system.
| Model | Name | β̂ means | Say it like this |
|---|---|---|---|
| y = β₀ + β₁x | Level-level | A one-unit rise in x changes y by β̂ units | “One more year of schooling is associated with £1,240 more annual pay” |
| log(y) = β₀ + β₁x | Log-level | A one-unit rise in x changes y by β̂ × 100 percent | “One more year of schooling is associated with 8.7% higher pay” |
| y = β₀ + β₁log(x) | Level-log | A 1% rise in x changes y by β̂ / 100 units | “A 1% rise in firm size is associated with £34 more pay” |
| log(y) = β₀ + β₁log(x) | Log-log | A 1% rise in x changes y by β̂ percent | “A 1% rise in price is associated with a 1.4% fall in quantity” |
So the answer to the question we opened with: 0.087 on a log-level model means roughly 8.7% higher wages per additional year of education.
Not 0.087%. Not 8.7 percentage points. About 8.7 percent.
💡 Insight — the memory trick
A log on a variable turns “units” into “percent” for that variable.
That’s the entire system in one sentence. Log on y? y is now in percent. Log on x? x is now in percent. Logs on both? Both in percent — and a percent-per-percent is an elasticity, which is why the log-log slope is the elasticity exactly.
Where does the ×100 come from? Nowhere magical. Software reports the coefficient as a proportion (0.087) and humans prefer percentages (8.7%). The 100 is a units conversion, nothing more. If you remember “logged means percent,” you’ll reconstruct the rest under exam pressure.
Percent vs percentage points (the error that costs the most marks)
Before we go further, we have to kill the mistake that markers see more than any other in this topic.
Percent is relative. Percentage points are absolute. They are not interchangeable, and confusing them can turn a correct number into a wrong sentence.
If wages are £30,000 and rise 8.7 percent, they go to £32,610.
If an unemployment rate is 5% and rises by 8.7 percentage points, it goes to 13.7%. If it rises by 8.7 percent, it goes to 5.44%. Those are wildly different claims about the world, and one of them is a recession.
🔑 Key term
Percentage points — the arithmetic difference between two percentages. Used when your variable is already measured as a percentage or a rate. If your dependent variable is an unemployment rate, an interest rate, or a share, its coefficient in a level-level model is in percentage points, and calling it “percent” is wrong even though it looks like a percentage.
Here’s the rule that resolves it: ask what the variable is, not what the coefficient looks like.
• log(wage) regressed on schooling → coefficient is in percent (the log did that)
• unemployment_rate regressed on GDP growth → coefficient is in percentage points (the variable is already a rate; no log was taken)
• log(unemployment_rate) regressed on GDP growth → coefficient is in percent change of the rate, which is a genuinely confusing object and one reason people rarely log rates
• unemployment_rate regressed on GDP growth → coefficient is in percentage points (the variable is already a rate; no log was taken)
• log(unemployment_rate) regressed on GDP growth → coefficient is in percent change of the rate, which is a genuinely confusing object and one reason people rarely log rates
Why economists log absolutely everything
Open any issue of any economics journal. Count the logs. It looks like a stylistic tic. It isn’t. There are four reasons, and they’re all good.
Reason 1 — Elasticities fall out for free
Economic theory doesn’t predict “a £1 price rise cuts demand by 40 units.” Theory predicts elasticities — percent responses to percent changes — because they’re unit-free and comparable across contexts. A log-log regression hands you the elasticity as the slope, with no further work.
📐 Formula — why the log-log slope is the elasticity
Start with the model:
log(y) = β₀ + β₁ log(x)
Differentiate both sides with respect to x:
(1/y)(dy/dx) = β₁(1/x)
Rearrange:
β₁ = (dy/y) / (dx/x)
And that is the definition of elasticity: a proportional change in y over a proportional change in x. Not an approximation. Not a convention. The log-log slope is the elasticity, by construction.
When to reach for it: any time theory speaks in elasticities — demand, supply, labour supply, production functions.
