Chapter 3.3 — Selection on Observables II: Entropy Balancing & Doubly-Robust Estimation

Maya has a propensity score and a problem with it.

In Ch 3.2 she did everything right. She is studying a free financial-literacy program — a six-week course on budgeting and credit that some students at her old high school enrolled in and some did not — and she wants to know whether enrolling later lowered a student’s interest rate on their first installment loan. Enrollment was not random: the students who signed up were already a little more financially engaged, came from households with slightly higher income, had marginally better starting credit, and skewed older. That is selection — exactly the bias Ch 3.1 decomposed and Ch 3.2 set out to defeat. Following Ch 3.2, Maya estimated a propensity score e(\mathbf{X}) = \Pr(D=1 \mid \mathbf{X}) — each student’s probability of enrolling given their observed covariates \mathbf{X} (income, starting credit score, age, prior account history) — and matched each enrollee to the non-enrollee with the nearest score.

It worked, sort of. But matching threw away a third of her control group (no good match within the caliper), the balance table still showed her income covariate off by a noticeable margin, and when she nudged the propensity-model specification — added an interaction here, a quadratic there — her estimated effect wobbled by half a percentage point. She has a result. She does not yet have a result she trusts.

This chapter is about three tools that all start from the same place Ch 3.2 left us — the propensity score and the conditional-independence assumption — and push past the limitations of matching. The first, inverse-probability weighting (IPW), uses the propensity score not to pair units but to reweight them, manufacturing a pseudo-population in which treatment looks random. The second, entropy balancing, skips the propensity model’s middleman entirely and solves directly for weights that force the covariates into balance. The third, doubly-robust / augmented IPW (AIPW), combines a propensity model and an outcome model so that you are protected if either one is right — a genuinely surprising and useful guarantee.

The one-sentence result of this chapter: Under the same conditional-independence and overlap assumptions as Ch 3.2, you can recover the causal effect either by reweighting units by the inverse of their propensity score (IPW), by solving directly for weights that balance the covariates (entropy balancing), or — best of all — by combining a weighting model and an outcome model so the answer is right if either model is correct (doubly robust).

We are not going to re-derive the conditional-independence assumption (CIA) — that \{Y_i(0), Y_i(1)\} \perp D_i \mid \mathbf{X}_i, the assumption that once you condition on the observed covariates \mathbf{X}, who got treated is as good as random — nor the overlap (or common-support) requirement that 0 < e(\mathbf{X}) < 1 for everyone. Ch 3.2 earned those. This chapter takes them as given and asks: given that they hold, what is the smartest way to use them? And the recurring theme — the thread to keep your eye on — is stability. All three methods are consistent in theory. They differ enormously in how badly they misbehave when the propensity model is even slightly wrong, which in real data it always is.

A reminder of the target, in the notation from Ch 3.1. We want a causal contrast — usually the average treatment effect on the treated (ATT),

\tau_{\text{ATT}} = \mathbb{E}[\,Y_i(1) - Y_i(0) \mid D_i = 1\,],

the average effect for the kind of student who actually enrolled — or the average treatment effect (ATE), \tau_{\text{ATE}} = \mathbb{E}[Y_i(1) - Y_i(0)], over everyone. The fundamental problem of causal inference from Ch 3.1 is that for each student we see only one of the two potential outcomes Y_i(1) (rate if enrolled) or Y_i(0) (rate if not); the other is a missing counterfactual. Every method in this chapter is, at heart, a different clever way to fill in the missing column using the units we do observe.

1. Inverse-probability weighting: building a pseudo-population

1.1 The trick, stated plainly

Here is the whole idea of IPW in one image. Maya’s enrollees are over-represented among high-income, high-credit students and under-represented among low-income ones — that is what selection means. Suppose a particular high-income student had a propensity score of e(\mathbf{X}) = 0.8: students like her enrolled 80% of the time. Then among the treated group, students like her are over-counted relative to the population, because most of them ended up treated. To undo that, we should count each treated student not as one person but as 1/e(\mathbf{X}) = 1/0.8 = 1.25 people — slightly up-weighting her, because she stands in for the few students like her who happen not to have enrolled. Symmetrically, a low-income student with e(\mathbf{X}) = 0.1 who did enroll is rare and precious: she represents the many students like her who did not enroll, so we count her as 1/0.1 = 10 people.

Do the mirror image for the controls. A control student with e(\mathbf{X}) = 0.8 (the kind who usually enrolls but didn’t) is rare among controls and gets up-weighted by 1/(1 - e(\mathbf{X})) = 1/0.2 = 5; a control with e(\mathbf{X}) = 0.1 is common and gets a weight of 1/0.9 \approx 1.11.

IPW in one sentence: weight each treated unit by 1/e(\mathbf{X}_i) and each control unit by 1/(1 - e(\mathbf{X}_i)), and the reweighted treated and control groups have the same covariate distribution — so comparing their average outcomes is no longer apples-to-oranges.

The reweighted dataset is called a pseudo-population. In it, by construction, the covariates \mathbf{X} are independent of treatment D: treatment is as good as randomly assigned, which is exactly the world a randomized experiment lives in. And in that world the naive difference-in-means is an unbiased causal estimate. We have not run an experiment; we have reweighted our way into a fake one.

1.2 Why it works — the algebra behind the magic

Why does dividing by the propensity score recover the right average? The key is one line of expectation algebra. Take the treated arm. We want \mathbb{E}[Y_i(1)], the average outcome if everyone were treated. We only observe Y_i for the units with D_i = 1. Consider the weighted quantity

\mathbb{E}\!\left[\frac{D_i\, Y_i}{e(\mathbf{X}_i)}\right].

The factor D_i zeroes out every control (they contribute nothing), so the expectation only runs over treated units, where Y_i = Y_i(1). Now use the law of iterated expectations, conditioning on \mathbf{X}_i first:

\mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)}\right] = \mathbb{E}\!\left[\, \mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)} \,\Big|\, \mathbf{X}_i\right]\right].

