Lasso — L1 regularized regression

Lasso (Least Absolute Shrinkage and Selection Operator) adds an L1 penalty to OLS. Unlike Ridge, Lasso can drive some coefficients exactly to zero — i.e. automatic variable selection, yielding a sparse, interpretable model.

When to use

Use Lasso when you have many regressors and want to select the important subset. When groups of variables are highly correlated, consider Elastic Net.

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Model specification

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

The L1 penalty ($\sum |\beta_j|$) produces corner solutions ⇒ many $\beta_j = 0$. $\lambda$ controls the sparsity level.

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Notes

• Choose $\lambda$ by cross-validation; standardize variables first.
• With highly correlated variables, Lasso tends to pick one and drop the rest (unstable) ⇒ Elastic Net fixes this.
• Post-selection inference requires care.

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Running in EcoLab

1. Modeling module → Regularized regression family → Lasso.
2. Select $Y$, the $X$ variables; enable standardization; choose $\lambda$ (CV).
3. Read the retained variables (non-zero coefficients) and the path; export the replication code.

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Replication code

Stata

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

lasso linear y x1-x20, selection(cv)

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

R

# ---- Lasso (alpha = 1) 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

# Lasso with CV
cv_lasso <- cv.glmnet(X, y, alpha = 1)
plot(cv_lasso)

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

Python

# ---- Lasso with cross-validation ----
from sklearn.linear_model import LassoCV
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)

# LassoCV selects lambda automatically
model = LassoCV(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

• Unstable when variables are highly correlated.
• Selects at most $n$ variables when $p > n$.

Video tutorial

Video Tutorial: Running Lasso regression in EcoLab

See also

• Ridge · Elastic Net · Adaptive Lasso · Catalog