Lesson M05.L01: Log Transformations in Regression

Field	Detail
Module	05 — Functional Forms & Dummy Variables
Level	Graduate
Duration	30–35 min
Learning Objective	Understand the four log/level functional forms, interpret coefficients correctly in each case, and know when to apply logarithmic transformations.
Provenance	Wooldridge Introductory Econometrics, Ch 6.2

Explanation

Why Functional Form Matters

OLS assumes the relationship between $y$ and $x$ is linear in parameters — but it does not require linearity in the original variables. By transforming variables before estimating, we can capture a wide range of non-linear economic relationships while keeping the machinery of OLS intact.

Logarithmic transformations are by far the most common such device in applied econometrics.

Convention: In econometrics, $\log$ always means the natural logarithm $\ln$ (base $e \approx 2.718$). It never means $\log_{10}$. Software commands like log() in Stata and R also default to the natural log.

The Four Functional Forms

1. Level-Level

\[y = \beta_0 + \beta_1 x + u\]

Interpretation: A one-unit increase in $x$ is associated with a $\beta_1$-unit change in $y$.

\[\Delta y = \beta_1 \,\Delta x\]

Use when both variables are naturally measured in levels and the relationship is approximately linear (e.g. test score on class size).

2. Log-Level (Semi-Elasticity)

\[\log(y) = \beta_0 + \beta_1 x + u\]

Interpretation: A one-unit increase in $x$ is associated with an approximate $100\beta_1$ percent change in $y$.

\[\% \Delta y \approx 100 \beta_1 \,\Delta x\]

This is called a semi-elasticity — $x$ is in levels, $y$ is in logs. Use when $y$ is right-skewed (wages, income, prices) and the effect of $x$ is proportional rather than additive.

Example: If $\hat{\beta}_1 = 0.082$ in a log-wage regression on years of education, then each additional year of education is associated with approximately an 8.2% increase in wages.

3. Level-Log

\[y = \beta_0 + \beta_1 \log(x) + u\]

Interpretation: A 1% increase in $x$ is associated with a $\beta_1/100$ unit change in $y$.

\[\Delta y \approx \frac{\beta_1}{100}\,\% \Delta x\]

Use when $x$ exhibits diminishing marginal impact — each extra unit of $x$ matters less than the previous (e.g. $y$ = test score, $x$ = expenditure per student).

4. Log-Log (Constant Elasticity)

\[\log(y) = \beta_0 + \beta_1 \log(x) + u\]

Interpretation: A 1% increase in $x$ is associated with a $\beta_1$% change in $y$. Here $\beta_1$ is the elasticity of $y$ with respect to $x$.

\[\% \Delta y \approx \beta_1 \,\% \Delta x\]

This is the workhorse model for demand equations and production functions. The elasticity is constant across all values of $x$.

Example: If $\hat{\beta}_1 = -0.5$ in a log-log demand equation, the price elasticity of demand is $-0.5$ (inelastic).

Summary Table

Model	Dep. Var.	Indep. Var.	Interpretation of $\hat{\beta}_1$
Level-level	$y$	$x$	$\Delta y = \hat{\beta}_1 \Delta x$ (units)
Log-level	$\log(y)$	$x$	$\% \Delta y \approx 100\hat{\beta}_1 \Delta x$ (semi-elasticity)
Level-log	$y$	$\log(x)$	$\Delta y \approx (\hat{\beta}_1/100)\% \Delta x$
Log-log	$\log(y)$	$\log(x)$	$\% \Delta y \approx \hat{\beta}_1 \% \Delta x$ (elasticity)

When to Use Logs

Use logarithmic transformations when:

The variable is right-skewed — wages, income, housing prices, firm revenues all tend to have long right tails. Taking logs compresses the distribution and makes OLS better behaved.
You want an elasticity interpretation — log-log gives a clean, scale-free elasticity.
The marginal effect is proportional rather than additive — if a $1 increase in $x$ has a larger absolute effect at low values of $y$ than at high values, a log-level model is more appropriate.
To reduce heteroskedasticity — log transformations often stabilise variance across observations.

