Double LASSO for Causal Inference

Does abortion reduce crime? A disciplined replication

−0.096rigorous Double LASSO

+0.019cross-validated · sign flip

284candidate controls

Carlos Mendez

Nagoya University (GSID)

June 11, 2026

The Tension

Act I

With 284 candidate controls, the answer depends on which ones you keep

Donohue & Levitt found more abortion access tracked less crime. Belloni–Chernozhukov–Hansen then expanded 8 controls into 284.

Keep too few and you risk confounding. Keep too many and the signal drowns. Which subset?

This is the central tension of high-dimensional causal inference: variable selection decides the result. The same data will give opposite answers depending on the selection rule.

Five estimators, three crimes — wildly different answers from one dataset

\(\hat\alpha \pm 95\%\) CI · First-diff, kitchen-sink OLS, PSL, Double LASSO (rigorous), Double LASSO (CV). Dashed line = zero.

Spoiler figure. Don’t explain every bar yet — just plant that the estimators disagree, sometimes flipping sign. We earn each one in Act II and come back to this picture.

Where we’re going

• The data: a 48-state, 12-year panel with 284 candidate controls
• Five estimators, escalating discipline
• Double LASSO — selecting on the outcome and the treatment
• The lesson: theory-tuned vs prediction-tuned penalties

The Investigation

Act II

The lab: 48 states × 12 years, 576 rows, 284 candidate controls

• Outcome — one of three crime rates (violent, property, murder)
• Treatment — the “effective abortion rate”
• Controls — 8 original covariates expanded to 284 (lags, interactions, trends)

State fixed effects absorbed by first-differencing; year effects partialled out (Frisch–Waugh–Lovell). \(p/n \approx 0.49\) — the high-dimensional regime where Double LASSO is meant to help.

Five estimators ask the same question with escalating discipline

• First-difference OLS — no controls (the Donohue–Levitt baseline)
• Kitchen-sink OLS — all 284 controls at once
• PSL — one LASSO, treatment forced in
• Double LASSO (rigorous) — two LASSOs, theory-chosen penalty
• Double LASSO (CV) — two LASSOs, cross-validated penalty

With zero controls, more abortion tracks less crime: −0.152

Outcome	\(\hat\alpha\)	SE	Sig. 5%?
Violent	−0.152	0.034	yes
Property	−0.108	0.022	yes
Murder	−0.204	0.067	yes

This is the result the four LASSO methods stress-test — not one they generate.

All three negative and significant with state-clustered standard errors (48 clusters). A one-unit rise in the differenced abortion rate ≈ a 15% fall in violent crime. Does this survive adding 284 controls?

Throw in all 284 controls and OLS claims abortion raises murder by 234%

+2.34

Kitchen-sink OLS, murder (\(\hat\alpha\)); violent crime flips sign to +0.014

p = 284 < n = 576, so OLS technically inverts — but near-collinear controls make the estimate nonsense and the SEs balloon (R silently drops 3 rank-deficient columns). The cautionary tale that motivates disciplined selection.

Double LASSO selects on the outcome and the treatment, then runs OLS

\[\hat\beta(\lambda)=\arg\min_\beta\ \frac{1}{2n}\sum_{i=1}^{n}\big(y_i-x_i^\top\beta\big)^2+\lambda\sum_{j=1}^{p}|\beta_j|\]

Run it twice — once for \(y\) on \(X\) (set \(I_y\)), once for \(d\) on \(X\) (set \(I_d\)) — then OLS of \(y\) on \(d\) and the union \(I_y\cup I_d\).

The L1 penalty \(\lambda\sum_j|\beta_j|\) zeroes weak controls; the union keeps anything that predicts either side.

PSL selects only on y, so it can drop a control that predicts the treatment but not the outcome — exactly the omitted variable that biases α. Double LASSO’s second LASSO (d on X) catches it. Frisch–Waugh–Lovell guarantees the post-OLS α is unbiased if the union captures the confounding.

Six lines fit the rigorous Double LASSO in R

library(hdm); library(sandwich); library(lmtest)
Iy <- which(rlasso(X, y)$index)     # controls that predict crime
Id <- which(rlasso(X, d)$index)     # controls that predict abortion
S  <- union(Iy, Id)                  # the union is the Double LASSO safeguard
fit <- lm(y ~ d + X[, S])            # post-OLS on the selected support
coeftest(fit, vcov = vcovCL, cluster = state)["d", ]

“Rigorous” means hdm::rlasso chooses λ from Belloni et al.’s theory (c = 1.1, γ = 0.05), not from cross-validation. The post-OLS step is load-bearing: α comes from unshrunk OLS on the selected support, never from LASSO’s shrunk coefficients.

Theory keeps 8 controls; cross-validation keeps 150

\(|I_y|\), \(|I_d|\), intersection, union out of 284 candidates — rigorous (teal) vs CV (orange).

For violent crime the rigorous union is 8; the CV union is 150 — twenty times more. |I_y| = 0 for violent and murder: no control predicts crime well enough to survive the rigorous threshold, while |I_d| = 8–9 says abortion is predictable. That asymmetry is exactly when DL helps most.

Theory-tuned \(\lambda\) protects the causal signal; prediction-tuned \(\lambda\) flips it

Rigorous (theory)

• \(\lambda\) from Belloni et al. theory
• 8–12 controls kept
• violent \(\hat\alpha = -0.096\)
• selection matches the paper exactly

CV (prediction)

• \(\lambda\) minimises prediction MSE
• 109–161 controls kept
• violent \(\hat\alpha = +0.019\) (sign flip)
• murder \(\hat\alpha = -1.11\) (explodes)

Same three-step recipe, same data — only λ differs. CV optimises prediction, so it includes marginally-useful controls; cumulatively they soak up the treatment’s variation.

Cross-validation’s \(\lambda\) is so small that 143 of 284 controls survive

Coefficient paths, \(d\)-equation (violent panel). Dashed line = \(\log(\lambda_{\min})\); 143 paths nonzero there.

Even in the treatment equation alone, CV’s MSE-optimal λ keeps 143 controls. Prediction-optimal is not selection-optimal for a causal target — the whole lesson in one picture.

The Resolution

Act III

Rigorous Double LASSO restores a sensible −0.096 for violent crime

−0.096

\(\hat\alpha\), rigorous Double LASSO (SE 0.051) · matches the paper’s −0.104; selection counts match exactly

Between the no-controls baseline (−0.152) and the kitchen-sink mess (+0.014), the rigorous DL lands at −0.096 off just 8 controls. The 95% CI [−0.197, +0.004] barely contains zero — one notch below 5% significance, an honest result.

Does LASSO make this causal? No — two assumptions still carry the weight

Objection. Machine-selecting controls can’t manufacture identification.

Response. Correct. \(\alpha\) is identified only under conditional independence given X and parallel trends. LASSO just chooses controls flexibly; it can’t rule out collider bias or bias amplification. The paper evaluates a method, not the abortion–crime claim.

Steelman, don’t strawman. DL disciplines selection; it does not relax the identifying assumptions. Fitzgerald et al. explicitly disclaim a substantive causal reading.

Let the theory, not the cross-validator, choose your controls.

The single takeaway. In the high-dimensional, small-sample regime, a theory-tuned penalty under-selects on purpose to leave the causal signal undisturbed — the difference between −0.096 and a sign flip.