Log & Polynomial Models

ECON 3209 · Week 14, Lecture 1 · Kerala Agricultural University

Department of Development Economics, KAU

Autumn 2026

Learning Outcomes

By the end of this lecture, you will be able to:

1. distinguish linear, log, and polynomial specifications
2. interpret coefficients in log-log, log-lin, and lin-log models
3. recognize diminishing returns using quadratic terms
4. estimate transformed models in Python
5. compare alternative functional forms using economic reasoning and fit statistics

Why Move Beyond Straight Lines?

Many economic relationships are nonlinear: - fertilizer may show diminishing returns - income may rise proportionally with land only at low levels - percentage changes may matter more than absolute changes

A model can still be linear in parameters while being nonlinear in variables.

Log Models and Interpretation

\[\ln Y_i = \beta_0 + \beta_1 \ln X_i + \varepsilon_i\]

In a log-log model, \(\beta_1\) is an elasticity.

• 1% increase in \(X\) is associated with a \(\beta_1\)% change in \(Y\).
• Useful when scale differs greatly across observations.

Other Common Transformations

Model	Equation	Interpretation of \(\beta_1\)
log-log	\(\ln Y = \beta_0 + \beta_1 \ln X + u\)	elasticity
log-lin	\(\ln Y = \beta_0 + \beta_1 X + u\)	one-unit rise in \(X\) changes \(Y\) by about \(100\beta_1\)%
lin-log	\(Y = \beta_0 + \beta_1 \ln X + u\)	1% rise in \(X\) changes \(Y\) by about \(0.01\beta_1\) units
quadratic	\(Y = \beta_0 + \beta_1 X + \beta_2 X^2 + u\)	slope depends on \(X\)

Quadratic Model and Diminishing Returns

\[Y = \beta_0 + \beta_1 X + \beta_2 X^2 + \varepsilon\]

If \(\beta_2 < 0\), the curve bends downward and marginal returns fall as \(X\) increases.

\[\frac{\partial Y}{\partial X} = \beta_1 + 2\beta_2 X\]

Python Demo: Estimate a Log-Log Model

Python Demo: Compare Functional Forms on the Original Income Scale

• AIC and BIC are directly comparable only when the dependent variable is the same across models.
• Here we compare fitted values on the original income scale instead.

Python Demo: Quadratic Fertilizer Response

Python Demo: Plot the Quadratic Fit

Turning Point of a Quadratic

The slope becomes zero at

\[X^* = -\frac{\beta_1}{2\beta_2}\]

When \(\beta_2 < 0\), this is the peak of the fitted parabola.

Python Demo: Compute the Turning Point

Practical Warnings

• Logs require positive values.
• Polynomial terms can overfit at the extremes.
• Functional form should reflect both theory and data patterns.

Choose nonlinear forms because they make economic sense, not just because they raise fit statistics.

🏋️ Exercise

1. Estimate a log-log income model using land as the regressor.
2. Interpret the elasticity estimate.
3. Simulate a quadratic fertilizer response and estimate a polynomial model.
4. Compute and interpret the turning point.

Summary

✅ Log models are useful when proportional or percentage effects are more natural than level effects.

✅ In a log-log model, the slope is an elasticity.

✅ Quadratic models allow marginal effects to vary with the level of the regressor.

✅ A negative squared term captures diminishing returns.

✅ Functional-form choice should combine theory, interpretation, and empirical fit.

Next Lecture

Lecture 2 — Splines & Piecewise

We will cover: - piecewise linear regression - spline intuition - knot selection - interpreting hinge terms