9.2 The four functional forms
Logarithms of what, and what happens to the interpretation?
Either variable may be entered in levels or in logarithms, giving four combinations. Everything else about the regression is unchanged: it is still linear in the parameters, still fitted by least squares, still tested with the same \(t\) and \(F\) statistics.
What changes is the meaning of \(\beta_1\).
| Model | Equation | Interpretation of \(\beta_1\) |
|---|---|---|
| Level–level | \(y = \beta_0 + \beta_1 x + u\) | A one-unit rise in \(x\) changes \(y\) by \(\beta_1\) units |
| Log–level | \(\ln y = \beta_0 + \beta_1 x + u\) | A one-unit rise in \(x\) changes \(y\) by \(100\beta_1\) per cent |
| Level–log | \(y = \beta_0 + \beta_1 \ln x + u\) | A 1% rise in \(x\) changes \(y\) by \(\beta_1/100\) units |
| Log–log | \(\ln y = \beta_0 + \beta_1 \ln x + u\) | A 1% rise in \(x\) changes \(y\) by \(\beta_1\) per cent |
The last is an elasticity, and it is unit-free: it does not matter whether weight is in kilograms or tonnes, or price in rupees or lakhs.
All four interpretations follow from one fact of calculus. For small changes,
\[\Delta \ln y \;\approx\; \frac{\Delta y}{y}\]
that is, a change in the logarithm is approximately a proportional change in the variable. Multiplying by 100 turns it into a percentage.
Log–level. Differentiating \(\ln y = \beta_0 + \beta_1 x\) with respect to \(x\),
\[\frac{1}{y}\frac{dy}{dx} = \beta_1 \qquad\Longrightarrow\qquad \frac{dy/y}{dx} = \beta_1\]
so a one-unit rise in \(x\) raises \(y\) by a proportion \(\beta_1\), which is \(100\beta_1\) per cent.
Level–log. From \(y = \beta_0 + \beta_1 \ln x\),
\[dy = \beta_1 \, d(\ln x) = \beta_1 \frac{dx}{x}\]
A 1% rise in \(x\) means \(dx/x = 0.01\), so \(dy = 0.01\beta_1 = \beta_1/100\) units.
Log–log. From \(\ln y = \beta_0 + \beta_1 \ln x\),
\[\frac{dy/y}{dx/x} = \beta_1\]
which is the definition of an elasticity: the ratio of the proportional change in \(y\) to the proportional change in \(x\).
The rule for reading a log coefficient as a percentage.
Whenever \(y\) is in logarithms — whether the coefficient belongs to a continuous variable or to a 0/1 indicator — the exact percentage change is
\[100\left(e^{\beta} - 1\right)\]
If \(|\beta| < 0.2\), the approximation \(100\beta\) is close enough to report. Above that, use the exact expression.
The approximation degrades quickly once the coefficient grows.
For \(\beta_1 = 0.0942\) it says 9.42% and the exact figure is \(100(e^{0.0942} - 1) = 9.88\)%. For \(\beta_1 = 0.7\) it says 70% and the truth is \(101\)%. The error is always in the same direction: the approximation understates the true percentage.