What it assumes: that the elasticity is constant across the whole range of x. That’s a real assumption and it’s often wrong at the extremes.
What it assumes: that the elasticity is constant across the whole range of x. That’s a real assumption and it’s often wrong at the extremes.
Reason 2 — It fixes the skew
Income, wealth, firm size, city population, house prices — all violently right-skewed. Take logs and they become roughly symmetric, often near-normal. That’s not cosmetic: it stabilises variance, tames the influence of the one billionaire in your sample, and makes your residuals behave.
Why does this keep happening? Because the normal distribution is addition’s favourite shape, not nature’s. Anything built from many small additive influences drifts towards normal. Anything built from many small multiplicative influences drifts towards lognormal — and taking logs turns multiplication into addition, handing you normality back.
Height is additive, so height is normal-ish. Wealth is multiplicative — returns compound on what you already have — so wealth is lognormal. This is why heights don’t have billionaires.
Reason 3 — Percentage effects are usually the sensible model
Does an extra year of schooling add a fixed £1,240 to everyone’s salary, whether they’re a cleaner or a consultant? Almost certainly not. Does it add a roughly constant percentage? That’s far more plausible. The log-level model isn’t a transformation of convenience — it’s a claim that effects are proportional, and for wages that claim is usually more defensible than the alternative.
Reason 4 — Log differences are growth rates
For small changes, log(y₂) − log(y₁) ≈ the proportional change in y. So a first difference of a logged series is the growth rate. That’s why every macro time series you’ll ever meet arrives pre-logged.
💡 Insight — the choice of functional form is a claim about the world
Students often treat “should I log it?” as a technical question with a technical answer. It isn’t. It’s a substantive question about how you think the world works.
Level-level says: the effect is the same in absolute terms for everyone.
Log-level says: the effect is the same in percentage terms for everyone.
Log-level says: the effect is the same in percentage terms for everyone.
Those are different theories of the phenomenon. Pick the one your economics supports, then check the residuals to see whether the data agrees. Don’t pick whichever gives you a better R² — that’s not a reason, that’s a symptom.
When ×100 breaks (and what to use instead)
Here’s the footnote your lecturer mumbled.
In a log-level model, β̂ × 100 is an approximation to the percentage effect. It comes from the fact that for small values, log(1 + r) ≈ r.
📐 Formula — the exact percentage effect
% change in y = (e^β̂ − 1) × 100
When to reach for it: whenever β̂ is larger than about 0.1 in absolute value — which in practice means almost every dummy variable coefficient.
When the approximation is fine: small coefficients, roughly |β̂| < 0.1. Most continuous regressors land here.
When the approximation is fine: small coefficients, roughly |β̂| < 0.1. Most continuous regressors land here.
Watch how fast it diverges:
| β̂ | Approximation (β̂ × 100) | Exact ((e^β̂ − 1) × 100) | Error |
|---|---|---|---|
| 0.02 | 2.0% | 2.02% | negligible |
| 0.087 | 8.7% | 9.09% | small |
| 0.20 | 20.0% | 22.14% | getting real |
| 0.50 | 50.0% | 64.87% | badly wrong |
| 0.69 | 69.0% | 99.37% | you’ve doubled y, not added 69% |
That last row matters. A coefficient of 0.69 on a dummy means the treated group’s y is roughly double the control group’s, because log(2) ≈ 0.693. If you’d reported “69% higher,” you’d have understated the effect by a third.
⚠ Common error — the dummy variable trap in logs
What the marker sees: “The coefficient on female is −0.31, so women earn 31% less.”
Why it’s wrong: the approximation is being applied to a coefficient far outside its safe range. The exact figure is (e^(−0.31) − 1) × 100 = −26.7%. You’ve overstated the gap by more than four percentage points, and dummy coefficients are almost always big enough for this to bite.
The fix: for any dummy, use the exact formula. It’s one line. Some econometricians go further and apply Kennedy’s (1981) correction, which adjusts for the fact that β̂ is itself estimated — worth a footnote in a dissertation, overkill in an assignment.