Inside the inner expectation, e(\mathbf{X}_i) is a constant (we have fixed \mathbf{X}_i), so it comes out front. And under CIA, D_i is independent of Y_i(1) given \mathbf{X}_i, so the expectation of their product factors:

\mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)} \,\Big|\, \mathbf{X}_i\right] = \frac{1}{e(\mathbf{X}_i)}\, \mathbb{E}[D_i \mid \mathbf{X}_i]\, \mathbb{E}[Y_i(1) \mid \mathbf{X}_i] = \frac{1}{e(\mathbf{X}_i)}\, e(\mathbf{X}_i)\, \mathbb{E}[Y_i(1) \mid \mathbf{X}_i].

The propensity scores cancel — that is the whole trick — leaving \mathbb{E}[Y_i(1) \mid \mathbf{X}_i], whose outer expectation over \mathbf{X} is exactly \mathbb{E}[Y_i(1)]. The same argument with (1 - D_i)/(1 - e(\mathbf{X}_i)) recovers \mathbb{E}[Y_i(0)]. Subtract, and you have the ATE. The cancellation of e(\mathbf{X}_i) in that middle step is the entire reason inverse-probability weighting works: dividing by the probability of selection exactly undoes the over- or under-sampling that selection created.

1.3 Two estimators: Horvitz–Thompson and Hájek

Turning that expectation into a formula you can run on data gives the Horvitz–Thompson estimator (the name comes from 1950s survey sampling, where the same inverse-probability idea was invented to handle unequal sampling rates):

\hat\tau_{\text{HT}} = \frac{1}{N}\sum_{i=1}^{N} \frac{D_i\,Y_i}{\hat e(\mathbf{X}_i)} - \frac{1}{N}\sum_{i=1}^{N} \frac{(1-D_i)\,Y_i}{1 - \hat e(\mathbf{X}_i)},

where \hat e(\mathbf{X}_i) is the estimated propensity score from Ch 3.2 (typically a logit). This is a direct translation of the two expectations above, with sample averages standing in for \mathbb{E}[\cdot].

There is a subtle defect, though. The weights 1/\hat e(\mathbf{X}_i) do not, in any given sample, add up to N — they add up to whatever they happen to add up to. So the Horvitz–Thompson “average” is dividing the weighted sum of outcomes by N rather than by the actual total weight, which can make it numerically wild and even push an estimated mean outside the range of the data. The fix is to normalize the weights so they sum to one within each arm, giving the Hájek estimator:

\hat\tau_{\text{H\'ajek}} = \frac{\sum_i \dfrac{D_i\,Y_i}{\hat e(\mathbf{X}_i)}}{\sum_i \dfrac{D_i}{\hat e(\mathbf{X}_i)}} - \frac{\sum_i \dfrac{(1-D_i)\,Y_i}{1 - \hat e(\mathbf{X}_i)}}{\sum_i \dfrac{1-D_i}{1 - \hat e(\mathbf{X}_i)}}.

The numerators are the same; the denominators are now the sum of the weights in each arm rather than N. This is a self-normalizing weighted mean, it is far more stable in practice, and it is what essentially every software package means when it says “IPW.” Use Hájek, not raw Horvitz–Thompson. The distinction is the IPW version of the lesson from Ch 1: an estimator can be unbiased in expectation (Horvitz–Thompson) and still be a bad idea in finite samples because its variance is enormous.

1.4 A worked number

Make it concrete with a tiny pseudo-population. Maya has six students; three enrolled (D=1), three did not. The outcome Y is the interest rate (in percentage points); \hat e is the estimated propensity score.

Student	D	\hat e	Y (rate)	weight
A	1	0.8	7.0	1/0.8 = 1.25
B	1	0.5	8.0	1/0.5 = 2.00
C	1	0.2	9.0	1/0.2 = 5.00
D	0	0.8	9.5	1/0.2 = 5.00
E	0	0.5	9.0	1/0.5 = 2.00
F	0	0.2	8.5	1/0.8 = 1.25

The Hájek treated mean is the weighted average of A, B, C: \frac{1.25(7) + 2(8) + 5(9)}{1.25 + 2 + 5} = \frac{8.75 + 16 + 45}{8.25} = \frac{69.75}{8.25} \approx 8.45. The Hájek control mean is the weighted average of D, E, F: \frac{5(9.5) + 2(9) + 1.25(8.5)}{5 + 2 + 1.25} = \frac{47.5 + 18 + 10.625}{8.25} = \frac{76.125}{8.25} \approx 9.23. The IPW estimate is 8.45 - 9.23 = -0.78 percentage points — enrolling lowered the rate by about three-quarters of a point.

Notice what the weights did. Student C (a low-propensity enrollee — the rare student “like a control” who enrolled anyway) got a weight of 5 and dominates the treated mean, pulling it toward the higher rate she carries. Student D (a high-propensity control — the rare student “like a treated” who didn’t enroll) likewise got weight 5 and dominates the control mean. The estimator leans hardest on exactly the units that look like the other group, because those are the units that tell us about the missing counterfactual. The simple unweighted difference, by contrast, would have been 8.0 - 9.0 = -1.0 — a different answer, because the unweighted version lets selection contaminate the comparison.

1.5 When IPW explodes — and what to do about it

Now the failure mode, because reveal-the-trick demands we say what breaks. Look at student C’s weight of 5, and imagine her propensity score had been estimated at \hat e = 0.02 instead of 0.2. Her weight becomes 1/0.02 = 50. One student would carry the weight of fifty, and the entire treated mean would be hostage to that single observation’s outcome and to the accuracy of that one tiny estimated probability. This is the IPW explosion problem, and it is the central practical hazard of the method: as \hat e(\mathbf{X}_i) \to 0 for a treated unit (or \to 1 for a control), the weight \to \infty, and the estimator’s variance blows up.

This is not a rare edge case; it is the typical situation whenever overlap is thin. Recall from Ch 3.2 that overlap requires 0 < e(\mathbf{X}) < 1 for everyone. IPW is exquisitely sensitive to near-violations of overlap: a unit whose covariates make treatment nearly certain or nearly impossible gets an astronomically large weight, and a handful of such units can swamp everything. A single mis-estimated propensity score near zero can move your headline number by a percentage point. Worse, the symptom is quiet — you get an estimate and a standard error; nothing throws an error — so you have to look for it.

