Nonlinear Engel curve (GAM)

This illustrates GAM: the relationship between the food expenditure share and income (the Engel curve) is typically nonlinear — declining as income rises (Engel's law). GAM captures this curvature with a smooth function without pre-specifying the form. Figures are illustrative.

Summary: use a smooth function $f(\text{income})$ to model the Engel curve, versus a linear OLS.

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Step 1 — Ideation

• Question: how does the food expenditure share change nonlinearly with income?

Step 2 — Literature Review

Engel's law; nonparametric/semiparametric Engel-curve estimation.

Step 3 — Data Collection

Household data (VHLSS): food_share (food expenditure share), lninc (log income/expenditure), controls for household size and region.

Step 4 — Modeling

Choose the Non-linear & semi-parametric family → GAM; mark lninc as a smooth term:

$$food\_share_i = \beta_0 + f(lninc_i) + X_i\gamma + \varepsilon_i$$

Illustrative results (format — not real results):

	Linear OLS	GAM
Relationship shape	straight line	declining curve (concave)
Adjusted $R^2$	0.41	0.52
edf of f(lninc)	—	3.8 (clearly nonlinear)

Sample interpretation: the smooth-function plot shows the food share falling quickly at low income then flattening — exactly Engel's law; GAM fits better than linear OLS (edf > 1 ⇒ nonlinear).

Stata

* ── GAM: Engel curve ──────────────────────────────
* Smooth nonlinear relationship: food_share ~ f(income)
npregress series food_share income, basis(bspline 4)

* Marginal effect plot
margins, at(income=(1(0.5)10)) post
marginsplot

R

# ── GAM: Engel curve ──────────────────────────────
library(mgcv)

model_engel <- gam(
  food_share ~ s(log_income) + household_size,
  data   = df,
  method = "REML"
)

summary(model_engel)

# Smooth-function plot with partial residuals
plot(model_engel, residuals = TRUE, pch = 20,
     main = "Engel curve — smooth f(log income)")

Python

# ── GAM: Engel curve ──────────────────────────────
from pygam import LinearGAM, s, l
import matplotlib.pyplot as plt

X = df[["log_income", "household_size"]].values
y = df["food_share"].values

gam = LinearGAM(s(0) + l(1)).fit(X, y)
print(gam.summary())

# Plot the smooth function for log_income
gam.plot(term=0)
plt.title("Engel curve — smooth f(log income)")
plt.show()

Step 5 — Reporting

Export a report + the smooth-function plot + replication code.

Video tutorial

Video Tutorial: Guide to running GAM in EcoLab

See also

• GAM · NLS · Catalog