📊 Case study — the Mincer equation, and the most-estimated regression in economics
In 1974, Jacob Mincer proposed a wage equation that has since been estimated on more datasets, in more countries, by more researchers than probably any other specification in the social sciences:
log(wage) = β₀ + β₁·schooling + β₂·experience + β₃·experience² + ε
Look at the left-hand side. Logged. Deliberately.
Mincer’s argument wasn’t statistical convenience — it was theoretical. He derived the log form from a human capital model in which each year of schooling raises earnings capacity by a proportional amount, because you’re investing in a stock that compounds. The functional form is the theory. The log isn’t a transformation applied to the model; it’s a prediction of the model.
And β₁ has a name: the return to schooling. Across hundreds of studies, in dozens of countries, it lands remarkably consistently somewhere around 6–10% per year. That’s the 0.087 we opened with.
Two things to notice, because they’re the difference between reciting a result and understanding one.
First, the experience term is quadratic, not logged. Why? Because theory says earnings rise with experience and then flatten — and eventually fall — as human capital depreciates and hours decline. That’s a hump, not a proportional effect, and a quadratic gives you a hump. Different variable, different theory, different functional form. Mincer chose each one deliberately.
Second, and this is the part that should make you uncomfortable: β₁ is almost certainly biased upward. People who are more able tend to get more schooling and earn more regardless. Ability isn’t in the data — it isn’t in anybody’s data — so it sits in the error term, correlated with schooling. Which means the zero conditional mean assumption fails and OLS overstates the return.
Everyone knows this. The entire literature knows this. It’s why economists spent thirty years attacking the problem with twins studies, compulsory schooling reforms, distance-to-college instruments, and quarter-of-birth quirks — trying to find variation in schooling that has nothing to do with ability.
So when you read “the return to schooling is 8.7%,” you now know three things at once: what the number means, why it’s logged, and why it’s probably too big. That’s what reading econometrics looks like.
🔬 Research spotlight — Halvorsen & Palmquist (1980)
The paper: Halvorsen, R. & Palmquist, R. (1980), “The Interpretation of Dummy Variables in Semilogarithmic Equations,” American Economic Review, 70(3), 474–475.
Length: two pages. Genuinely — go and read it, it’s shorter than this section.
What it does: it points out, politely, that a substantial chunk of the published applied economics literature had been misinterpreting its own dummy variable coefficients. Everyone had been reading β̂ × 100 as the percentage effect. For dummies, where coefficients are typically large, that approximation is materially wrong. The correct figure is (e^β̂ − 1) × 100.
Why it’s worth your time as an undergraduate: it’s a two-page paper in a top-five journal that fixed a widespread error, and the mathematics in it is one line of algebra you could have done yourself. It’s a useful corrective to the idea that contributing to a field requires a new technique. Sometimes it requires reading carefully.
The follow-up: Kennedy, P. (1981) refined it further, noting that because β̂ is an estimate, the exponential transformation introduces a small bias of its own, and proposing a correction. The literature on how to interpret one coefficient ran to several papers. That should tell you the question isn’t trivial.
What to actually write in your assignment
Markers award interpretation marks mechanically. Give them the sentence they’re looking for.
The template:
“A one-[unit] increase in [x, named, with units] is associated with a [β̂-derived figure, with the right units] change in [y, named], holding [controls] constant.”
Filled in, for each of the four cases:
• Level-level: “A one-year increase in schooling is associated with a £1,240 increase in annual earnings, holding experience and region constant.”
• Log-level: “A one-year increase in schooling is associated with approximately 8.7% higher earnings, holding experience and region constant.”
• Level-log: “A 1% increase in firm size is associated with a £34 increase in annual earnings, holding tenure constant.”
• Log-log: “A 1% increase in price is associated with a 1.4% decrease in quantity demanded — that is, the estimated price elasticity of demand is −1.4.”
• Log-level: “A one-year increase in schooling is associated with approximately 8.7% higher earnings, holding experience and region constant.”
• Level-log: “A 1% increase in firm size is associated with a £34 increase in annual earnings, holding tenure constant.”