Three standard defenses, in increasing order of how much you should reach for them:

Trimming. Drop (or set aside) units with extreme estimated propensity scores — a common rule is to discard observations with \hat e < 0.01 or \hat e > 0.99, or to restrict to the region of common support where treated and control scores overlap. Trimming changes the estimand slightly (you are now estimating the effect on the trimmable subpopulation), so you must report how many units you dropped and re-run without trimming as a robustness check.

Stabilized weights. Instead of 1/\hat e(\mathbf{X}_i), use \hat p \,/\, \hat e(\mathbf{X}_i) for the treated and (1-\hat p)\,/\,(1 - \hat e(\mathbf{X}_i)) for controls, where \hat p = \Pr(D=1) is the overall (marginal) treatment share. Multiplying by the marginal probability shrinks the variance of the weights without changing what they estimate — the giant weights get tamed by a constant factor — and it is essentially free. Most modern IPW uses stabilized weights by default.

Just diagnose the weights. Always print the distribution of your weights: the max, the 99th percentile, and the effective sample size \big(\sum_i w_i\big)^2 / \sum_i w_i^2, which tells you how many “real” observations your weighted sample is worth. If your 600-student sample has an effective sample size of 40 because three units carry half the weight, no amount of trimming will save you — your data simply do not contain a credible counterfactual, and you should say so.

The honest summary: IPW is beautiful in theory and treacherous in practice, and the treachery is entirely about extreme weights born of thin overlap. That treachery is precisely what the next method was invented to avoid.

2. Entropy balancing: solve for the weights directly

2.1 The complaint that motivates it

Step back and notice something odd about the whole propensity-score pipeline, IPW and matching alike. We do not actually care about the propensity score. It is a means to an end: a correctly specified propensity model would, if we reweight by it, balance the covariates. But balancing the covariates is the goal; the propensity model is just an indirect route to it. We fit a logit, hope it is the right functional form, convert its predictions into weights, and then — this is the embarrassing part — check afterward whether the covariates actually came out balanced (the balance diagnostics of Ch 3.2). If they did not, we go back and add interactions and polynomials to the propensity model and try again, fishing for a specification that produces balance.

Jens Hainmueller’s entropy balancing (2012) asks the obvious question: if balance is what we want, why not just solve for it directly?¹ Skip the propensity model. State the balance conditions you want — “the reweighted control group must have the same mean income, mean credit score, and mean age as the treated group” — as exact mathematical constraints, and find weights on the controls that satisfy them. There will be many such weight vectors (the constraints are far fewer than the number of control units), so among all weight vectors that achieve exact balance, pick the one that is closest to uniform — that disturbs the data the least. “Closest to uniform” is made precise by maximizing entropy, which is where the name comes from.

2.2 The optimization, in plain terms

Set up to estimate the ATT (the most common use). We will reweight the control units to look like the treated group, then compare. Let w_i be the weight on control unit i. The problem is

\min_{\{w_i\}} \; \sum_{i \in \text{control}} w_i \log\!\frac{w_i}{q_i} \quad\text{subject to balance and normalization constraints,}

where q_i is a base weight (uniform, q_i = 1/N_c, unless you have a reason otherwise). That objective \sum w_i \log(w_i / q_i) is the Kullback–Leibler divergence of the weights from uniform — a standard measure of how far one distribution is from another. Minimizing it is the same as maximizing entropy: it keeps the weights as flat and as close to the base weights as possible. Flat weights are good weights — they mean no single unit dominates, which is exactly the explosion problem IPW could not control.

The constraints are where you say what balance means. For each covariate X_k you want to match, you impose

\sum_{i \in \text{control}} w_i\, X_{ik} \;=\; \bar X_{k}^{\text{treated}},

i.e. the weighted mean of covariate k among controls must exactly equal the treated-group mean of that covariate. You can add a second moment to match variances — $i w_i (X{ik} - X_k^{})2 = $ treated variance — or cross-moments to match covariances. Plus the normalization \sum_i w_i = 1 and non-negativity w_i \ge 0.

Two things make this both powerful and unusual. First, balance is exact, not approximate: the income means do not come out “close,” they come out equal, by construction, to as many decimals as you like. There is no post-hoc balance check that can fail, because balance is the constraint, not the hope. Second, you choose which moments to balance — means only, or means and variances, or higher moments — making it transparent and tunable in a way the black-box logit is not. You are stating your balance requirements as an explicit, inspectable list.

The solution has a tidy closed form. Because of the entropy objective, the optimal weights turn out to be exponential in the covariates, w_i \propto q_i \exp\!\big(-\sum_k \lambda_k X_{ik}\big), where the \lambda_k are Lagrange multipliers chosen to satisfy the constraints — found quickly by a low-dimensional convex optimization (one \lambda per balance condition, not one per unit). You do not need to implement this by hand; the point is that it is a small, well-behaved problem with a unique answer, not a fishing expedition.

2.3 Why it is usually more stable than PSM and IPW

Here is the punchline, and the reason entropy balancing has become a workhorse in empirical finance. Plain IPW gets balance only if the propensity model is correctly specified, and even then only approximately in any finite sample — and it pays for misspecification with exploding weights. Entropy balancing guarantees exact balance on the moments you chose, in every sample, by construction, while simultaneously keeping the weights as close to uniform as the constraints allow. It gives you the thing IPW only promises (balance) without the thing IPW inflicts (wild weights).

There is even a deep connection: entropy balancing is equivalent to a particular logistic propensity model — but one fit by a loss function that targets covariate balance directly rather than predictive likelihood. So it is not magic from nowhere; it is what you get when you stop asking the propensity model to predict treatment well and start asking it to balance covariates well, which is what you actually needed all along.

The practical contrast with matching (Ch 3.2) is just as sharp. Matching discards units (the unmatched controls, the ones outside the caliper) — Maya lost a third of her control group. Entropy balancing keeps every unit and simply gives some of them small weights; nothing is thrown away, so you retain more information and more precision. And it sidesteps the specification-search problem: there is no “which matching algorithm, which caliper, with or without replacement” menu to agonize over and to invite reviewer suspicion that you picked the spec that gave the prettiest answer.

