An Introduction to Double / Debiased Machine Learning

A practical guide for robust causal inference with modern data

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🗺️ Expanded Road Map

1. Motivation – why predictive ML alone cannot deliver causal answers
2. Two pillars of DML – Neyman orthogonality & cross‑fitting
3. The algorithm – partitioning, nuisance fitting, solving, standard errors
4. Diagnostics & model choice – loss metrics, CVC, residual checks, stacking
5. Three empirical case studies – retirement savings, health shocks, online monopsony
6. Hands‑on tips – folds, repetitions, transparent reporting
7. Recent extensions – automatic scores, set identification, HD targets
8. Key take‑aways & open questions

❓ Motivation: The Inference Gap

• Objective: quantify causal effects with valid uncertainty
• Obstacle: nuisance objects—propensity scores, outcome models, fixed effects—are high‑dimensional
• Naïve ML plug‑ins induce regularization bias and *over‑fitting bias
• Classical asymptotics fail ⇒ CIs undercover, p‑values unreliable

⚙️ Generic Semi‑Parametric Setup

• (Y): scalar outcome
• (D): treatment or regressors
• (X): high‑dimensional controls

🎯 Neyman Orthogonality (Pillar 1)

A score \(\psi\) is orthogonal if $$\mathbb E[\psi(W;\theta_0,\eta_0)]=0$$ and $$\frac{\partial}{\partial\lambda},\mathbb E\big[\psi\big(W;\theta_0,\eta_0+\lambda(\tilde\eta-\eta_0)\big)\big]\Big|_{\lambda=0}=0\quad (2)$$ 

• First‑order bias from estimating \(\eta\) cancels
• Only need \(n^{-1/4}\) convergence of \(\hat\eta\)

Think of a Taylor series: the linear term vanishes, leaving only a tiny quadratic remainder.

Implications

🔄 Cross‑Fitting (Pillar 2)

“Don’t grade your own homework.”

1. Randomly split data into (K) folds \(I_1,\dots,I_K\)
2. Fit \(\hat\eta_{-k}\) on the other \(K-1\) folds
3. Evaluate score on held‑out fold
4. Aggregate all folds and solve for \(\hat\theta\)

Breaks correlation between nuisance errors and scores → kills over‑fitting bias while using every observation efficiently.
Default: (K=5) – 10 (larger if cheap).

📐 Asymptotics & Variance

With pillars 1 + 2: $$\sqrt n,(\hat\theta-\theta_0);\rightarrow_d;\mathcal N(0,V)$$ Sandwich variance is plug‑in: $$\hat V=\hat J^{-1}\Big(\tfrac1n\sum_{i}\psi_i^2\Big)\hat J^{-1},\qquad \hat J=\tfrac1n\sum_i\partial_\theta\psi_i$$ No fragile bootstrap needed.

🧮 Example 1 – Average Treatment Effect

Parameter: $$\theta_0=\mathbb E[Y(1)-Y(0)]$$

Doubly‑robust score:

$$\psi_{ATE}=\Big[\tfrac{D}{r(X)}-\tfrac{1-D}{1-r(X)}\Big]\big(Y-\ell(D,X)\big)+\ell(1,X)-\ell(0,X)-\theta$$

• Consistent if either \(r\) or \(\ell\) correct
• Simulations: 95 % coverage vs 71 % for naïve IPW

🧮 Example 2 – Partially Linear Regression

Model: (Y=D\theta_0+g(X)+\varepsilon) $$\psi_{PLR}=\big(D-r(X)\big)\big(Y-\ell(X)\big)-\big(D-r(X)\big)^2\theta$$ Effective with text/image controls when (g) is ML‑fitted (RF, XGB, BERT …)

🧮 Example 3 – Group‑Time ATT (Staggered DiD)

For units first treated in (g) and evaluated in (t): $$ATT_{g,t}=\mathbb E[\Delta Y_{it}(1)-\Delta Y_{it}(0)\mid G=g]$$ Orthogonal score from Chang & Santonastaso (2021) gives valid DiD with heterogenous effects.

💼 Application 1 — 401(k) Eligibility

• 1991 SIPP, (n≈10{,}000); outcome: net assets
• Random‑forest nuisances
• DML: −$6.9 k (SE 1.3 k)
• IPW & RA diverge → DML preferred

🏥 Application 2 — Hospital Admission

• HRS, 656 households, 5 waves
• 15‑fold DML per (g,t), median aggregation
• Spending jumps ≈ $2.4 k in first post‑admission wave, no pre‑trend

🛒 Application 3 — MTurk Monopsony

• Outcome: log fill‑time; regressor: log payment
• 30 k text features + BERT embeddings
• Nine learners tested; diagnostics via $R^2$ & CVC
• Stacked DML: elasticity ≈ −0.024 (SE 0.005) → moderate monopsony

🧰 Practical Guide & Rules of Thumb

🌱 Recent Extensions

• Automatic score generators (Chernozhukov et al., 2022)
• Set‑identified and sensitivity bounds
• Orthogonal learners for function targets
• Higher‑order theory: finite‑sample tweaks
• Causal AI pipelines with LLM embeddings

🚀 Key Take‑Aways

• DML = Orthogonality ➕ Cross‑fitting → valid inference with ML
• Mitigates bias without restrictive functional forms
• Requires thoughtful learner tuning and transparent diagnostics
• A robust baseline for modern empirical work

🙏 Thank You!

Questions or feedback? → email / GitHub
Code & resources: repository link