• Log-log: “A 1% increase in price is associated with a 1.4% decrease in quantity demanded — that is, the estimated price elasticity of demand is −1.4.”
Three things that lose marks every time:
1. “Causes.” Unless you have randomisation or a credible identification strategy, write associated with. Markers circle “causes” faster than they circle arithmetic.
2. No units. “Increases by 0.087” is not an interpretation. It’s a reading of the table.
3. No “holding constant.” In a multiple regression, the coefficient is a partial effect. Say so.
2. No units. “Increases by 0.087” is not an interpretation. It’s a reading of the table.
3. No “holding constant.” In a multiple regression, the coefficient is a partial effect. Say so.
Common errors — the full list
⚠ The six that cost marks
Error 1 — Reporting the raw coefficient as a percentage.
“log(wage) coefficient is 0.087, so wages rise 0.087%.” No. Multiply by 100. It’s 8.7%.
“log(wage) coefficient is 0.087, so wages rise 0.087%.” No. Multiply by 100. It’s 8.7%.
Error 2 — Approximating a large coefficient.
Any |β̂| above ~0.1 needs (e^β̂ − 1) × 100. Dummies almost always qualify.
Any |β̂| above ~0.1 needs (e^β̂ − 1) × 100. Dummies almost always qualify.
Error 3 — Percent vs percentage points.
If your y is already a rate, its coefficient is in percentage points and no log was involved. If you logged it, you’re now talking about the percent change of a rate, which is a strange object. Be sure you meant to.
If your y is already a rate, its coefficient is in percentage points and no log was involved. If you logged it, you’re now talking about the percent change of a rate, which is a strange object. Be sure you meant to.
Error 4 — Logging a variable with zeros.
log(0) is undefined and your software will silently drop those rows. If 400 firms reported zero exports and your n fell from 2,000 to 1,600, you haven’t transformed your data — you’ve deleted a quarter of it, and probably the most interesting quarter. Check your n before and after. Every time.
log(0) is undefined and your software will silently drop those rows. If 400 firms reported zero exports and your n fell from 2,000 to 1,600, you haven’t transformed your data — you’ve deleted a quarter of it, and probably the most interesting quarter. Check your n before and after. Every time.
Error 5 — Comparing R² across logged and unlogged models.
An R² from a model with y as the outcome and one with log(y) as the outcome are measuring variation in different quantities. They are not comparable. This is not a subtle point and it appears in published work anyway.
An R² from a model with y as the outcome and one with log(y) as the outcome are measuring variation in different quantities. They are not comparable. This is not a subtle point and it appears in published work anyway.
Error 6 — Forgetting to back-transform predictions.
If you predict log(y) and then exponentiate, you get the predicted median of y, not the mean. For the mean you need a smearing correction. Most undergraduates don’t need this; everyone should know it exists.
If you predict log(y) and then exponentiate, you get the predicted median of y, not the mean. For the mean you need a smearing correction. Most undergraduates don’t need this; everyone should know it exists.
Practice questions
Try these before you look. Full workings below.
Q1. A regression of log(price) on square_metres gives a coefficient of 0.0042. Interpret it.
Q2. A regression of log(quantity) on log(price) gives a coefficient of −0.86. Interpret it. Is demand elastic or inelastic here, and how do you know?
Q3. A regression of log(wage) on a union dummy gives a coefficient of 0.24. A student writes: “Union members earn 24% more.” Correct them, with a number.
Q4. A regression of unemployment_rate on log(GDP) gives a coefficient of −3.1. Interpret it carefully. What are the units of the effect?
Q5. Your regression of log(exports) on log(GDP) runs on 2,400 firms. The output reports n = 1,850. What has almost certainly happened, and what should you do about it?
Worked answers
A1. Log-level. A one-square-metre increase in floor area is associated with approximately 0.42% higher price (0.0042 × 100), holding other factors constant. The coefficient is well below 0.1, so the approximation is fine; the exact figure is 0.4209%, which rounds to the same thing.