2.4 Maya, reweighted

Back to Maya. She wants the ATT: the effect of enrolling, for the kind of student who enrolled. She reweights her non-enrollees so that their weighted means of income, starting credit score, age, and prior-account count exactly match the enrollees’ means, while keeping the weights as flat as possible. After solving, her balance table is perfect — every covariate difference is mechanically zero — and her effective sample size among controls is, say, 410 out of 600, a mild and honest cost. The reweighted control mean rate is 9.1; the (unweighted, since this is ATT) enrollee mean is 8.4; her entropy-balanced ATT is 8.4 - 9.1 = -0.7 percentage points. Crucially, when she re-runs adding variance matching, the number barely moves — a reassurance the IPW wobble never gave her, because the estimate no longer rides on a fragile functional form. This is the kind of robustness that turns “I have a result” into “I have a result I trust.”

A Priya variant to show it generalizes beyond Maya’s loans. Priya studies climate insurance: she wants to know whether buying parametric drought insurance changed a smallholder farm’s investment in irrigation. Insured and uninsured farms differ on rainfall variability, farm size, distance to a bank branch, and prior losses — classic selection. Rather than match or fit a logit, Priya entropy-balances the uninsured farms to share the insured farms’ means (and variances) of those four covariates, then compares irrigation spending. The weights stay tame even though a few insured farms sit in unusual rainfall zones (the units that would have blown up an IPW estimator), precisely because the entropy objective refuses to let any one farm’s weight run away. Same estimand, same assumptions as IPW — far steadier in hand.

3. Doubly-robust / Augmented IPW: two shots at the truth

3.1 The setup: two models, two ways to be right

So far we have leaned entirely on a model of treatment — the propensity score, or the balancing weights, which is the same job in different clothes. There is a completely different route to the counterfactual that we have so far only mentioned: a model of the outcome. Fit a regression of Y on \mathbf{X} separately in the treated and control groups, getting predicted outcomes \hat\mu_1(\mathbf{X}) = \hat{\mathbb{E}}[Y \mid \mathbf{X}, D=1] and \hat\mu_0(\mathbf{X}) = \hat{\mathbb{E}}[Y \mid \mathbf{X}, D=0]. Then impute each unit’s missing counterfactual with the model’s prediction and average the difference: \hat\tau_{\text{reg}} = \frac{1}{N}\sum_i [\hat\mu_1(\mathbf{X}_i) - \hat\mu_0(\mathbf{X}_i)]. This is regression imputation (sometimes “g-computation”), and under CIA it is also consistent — if the outcome model is correctly specified.

So we have two strategies, each consistent under CIA but each leaning on a different modeling assumption:

IPW / weighting is consistent if the propensity model \hat e(\mathbf{X}) is correct. It does not care whether the outcome is linear in \mathbf{X}.
Regression imputation is consistent if the outcome model \hat\mu_d(\mathbf{X}) is correct. It does not care whether you got the propensity score right.

In real data you are never sure either model is correct. Wouldn’t it be nice to hedge — to use both, in a way that is forgiving if one of them is wrong?

3.2 The estimator, and why “doubly”

That hedge exists. It is the augmented inverse-probability-weighted (AIPW) estimator, also called doubly robust, developed by Robins, Rotnitzky, and Zhao (1994).² For the ATE it is, per arm,

\hat\mu_1^{\text{DR}} = \frac{1}{N}\sum_{i=1}^N \left[ \hat\mu_1(\mathbf{X}_i) + \frac{D_i\,\big(Y_i - \hat\mu_1(\mathbf{X}_i)\big)}{\hat e(\mathbf{X}_i)} \right],

with the symmetric formula for \hat\mu_0^{\text{DR}} (replace D_i with 1-D_i, \hat\mu_1 with \hat\mu_0, and \hat e with 1-\hat e), and \hat\tau_{\text{DR}} = \hat\mu_1^{\text{DR}} - \hat\mu_0^{\text{DR}}.

Read the bracket as a sum of two pieces. The first piece, \hat\mu_1(\mathbf{X}_i), is the outcome-regression prediction — the regression-imputation estimate. The second piece is an IPW correction applied to the regression’s residuals: it takes the part the outcome model got wrong on the treated units, Y_i - \hat\mu_1(\mathbf{X}_i), and reweights it by 1/\hat e(\mathbf{X}_i). So AIPW starts from the outcome model’s best guess and then patches it using inverse-probability-weighted residuals. The word augmented is literal: it is IPW augmented with an outcome model, or equivalently the outcome model augmented with an IPW correction.

Now the “doubly” — the surprising part that earns the method its fame. The estimator is consistent if EITHER the propensity model OR the outcome model is correct; you do not need both. You get two independent shots at the truth, and you only have to hit with one. Here is why, informally, by checking the two cases:

Case 1 — the outcome model is right, the propensity model is wrong. Then \hat\mu_1(\mathbf{X}_i) already estimates \mathbb{E}[Y(1) \mid \mathbf{X}_i] correctly. The correction term has expectation zero because, with a correct outcome model, the residual Y_i - \hat\mu_1(\mathbf{X}_i) averages to zero within each value of \mathbf{X} — so no matter what (possibly wrong) weights 1/\hat e you multiply those residuals by, they still average out to zero. The wrong propensity score is multiplied into something that vanishes. The first term carries the day.

Case 2 — the propensity model is right, the outcome model is wrong. Then \hat\mu_1 is some incorrect prediction. But the second term, with the correct e(\mathbf{X}), is exactly the IPW estimator applied to Y_i — which we proved in §1.2 is consistent — minus an IPW-weighted term in \hat\mu_1 that, with correct propensity scores, precisely cancels the wrong \hat\mu_1 sitting in the first piece. The bad outcome model gets added in by the first term and subtracted back out by the second, leaving the consistent IPW estimate standing. The IPW machinery carries the day.

Either way, one correct model is enough; the other model’s errors get neutralized. That is the doubly-robust property, and it is genuinely remarkable: a kind of statistical insurance policy where you only need one of two claims to be valid.

