Module 5 — Modern Causal Analysis

Double Machine Learning

Estimate low-dimensional causal parameters with orthogonal scores, cross-fitting and flexible nuisance models

Double Machine Learning

Prediction enters as a nuisance task

Consider the partially linear model

Y=\theta D+g(X)+\varepsilon,\qquad D=m(X)+v,

with $E[\varepsilon\mid X,D]=0$ and $E[v\mid X]=0$ . The target is the scalar $\theta$ ; $g$ and $m$ may be complex nuisance functions.

Residualise both sides:

\tilde Y=Y-\hat E[Y\mid X],\qquad \tilde D=D-\hat E[D\mid X],qquad \hat\theta=\frac{\sum_i\tilde D_i\tilde Y_i}{\sum_i\tilde D_i^2}.

This resembles FWL, but flexible learners create overfitting bias if a record helps train the function used to residualise itself.

Cross-fitting gives observations separate jobs

For two folds:

fit both nuisance functions on fold A;
predict and residualise fold B;
reverse A and B;
combine out-of-fold residuals;
estimate $\theta$ and design-aligned uncertainty.

All tuning must occur inside the training data for each fold. Preprocessing on the full sample leaks information.

Cross-fit nonlinear nuisance functions

The polynomial basis is only a transparent learner for teaching. In practice, choose learners through out-of-fold performance and substantive structure, then inspect whether the causal result is stable across reasonable learner libraries and splits.

Orthogonality buys local robustness, not immunity

The score

\psi(W;\theta,\eta)= [D-m(X)]\{Y-\ell(X)-\theta[D-m(X)]\}

is constructed so small first-order nuisance errors have limited effect on $\theta$ . Chernozhukov et al. (2018) formalise this orthogonality and cross-fitting framework.

It still needs:

an identifying model such as conditional exchangeability;
overlap and residual treatment variation;
suitable nuisance convergence and moments;
clustering/splitting that respects grouped or time-dependent data;
a low-dimensional target defined before model search.

Practical audit

Report the learner library, tuning process, fold construction, random seed, overlap, out-of-fold nuisance performance, repeated-split stability and final score/inference formula. The DoubleML documentation is useful implementation guidance, not a replacement for this record.

Quick check

A powerful learner predicts scholarship receipt almost perfectly. Is this ideal for DML?

Answer

Not necessarily. Near-perfect treatment prediction can reveal poor overlap: residual treatment variation becomes tiny and the effect is weakly supported. Inspect propensity distributions and the denominator

\sum\tilde D_i^2

Next: Heterogeneity and Policy Learning

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Module 5 — Modern Causal Analysis

Use flexible prediction, heterogeneity, sensitivity and transportability without weakening identification discipline

Heterogeneity and Policy Learning

Discover conditional effects honestly and turn them into constrained, out-of-sample policy decisions