GAM — Generalized Additive Models

GAM (Generalized Additive Models) model smooth nonlinear relationships between $Y$ and each regressor using smooth functions (splines) instead of linear coefficients — without pre-specifying the functional form as NLS requires. GAM bridges linear regression and nonparametric models.

When to use

Use GAM when you suspect a relationship is nonlinear but of unknown shape (e.g. a U-shaped effect of age/income). GAM produces interpretable smooth-function plots while keeping additivity.

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Model specification

$$g\big(E[Y_i]\big) = \beta_0 + f_1(X_{1i}) + f_2(X_{2i}) + \dots + f_k(X_{ki})$$

where $g(\cdot)$ is a link function (identity/logit/log…) and each $f_j$ is a smooth function (spline) estimated from data, with a roughness penalty to avoid overfitting; smoothness is chosen via GCV/REML.

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Running in EcoLab

1. Modeling module → Non-linear & semi-parametric family → GAM.
2. Select $Y$, the $X$ variables, mark which use a smooth function; choose the link.
3. Run; view the per-variable smooth-function plots + effective degrees of freedom; export the replication code.

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Replication code

Stata

* ── GAM estimation (semiparametric) ───────────────
* Stata does not have a built-in gam command;
* use npregress series or the community semipar command.

* Option 1: npregress series with B-spline basis
npregress series y x1, basis(bspline)

* Option 2: semipar (community-contributed)
* ssc install semipar
semipar y x2 x3, nonpar(x1)

R

# ── GAM estimation ────────────────────────────────
library(mgcv)

# Smooth terms s() for nonlinear, linear term for x3
model_gam <- gam(
  y ~ s(x1) + s(x2) + x3,
  data   = df,
  method = "REML"
)

# Summary: edf, significance of smooth terms
summary(model_gam)

# Smooth-function plots
plot(model_gam, pages = 1, residuals = TRUE)

Python

# ── GAM estimation ────────────────────────────────
from pygam import LinearGAM, s, l

# s(i) = smooth spline for column i
# l(i) = linear term for column i
X = df[["x1", "x2", "x3"]].values
y = df["y"].values

gam = LinearGAM(s(0) + s(1) + l(2)).fit(X, y)

# Summary: effective degrees of freedom, p-values
print(gam.summary())

# Plot partial dependence for each term
for i, term in enumerate(gam.terms):
    if term.isintercept:
        continue
    gam.plot(term=i)

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Limitations

• Interpreted via plots rather than a single coefficient; harder to fit into concise causal inference.
• Assumes additivity (does not automatically capture interactions unless specified).

Video tutorial

Video Tutorial: Guide to running GAM in EcoLab

See also

• NLS · Quantile Regression · Catalog