3.3 Why this matters beyond the safety net

Doubly robust is not just defensive. It also tends to be more efficient — lower variance — than plain IPW when both models are even roughly right, because the outcome model soaks up much of the variation in Y before the noisy inverse-probability weights ever touch it. The weights now act on residuals (small, after a decent regression) rather than on the raw outcomes (large), so the weight-explosion damage is muffled even when some \hat e are small. AIPW does not fully cure the extreme-weight problem — a \hat e near zero still hurts — but it leans on those weights far less.

There is a deeper modern reason AIPW is everywhere now. Because the estimator has a special “insensitivity” to small errors in each model (a property called Neyman orthogonality), you can plug machine-learning estimators — random forests, gradient boosting, neural nets — into both the propensity model and the outcome model, use sample-splitting (fit the models on one half, evaluate on the other) to avoid overfitting, and still get valid standard errors and confidence intervals for the treatment effect. This is the engine behind double/debiased machine learning (Chernozhukov et al., 2018),³ the dominant framework for high-dimensional causal estimation in current empirical economics and finance. You will not implement that this week, but it is worth knowing that the humble bracket above is the seed of the field’s most active methodology. AIPW is where selection-on-observables grows up.

4. The code

One runnable script ties the three methods together on a simulated dataset where we know the true effect, so you can watch which estimators land near it and which wander. We build Maya’s financial-literacy setup: enrollment depends on income and credit (so there is selection), and the true effect of enrolling is a -0.8 percentage-point reduction in the rate.

import numpy as np
import pandas as pd
import statsmodels.api as sm
import statsmodels.formula.api as smf

rng = np.random.default_rng(33)
N = 4000

# --- covariates ---
income = rng.normal(0, 1, N)          # standardized household income
credit = rng.normal(0, 1, N)          # standardized starting credit score

# --- selection into treatment (the propensity TRUTH) ---
lin = 0.9 * income + 0.7 * credit
p_true = 1 / (1 + np.exp(-lin))       # true propensity score
D = (rng.uniform(size=N) < p_true).astype(int)

# --- outcomes: TRUE treatment effect is -0.8; outcome depends on X too ---
tau_true = -0.8
y0 = 9.0 - 0.6 * income - 0.5 * credit + rng.normal(0, 1, N)   # rate if NOT enrolled
y1 = y0 + tau_true                                             # rate if enrolled
Y = np.where(D == 1, y1, y0)          # we observe only one potential outcome each

df = pd.DataFrame({"Y": Y, "D": D, "income": income, "credit": credit})

# --- (0) naive difference in means: biased by selection ---
naive = df.loc[df.D == 1, "Y"].mean() - df.loc[df.D == 0, "Y"].mean()

# --- estimate the propensity score (a logit), then build stabilized IPW weights ---
ps = smf.logit("D ~ income + credit", data=df).fit(disp=0)
ehat = ps.predict(df).clip(0.01, 0.99)        # clip = trimming extreme scores
p_marg = df.D.mean()
w = np.where(df.D == 1,
             p_marg / ehat,                    # stabilized treated weight
             (1 - p_marg) / (1 - ehat))        # stabilized control weight

# --- (1) IPW (Hajek): self-normalizing weighted means per arm ---
def hajek_mean(mask):
    ww = w[mask]
    return np.sum(ww * df.Y[mask]) / np.sum(ww)
ipw = hajek_mean(df.D == 1) - hajek_mean(df.D == 0)

# --- (2) Augmented IPW / doubly robust ---
m1 = smf.ols("Y ~ income + credit", data=df[df.D == 1]).fit()   # outcome model, treated
m0 = smf.ols("Y ~ income + credit", data=df[df.D == 0]).fit()   # outcome model, control
mu1 = m1.predict(df); mu0 = m0.predict(df)
dr1 = np.mean(mu1 + df.D * (df.Y - mu1) / ehat)
dr0 = np.mean(mu0 + (1 - df.D) * (df.Y - mu0) / (1 - ehat))
aipw = dr1 - dr0

print(f"true effect      : {tau_true:+.3f}")
print(f"naive diff       : {naive:+.3f}   (biased by selection)")
print(f"IPW (Hajek)      : {ipw:+.3f}")
print(f"AIPW (doubly rob): {aipw:+.3f}")
print(f"weights: max={w.max():.1f}  eff. N={w.sum()**2/np.sum(w**2):.0f} of {N}")

Run it and the naive difference comes out badly biased toward zero (selection makes enrollees and non-enrollees differ in the outcome for reasons that have nothing to do with the program), while both IPW and AIPW land close to the true -0.8. The printed weight diagnostics — the max weight and the effective sample size — are the habit to internalize: always look at your weights. For the entropy-balancing comparison, the EBweights/ebal route (or a short convex solve) is the subject of nb3.3, where you will see entropy weights deliver exact balance with a smaller max weight than IPW on the very same data.

To feel the double-robustness property, change the outcome models m1/m0 to deliberately wrong specifications (drop credit, or add a spurious term) and watch AIPW stay near -0.8 because the propensity model is still right; then instead corrupt the propensity formula (drop credit from the logit) while keeping the outcome models correct, and watch AIPW still stay near -0.8. Break both at once and it finally drifts. That experiment — one model wrong at a time is survivable, both wrong is not — is the whole “doubly” in one screen.

5. A practical decision guide: PSM vs. entropy balancing vs. AIPW

You now have, across Ch 3.2 and Ch 3.3, four ways to exploit selection-on-observables. Here is how to choose, stated as the kind of judgment you should be able to defend to a referee.

Reach for propensity-score matching (Ch 3.2) when you want a transparent, intuitive design and your treated and control groups overlap well. Matching’s appeal is that it mimics an experiment unit by unit and is easy to explain to a non-technical audience (“we compared each enrollee to a nearly identical non-enrollee”). Its costs are the ones Maya hit: it discards unmatched units, it forces a menu of arbitrary choices (caliper, with/without replacement, number of neighbors), and it only achieves approximate balance that you must check after the fact. Good for exposition and for a first look; rarely your final, most efficient estimator.

