Elastic Net — L1 + L2 regularization

Elastic Net combines the L1 penalty (Lasso) and the L2 penalty (Ridge). It both selects variables (like Lasso) and handles highly correlated variable groups well (like Ridge) — fixing Lasso's instability under correlation.

When to use

Use Elastic Net when you have many variables and there are groups of highly correlated variables that you want to keep/drop together rather than picking one at random.

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Model specification

$$\min_{\beta} \; \sum_{i=1}^{n} (Y_i - X_i \beta)^2 + \lambda \left[ \alpha \sum_{j} |\beta_j| + (1-\alpha) \sum_{j} \beta_j^2 \right]$$

• $\alpha \in [0,1]$ is the mixing ratio: $\alpha = 1$ ⇒ Lasso; $\alpha = 0$ ⇒ Ridge.
• $\lambda$ controls the total penalty.

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Running in EcoLab

1. Modeling module → Regularized regression family → Elastic Net.
2. Select $Y$, the $X$ variables; standardize; choose $\alpha$ and $\lambda$ (2-D grid CV).
3. Read the retained variables + coefficients; export the replication code.

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Replication code

Stata

* ---- Elastic Net with cross-validation ----
use "macro_data.dta", clear

elasticnet linear y x1-x20, selection(cv) alphas(0.5)

* Display selected coefficients
lassocoef, display(coef, standardized)

R

# ---- Elastic Net (alpha = 0.5) with cross-validation ----
library(glmnet)

# Load and prepare data (illustrative)
df <- read.csv("macro_data.csv")
X <- as.matrix(df[, paste0("x", 1:20)])
y <- df$y

# Elastic Net with alpha = 0.5 (midpoint between Ridge and Lasso)
cv_enet <- cv.glmnet(X, y, alpha = 0.5)
plot(cv_enet)

# Best lambda and coefficients
best_lambda <- cv_enet$lambda.min
coef(cv_enet, s = best_lambda)

Python

# ---- Elastic Net with cross-validation ----
from sklearn.linear_model import ElasticNetCV
from sklearn.preprocessing import StandardScaler
import pandas as pd

# Load data (illustrative)
df = pd.read_csv("macro_data.csv")
X = df[[f"x{i}" for i in range(1, 21)]]
y = df["y"]

# Standardize
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)

# Elastic Net: l1_ratio = alpha (0.5 = equal L1 + L2)
model = ElasticNetCV(l1_ratio=0.5, cv=5).fit(X_scaled, y)
print(f"Best alpha (lambda): {model.alpha_}")
print(f"Non-zero coefficients: {sum(model.coef_ != 0)}")
print(f"Coefficients: {model.coef_}")

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Limitations

• Has two tuning parameters ($\alpha$, $\lambda$) ⇒ heavier CV.
• Still a prediction-oriented method; causal interpretation requires care.

Video tutorial

Video Tutorial: Running Elastic Net in EcoLab

See also

• Ridge · Lasso · Adaptive Lasso · Catalog