A2. Log-log, so the coefficient is the elasticity. A 1% increase in price is associated with a 0.86% decrease in quantity demanded. The price elasticity of demand is −0.86. Since |−0.86| < 1, demand is inelastic — quantity responds less than proportionally, which means a price rise would increase total revenue. Note that this specification imposes a constant elasticity across the whole price range, which is an assumption you should state rather than assume.
A3. The student has applied the approximation to a coefficient that’s too large for it. The exact effect is (e^0.24 − 1) × 100 = 27.1%. So union members earn about 27% more, not 24% — the student has understated the union premium by three percentage points. And they should also have written “associated with,” not “earn,” unless they’ve solved the selection problem, which they haven’t: workers who join unions differ systematically from those who don’t.
A4. Level-log. A 1% increase in GDP is associated with a change of β̂/100 = −0.031 in the unemployment rate. Now the careful part: unemployment rate is already measured in percentage terms, so the units of the effect are percentage points. A 1% rise in GDP is associated with a fall of about 0.031 percentage points in unemployment. (If unemployment is coded as a proportion rather than a percentage, everything shifts by a factor of 100 — check your codebook, because this is exactly where people go wrong.)
A5. 550 firms have been silently dropped, almost certainly because they reported zero exports and log(0) is undefined. This isn’t a rounding issue — it’s a systematic deletion of the non-exporters, who are very likely the smallest and least productive firms in the sample. Your estimated GDP–exports elasticity is now conditional on already being an exporter, which is a completely different research question from the one you set out to answer.
What to do: first, report it. Second, choose deliberately: Poisson pseudo-maximum-likelihood handles zeros natively and is the standard fix in the trade literature (Santos Silva & Tenreyro 2006); the inverse hyperbolic sine transformation is another option with its own interpretive baggage; log(x + 1) is common but arbitrary and changes what your coefficient means. Whatever you pick, say what you picked and why. A supervisor will forgive a defensible choice. They will not forgive a silent n.
The short version
• Log on y → that variable is now in percent
• Log on x → that variable is now in percent
• Both logged → percent-per-percent → elasticity, exactly
• Software gives proportions; ×100 makes them percentages
• For |β̂| > 0.1 — which means nearly every dummy — use (e^β̂ − 1) × 100
• Percent ≠ percentage points, and knowing which you mean is half the marks
• Logs aren’t a technique. They’re a claim about whether effects are absolute or proportional. Make the claim on purpose.
• Log on x → that variable is now in percent
• Both logged → percent-per-percent → elasticity, exactly
• Software gives proportions; ×100 makes them percentages
• For |β̂| > 0.1 — which means nearly every dummy — use (e^β̂ − 1) × 100
• Percent ≠ percentage points, and knowing which you mean is half the marks
• Logs aren’t a technique. They’re a claim about whether effects are absolute or proportional. Make the claim on purpose.
References
1. Mincer, J. (1974) Schooling, Experience, and Earnings. New York: National Bureau of Economic Research.
2. Halvorsen, R. & Palmquist, R. (1980) “The Interpretation of Dummy Variables in Semilogarithmic Equations,” American Economic Review, 70(3), pp. 474–475.
3. Kennedy, P. (1981) “Estimation with Correctly Interpreted Dummy Variables in Semilogarithmic Equations,” American Economic Review, 71(4), p. 801.
4. Card, D. (1999) “The Causal Effect of Education on Earnings,” in Ashenfelter, O. & Card, D. (eds) Handbook of Labor Economics, Vol. 3A. Amsterdam: Elsevier, pp. 1801–1863.
5. Santos Silva, J.M.C. & Tenreyro, S. (2006) “The Log of Gravity,” Review of Economics and Statistics, 88(4), pp. 641–658.
6. Wooldridge, J.M. (2019) Introductory Econometrics: A Modern Approach. 7th edn. Boston: Cengage.
Struggling with regression interpretation?
This is one section of a chapter from Statistics Made Simple, our supplementary guide for undergraduates — the book that explains why the formula has that shape, not just what to type.