Reach for entropy balancing when you want guaranteed exact balance on chosen moments, maximum stability, and minimal specification angst — which in practice is most of the time in empirical finance. It keeps every unit, produces tame weights by construction, makes your balance targets explicit and inspectable, and is hard for a referee to accuse of cherry-picking because there is no algorithm-and-caliper menu to fish through. Its limitation is that you must choose which moments to balance, and it balances only those — if the true selection depends on a high-order interaction you did not include, you have not balanced it. State your moments and defend them.

Reach for AIPW / doubly-robust when you want the strongest consistency guarantee and the best efficiency, and especially when you are willing to bring an outcome model and a treatment model and let them cover for each other. This is the right default for a headline causal estimate you intend to publish: it is consistent if either model is right, it is more efficient than plain IPW, and it is the on-ramp to machine-learning-based estimation if your covariate set is large. Its cost is conceptual overhead (two models, not one) and that it still suffers — though less — from extreme weights.

The honest meta-advice: report more than one. If PSM, entropy balancing, and AIPW all point to roughly -0.8, your finding is robust to method choice and you can write it up with confidence. If they disagree, that disagreement is information — almost always a sign of thin overlap or a fragile specification — and the disagreement itself belongs in your paper, not swept under the rug of whichever method gave the prettiest number. This is the empirical-spec discipline from the Conventions applied to method choice: name the estimand (ATE or ATT), name the covariates you balanced on, name the overlap region you restricted to, and name the assumption — CIA — that the whole exercise rides on.

6. The ceiling on everything in this chapter — and the door out

Stand back and see the boundary of what Week 3 has built so far. IPW, entropy balancing, AIPW, and the matching of Ch 3.2 are four different engines bolted to the same chassis, and that chassis is the conditional-independence assumption: that once we condition on (or balance, or reweight on) the observed covariates \mathbf{X}, treatment is as good as random. Every method in this chapter is a way to use \mathbf{X} more cleverly, more stably, more efficiently. Not one of them can rescue you if CIA is false.

And CIA is false exactly when there is an unobserved confounder — a variable that drives both selection into treatment and the outcome, that you did not measure and therefore cannot put in \mathbf{X}. Suppose the students who enrolled in Maya’s program were also, on average, more conscientious in a way no dataset captures, and conscientiousness independently lowers interest rates. Then enrollees would have gotten lower rates even without the program, no amount of balancing on income and credit removes it, and every estimator in this chapter is biased — silently, with a clean-looking balance table and a confident standard error. Balancing the observables you have does nothing about the confounder you don’t. This is the unbreakable ceiling of selection-on-observables, and it is why a careful researcher always asks, “what’s the unobserved thing that could be driving both?”

That question is the doorway to Ch 3.4. When you cannot defend CIA — when you fear an unobserved confounder you cannot measure — you need a fundamentally different source of identifying variation: not a richer set of controls, but an instrument, a variable that shoves units into treatment for reasons unrelated to the outcome. Ch 3.4 introduces instrumental variables as the escape hatch for precisely the situation this chapter cannot handle. Everything in Week 3 up to now has assumed selection-on-observables; the rest of the week is about what to do when that assumption finally breaks.

Your Turn

Open nb3.3 — entropy balancing vs. IPW vs. AIPW. You will take a single simulated dataset with a known treatment effect and selection on observables, and run all three estimators side by side: build stabilized IPW (Hájek) weights and inspect their distribution and effective sample size; solve for entropy-balancing weights that exactly match covariate means (then means and variances) and confirm the balance table comes out mechanically perfect; and compute the AIPW estimate, then deliberately break the propensity model, then instead break the outcome model, and watch the doubly-robust estimator survive each single failure but not their conjunction. The goal is to see, with your own data, why entropy weights stay tame where IPW weights explode, and why “doubly robust” is more than a slogan.

Check questions.

A treated unit in Maya’s data has an estimated propensity score of \hat e = 0.04. What is its (unstabilized) IPW weight, and why should this single number make you nervous about the IPW estimate? Name two distinct things you could do about it.
Priya runs AIPW on her climate-insurance data. A colleague says, “your outcome regression is surely misspecified — irrigation spending isn’t linear in rainfall — so your estimate is garbage.” Explain why this objection is not necessarily fatal, and state precisely the condition under which Priya’s AIPW estimate is still consistent despite the wrong outcome model.
Maya’s entropy-balanced ATT and her IPW ATT agree (both about -0.7), but her colleague Sam warns that both could still be badly biased. Describe the one kind of variable whose existence would make that warning correct, explain why no method in this chapter can fix it, and name the chapter (and tool) that addresses it.

Hainmueller, J. (2012). Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies. Political Analysis, 20(1), 25–46.↩︎
Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). Estimation of Regression Coefficients When Some Regressors Are Not Always Observed. Journal of the American Statistical Association, 89(427), 846–866. The doubly-robust / AIPW form is developed and extended across this line of work by Robins and Rotnitzky and coauthors through the 1990s.↩︎
Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/Debiased Machine Learning for Treatment and Structural Parameters. The Econometrics Journal, 21(1), C1–C68.↩︎

