Specification & Data Problemsintermediate

Functional Form: Logs and Elasticities

Taking the natural log of a variable changes how a coefficient is read. With a log-level model $\log(y)=\beta_0+\beta_1 x+u$ , a one-unit rise in $x$ gives roughly a $\beta_1 \times 100$ percent change in $y$ (a semi-elasticity). With a log-log model, $\beta_1$ is an elasticity, the percent change in $y$ per percent change in $x$ . Logs are common for variables with skewed, positive distributions such as wages, prices, and firm size, because they pull in long right tails and make constant-percentage effects linear in parameters.

Try it yourself

Model specification sandbox

One model, many specifications. See how functional form, a squared term, and a dummy variable each change what a coefficient means and how the fitted line bends. The same seeded data sit under all three views.

Estimated slope β̂β̂ = 0.630

intercept β̂₀ 1.647slope β̂₁ 0.630

Functional form

ln(y) = β₀ + β₁·ln(x)
slope β̂₁ = 0.630

Elasticity. Both sides logged, so β̂₁ = 0.630 is the % change in y per % change in x, directly and unit-free. A 1% rise in x moves y by about 0.63%.

Try this

Discussion. The fitted curve barely moves, yet the slope's units change with every form. When would you report an elasticity rather than a dollar effect, and why is 100·β̂ only an approximation to the exact 100·(e^β̂ − 1)?

OLS is re-fit by least squares in the chosen form's transformed space, then mapped back to the raw axes. For log–level, a one-unit rise in x gives %Δy = 100·(e^β̂ − 1) exactly, with 100·β̂ as the first-order approximation. For log–log, β̂ is the elasticity directly (the % change in y per % change in x), exact at the margin.

Why it matters

Logs turn multiplicative stories into additive ones. Saying a year of schooling raises wages by about 8 percent is more natural than saying it adds a fixed dollar amount that is the same for a CEO and an intern. When a variable is strictly positive and spans several orders of magnitude, its log is usually closer to symmetric, the error term behaves better, and one slope describes a proportional effect that holds across the whole range.

Formulas

Four functional forms

\log(y)=\beta_0+\beta_1 \log(x)+u \quad\text{vs.}\quad \log(y)=\beta_0+\beta_1 x+u

Log-log:

\beta_1

is the elasticity. Log-level:

\beta_1 \times 100

is the percent change in

y

per one-unit rise in

x

Exact percent change (semi-elasticity)

\%\Delta y = 100\,\bigl(e^{\beta_1 \Delta x}-1\bigr)

The product

\beta_1 \times 100

is a first-order approximation. For larger

\hat{\beta}_1

use the exact exponential form, since the approximation overstates the true percent change.

Worked examples

Scenario

Estimate the return to education on a wage equation in Stata.

Solution

Run `regress lwage educ exper`, where `lwage = ln(wage)`. A coefficient of about 0.083 on `educ` means another year of schooling is associated with about an 8.3 percent higher wage, holding experience fixed. The exact effect is $(e^{0.083}-1)\times 100\approx 8.7$ percent.

NoteGenerate the dependent variable first with `gen lwage = ln(wage)`.

Scenario

A constant-elasticity demand or cost relationship between firm sales and R&D spending.

Solution

Run `regress lsales lrd`. A slope of about 1.08 says a 1 percent rise in R&D is associated with a 1.08 percent rise in sales. Because both sides are logged, the coefficient is a unit-free elasticity that does not depend on the currency or scale of either variable.

Common mistakes

✗Treating the log-level coefficient as a level effect. A value of 0.083 does not mean wages rise by 0.083 dollars or by 8.3 units. It means roughly 8.3 percent.
✗Thinking $\beta_1 \times 100$ is exact for any size of change. It is a linear approximation that drifts as $\hat{\beta}_1$ grows. Beyond about 0.1, prefer $(e^{\hat{\beta}_1}-1)\times 100$ .
✗Logging a variable that takes zero or negative values. $\log(0)$ is undefined, so dummies, net income, and returns that can be negative usually stay in levels rather than getting an ad hoc $\log(1+x)$ patch.
✗Believing you can compare $R^2$ across a level- $y$ model and a $\log(y)$ model. The dependent variable differs, so the two $R^2$ values are not comparable.

Revision bullets

•Log-log slope is an elasticity (percent per percent).
•Log-level slope times 100 is a semi-elasticity (percent per unit).
•Level-log slope divided by 100 is the unit change in $y$ per 1 percent rise in $x$ .
•Use logs for skewed, strictly positive variables (wages, prices, size).
•Exact percent change uses $(e^{\hat{\beta}_1}-1)\times 100$ , not just the linear approximation.

Quick check

In `regress lwage educ`, the coefficient on `educ` is 0.06. The best reading is:

A log-log regression of quantity on price gives a slope of -0.8. This means:

Connected topics

MLR model Poly & Inter.Dummies Units & Scale RESET

Sources

Wooldridge (2019), Ch. 2 & 6
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Section 2.4 and Chapter 6 develop the level-level, log-level, level-log, and log-log forms and the semi-elasticity versus elasticity reading.
Wooldridge (2019), §6.2
Wooldridge, Jeffrey M. Introductory Econometrics: A Modern Approach. 7th ed. Cengage, 2019.
Discusses why logs are used for skewed positive variables and the approximation error in the linear semi-elasticity.

How to cite this page

Dr. Phil's Quant Lab. (2026). Functional Form: Logs and Elasticities. Derivatives Atlas. https://phucnguyenvan.com/concept/efm-logs-elasticity

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