Limitations: What You Cannot Log

The natural log is undefined for zero and negative values:

\[\log(0) = -\infty, \quad \log(x) \text{ undefined for } x < 0\]

Alternatives when $y \geq 0$ with zeros:

$\log(1 + y)$: Adds 1 before taking logs. Simple, but coefficient interpretation is non-standard; the approximation $\%\Delta y \approx 100\hat{\beta}_1\Delta x$ holds only when $y$ is large.
Inverse Hyperbolic Sine (IHS): $\text{arcsinh}(y) = \log(y + \sqrt{y^2 + 1})$. Behaves like $\log(2y)$ for large $y$, equals 0 at $y=0$, and is defined for negative values. Increasingly preferred in modern applied work.
Two-part (Tobit-type) models: Separate models for the zero/non-zero decision and the level conditional on being positive.

Worked Example

Dataset: Cross-sectional sample of $n = 526$ workers (Wooldridge WAGE1).

Equation estimated:

\[\log(\widehat{wage}_i) = 0.584 + 0.082\, educ_i + 0.029\, exper_i - 0.0006\, exper_i^2\]

Coefficient on education: $\hat{\beta}_{educ} = 0.082$

Interpretation: Each additional year of education is associated with approximately 8.2% higher wages, holding experience constant.

This is a log-level model. The dependent variable is $\log(wage)$; education is measured in years (a level).

Why log wages? Wages are right-skewed: a few very high earners pull the distribution. Taking logs produces a more symmetric distribution and stabilises variance. The percentage interpretation is also more economically natural ("a 10% wage premium") than an additive one ("$0.82/hour more").

Exact vs approximate interpretation: The exact percentage effect of a one-unit change in $x$ in a log-level model is:

\[\% \Delta y = 100 \left(e^{\hat{\beta}_1} - 1\right)\]

For $\hat{\beta}_1 = 0.082$: $100(e^{0.082} - 1) = 100(1.0854 - 1) = 8.54\%$. The approximation ($8.2\%$) is close enough for small coefficients but use the exact formula for coefficients above $\approx 0.2$.

Common Misconception

"Log in a regression means log base 10."

Correction: In econometrics, $\log$ universally refers to the natural logarithm (base $e$), not $\log_{10}$. When Wooldridge writes $\log(wage)$, he means $\ln(wage)$. Stata's gen lwage = log(wage) and R's log(wage) both compute natural logs. If you use $\log_{10}$ by accident, your coefficients will be off by a factor of $\ln(10) \approx 2.303$ and your interpretations will be wrong. Always double-check which base your software uses (hint: it's always $e$ by default in econometrics packages).

Practice Prompts

Prompt 1 — Identification A researcher estimates:

\[\widehat{price} = 12.3 + 0.95\,\log(sqft)\]

where $price$ is in thousands of dollars and $sqft$ is house size in square feet. What functional form is this? Interpret $\hat{\beta}_1 = 0.95$. If house size increases by 10%, what is the predicted change in price (in dollars)?

Prompt 2 — Model choice You are studying the effect of advertising expenditure on firm sales. Sales range from $50,000 to $50 million across firms. Advertising expenditure ranges from $1,000 to $500,000. Which functional form would you choose and why? What does your choice imply about the marginal return to advertising?

Prompt 3 — Zero values A researcher wants to estimate the effect of government transfers on household savings. Savings takes the value zero for 35% of households in the sample. Why is $\log(savings)$ problematic here? Name two alternatives and briefly explain each.

Further Resources

Wooldridge, Introductory Econometrics (7th ed.), §6.2 — Functional Forms Involving Logarithms
Bellemare & Wichman (2020), "Elasticities and the Inverse Hyperbolic Sine Transformation", Oxford Bulletin of Economics and Statistics
Stata: gen lnwage = log(wage) then reg lnwage educ exper
R: lm(log(wage) ~ educ + exper, data = wage1)

Model	Dep. Var.	Indep. Var.	Interpretation of \(\hat{\beta}_1\)
Level-level	\(y\)	\(x\)	\(\Delta y = \hat{\beta}_1 \Delta x\) (units)
Log-level	\(\log(y)\)	\(x\)	\(\% \Delta y \approx 100\hat{\beta}_1 \Delta x\) (semi-elasticity)
Level-log	\(y\)	\(\log(x)\)	\(\Delta y \approx (\hat{\beta}_1/100)\% \Delta x\)
Log-log	\(\log(y)\)	\(\log(x)\)	\(\% \Delta y \approx \hat{\beta}_1 \% \Delta x\) (elasticity)