# Chapter 3.3 — Selection on Observables II: Entropy Balancing & Doubly-Robust Estimation Maya has a propensity score and a problem with it. In Ch 3.2 she did everything right. She is studying a free financial-literacy program — a six-week course on budgeting and credit that some students at her old high school enrolled in and some did not — and she wants to know whether enrolling later *lowered* a student's interest rate on their first installment loan. Enrollment was not random: the students who signed up were already a little more financially engaged, came from households with slightly higher income, had marginally better starting credit, and skewed older. That is **selection** — exactly the bias Ch 3.1 decomposed and Ch 3.2 set out to defeat. Following Ch 3.2, Maya estimated a **propensity score** $e(\mathbf{X}) = \Pr(D=1 \mid \mathbf{X})$ — each student's probability of enrolling given their observed covariates $\mathbf{X}$ (income, starting credit score, age, prior account history) — and matched each enrollee to the non-enrollee with the nearest score. It worked, sort of. But matching threw away a third of her control group (no good match within the caliper), the balance table still showed her income covariate off by a noticeable margin, and when she nudged the propensity-model specification — added an interaction here, a quadratic there — her estimated effect wobbled by half a percentage point. She has a result. She does not yet have a result she trusts. This chapter is about three tools that all start from the same place Ch 3.2 left us — the propensity score and the conditional-independence assumption — and push past the limitations of matching. The first, **inverse-probability weighting (IPW)**, uses the propensity score not to *pair* units but to *reweight* them, manufacturing a pseudo-population in which treatment looks random. The second, **entropy balancing**, skips the propensity model's middleman entirely and solves directly for weights that force the covariates into balance. The third, **doubly-robust / augmented IPW (AIPW)**, combines a propensity model and an outcome model so that you are protected if *either one* is right — a genuinely surprising and useful guarantee. > **The one-sentence result of this chapter:** Under the same conditional-independence and > overlap assumptions as Ch 3.2, you can recover the causal effect either by reweighting units > by the inverse of their propensity score (IPW), by solving directly for weights that balance > the covariates (entropy balancing), or — best of all — by combining a weighting model and an > outcome model so the answer is right if *either* model is correct (doubly robust). We are not going to re-derive the **conditional-independence assumption (CIA)** — that $\{Y_i(0), Y_i(1)\} \perp D_i \mid \mathbf{X}_i$, the assumption that once you condition on the observed covariates $\mathbf{X}$, who got treated is as good as random — nor the **overlap** (or **common-support**) requirement that $0 < e(\mathbf{X}) < 1$ for everyone. Ch 3.2 earned those. This chapter takes them as given and asks: given that they hold, what is the *smartest* way to use them? And the recurring theme — the thread to keep your eye on — is **stability**. All three methods are consistent in theory. They differ enormously in how badly they misbehave when the propensity model is even slightly wrong, which in real data it always is. A reminder of the target, in the notation from Ch 3.1. We want a causal contrast — usually the **average treatment effect on the treated (ATT)**, $$ \tau_{\text{ATT}} = \mathbb{E}[\,Y_i(1) - Y_i(0) \mid D_i = 1\,], $$ the average effect *for the kind of student who actually enrolled* — or the **average treatment effect (ATE)**, $\tau_{\text{ATE}} = \mathbb{E}[Y_i(1) - Y_i(0)]$, over everyone. The **fundamental problem of causal inference** from Ch 3.1 is that for each student we see only one of the two potential outcomes $Y_i(1)$ (rate if enrolled) or $Y_i(0)$ (rate if not); the other is a missing counterfactual. Every method in this chapter is, at heart, a different clever way to *fill in the missing column* using the units we do observe. --- ## 1. Inverse-probability weighting: building a pseudo-population ### 1.1 The trick, stated plainly Here is the whole idea of IPW in one image. Maya's enrollees are over-represented among high-income, high-credit students and under-represented among low-income ones — that is what selection *means*. Suppose a particular high-income student had a propensity score of $e(\mathbf{X}) = 0.8$: students like her enrolled 80% of the time. Then among the treated group, students like her are *over*-counted relative to the population, because most of them ended up treated. To undo that, we should count each treated student not as one person but as $1/e(\mathbf{X}) = 1/0.8 = 1.25$ people — slightly *up*-weighting her, because she stands in for the few students like her who happen *not* to have enrolled. Symmetrically, a low-income student with $e(\mathbf{X}) = 0.1$ who *did* enroll is rare and precious: she represents the many students like her who did not enroll, so we count her as $1/0.1 = 10$ people. Do the mirror image for the controls. A control student with $e(\mathbf{X}) = 0.8$ (the kind who usually enrolls but didn't) is rare among controls and gets up-weighted by $1/(1 - e(\mathbf{X})) = 1/0.2 = 5$; a control with $e(\mathbf{X}) = 0.1$ is common and gets a weight of $1/0.9 \approx 1.11$. > **IPW in one sentence:** weight each treated unit by $1/e(\mathbf{X}_i)$ and each control unit > by $1/(1 - e(\mathbf{X}_i))$, and the reweighted treated and control groups have the *same* > covariate distribution — so comparing their average outcomes is no longer apples-to-oranges. The reweighted dataset is called a **pseudo-population**. In it, by construction, the covariates $\mathbf{X}$ are independent of treatment $D$: treatment is *as good as randomly assigned*, which is exactly the world a randomized experiment lives in. And in that world the naive difference-in-means is an unbiased causal estimate. We have not run an experiment; we have *reweighted our way into a fake one.* ### 1.2 Why it works — the algebra behind the magic Why does dividing by the propensity score recover the right average? The key is one line of expectation algebra. Take the treated arm. We want $\mathbb{E}[Y_i(1)]$, the average outcome if *everyone* were treated. We only observe $Y_i$ for the units with $D_i = 1$. Consider the weighted quantity $$ \mathbb{E}\!\left[\frac{D_i\, Y_i}{e(\mathbf{X}_i)}\right]. $$ The factor $D_i$ zeroes out every control (they contribute nothing), so the expectation only runs over treated units, where $Y_i = Y_i(1)$. Now use the law of iterated expectations, conditioning on $\mathbf{X}_i$ first: $$ \mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)}\right] = \mathbb{E}\!\left[\, \mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)} \,\Big|\, \mathbf{X}_i\right]\right]. $$ Inside the inner expectation, $e(\mathbf{X}_i)$ is a constant (we have fixed $\mathbf{X}_i$), so it comes out front. And under CIA, $D_i$ is independent of $Y_i(1)$ given $\mathbf{X}_i$, so the expectation of their product factors: $$ \mathbb{E}\!\left[\frac{D_i\,Y_i(1)}{e(\mathbf{X}_i)} \,\Big|\, \mathbf{X}_i\right] = \frac{1}{e(\mathbf{X}_i)}\, \mathbb{E}[D_i \mid \mathbf{X}_i]\, \mathbb{E}[Y_i(1) \mid \mathbf{X}_i] = \frac{1}{e(\mathbf{X}_i)}\, e(\mathbf{X}_i)\, \mathbb{E}[Y_i(1) \mid \mathbf{X}_i]. $$ The propensity scores cancel — that is the whole trick — leaving $\mathbb{E}[Y_i(1) \mid \mathbf{X}_i]$, whose outer expectation over $\mathbf{X}$ is exactly $\mathbb{E}[Y_i(1)]$. The same argument with $(1 - D_i)/(1 - e(\mathbf{X}_i))$ recovers $\mathbb{E}[Y_i(0)]$. Subtract, and you have the ATE. The cancellation of $e(\mathbf{X}_i)$ in that middle step is the entire reason inverse-*probability* weighting works: dividing by the probability of selection exactly undoes the over- or under-sampling that selection created. ### 1.3 Two estimators: Horvitz–Thompson and Hájek Turning that expectation into a formula you can run on data gives the **Horvitz–Thompson estimator** (the name comes from 1950s survey sampling, where the same inverse-probability idea was invented to handle unequal sampling rates): $$ \hat\tau_{\text{HT}} = \frac{1}{N}\sum_{i=1}^{N} \frac{D_i\,Y_i}{\hat e(\mathbf{X}_i)} - \frac{1}{N}\sum_{i=1}^{N} \frac{(1-D_i)\,Y_i}{1 - \hat e(\mathbf{X}_i)}, $$ where $\hat e(\mathbf{X}_i)$ is the estimated propensity score from Ch 3.2 (typically a logit). This is a direct translation of the two expectations above, with sample averages standing in for $\mathbb{E}[\cdot]$. There is a subtle defect, though. The weights $1/\hat e(\mathbf{X}_i)$ do not, in any given sample, add up to $N$ — they add up to whatever they happen to add up to. So the Horvitz–Thompson "average" is dividing the weighted sum of outcomes by $N$ rather than by the actual total weight, which can make it numerically wild and even push an estimated mean outside the range of the data. The fix is to **normalize the weights so they sum to one within each arm**, giving the **Hájek estimator**: $$ \hat\tau_{\text{H\'ajek}} = \frac{\sum_i \dfrac{D_i\,Y_i}{\hat e(\mathbf{X}_i)}}{\sum_i \dfrac{D_i}{\hat e(\mathbf{X}_i)}} - \frac{\sum_i \dfrac{(1-D_i)\,Y_i}{1 - \hat e(\mathbf{X}_i)}}{\sum_i \dfrac{1-D_i}{1 - \hat e(\mathbf{X}_i)}}. $$ The numerators are the same; the denominators are now the *sum of the weights* in each arm rather than $N$. This is a self-normalizing weighted mean, it is far more stable in practice, and it is what essentially every software package means when it says "IPW." Use Hájek, not raw Horvitz–Thompson. The distinction is the IPW version of the lesson from Ch 1: an estimator can be unbiased in expectation (Horvitz–Thompson) and still be a bad idea in finite samples because its variance is enormous. ### 1.4 A worked number Make it concrete with a tiny pseudo-population. Maya has six students; three enrolled ($D=1$), three did not. The outcome $Y$ is the interest rate (in percentage points); $\hat e$ is the estimated propensity score. | Student | $D$ | $\hat e$ | $Y$ (rate) | weight | |---|---|---|---|---| | A | 1 | 0.8 | 7.0 | $1/0.8 = 1.25$ | | B | 1 | 0.5 | 8.0 | $1/0.5 = 2.00$ | | C | 1 | 0.2 | 9.0 | $1/0.2 = 5.00$ | | D | 0 | 0.8 | 9.5 | $1/0.2 = 5.00$ | | E | 0 | 0.5 | 9.0 | $1/0.5 = 2.00$ | | F | 0 | 0.2 | 8.5 | $1/0.8 = 1.25$ | The Hájek treated mean is the weighted average of A, B, C: $\frac{1.25(7) + 2(8) + 5(9)}{1.25 + 2 + 5} = \frac{8.75 + 16 + 45}{8.25} = \frac{69.75}{8.25} \approx 8.45$. The Hájek control mean is the weighted average of D, E, F: $\frac{5(9.5) + 2(9) + 1.25(8.5)}{5 + 2 + 1.25} = \frac{47.5 + 18 + 10.625}{8.25} = \frac{76.125}{8.25} \approx 9.23$. The IPW estimate is $8.45 - 9.23 = -0.78$ percentage points — enrolling lowered the rate by about three-quarters of a point. Notice what the weights did. Student C (a low-propensity enrollee — the rare student "like a control" who enrolled anyway) got a weight of 5 and dominates the treated mean, pulling it toward the higher rate she carries. Student D (a high-propensity control — the rare student "like a treated" who didn't enroll) likewise got weight 5 and dominates the control mean. The estimator leans hardest on exactly the units that look like the *other* group, because those are the units that tell us about the missing counterfactual. The simple unweighted difference, by contrast, would have been $8.0 - 9.0 = -1.0$ — a different answer, because the unweighted version lets selection contaminate the comparison. ### 1.5 When IPW explodes — and what to do about it Now the failure mode, because reveal-the-trick demands we say what breaks. Look at student C's weight of 5, and imagine her propensity score had been estimated at $\hat e = 0.02$ instead of $0.2$. Her weight becomes $1/0.02 = 50$. One student would carry the weight of fifty, and the entire treated mean would be hostage to that single observation's outcome and to the accuracy of that one tiny estimated probability. This is the **IPW explosion problem**, and it is the central practical hazard of the method: as $\hat e(\mathbf{X}_i) \to 0$ for a treated unit (or $\to 1$ for a control), the weight $\to \infty$, and the estimator's variance blows up. This is not a rare edge case; it is the *typical* situation whenever overlap is thin. Recall from Ch 3.2 that overlap requires $0 < e(\mathbf{X}) < 1$ for everyone. IPW is exquisitely sensitive to near-violations of overlap: a unit whose covariates make treatment nearly certain or nearly impossible gets an astronomically large weight, and a handful of such units can swamp everything. A single mis-estimated propensity score near zero can move your headline number by a percentage point. Worse, the symptom is quiet — you get an estimate and a standard error; nothing throws an error — so you have to *look* for it. Three standard defenses, in increasing order of how much you should reach for them: **Trimming.** Drop (or set aside) units with extreme estimated propensity scores — a common rule is to discard observations with $\hat e < 0.01$ or $\hat e > 0.99$, or to restrict to the region of common support where treated and control scores overlap. Trimming changes the estimand slightly (you are now estimating the effect on the trimmable subpopulation), so you must report how many units you dropped and re-run without trimming as a robustness check. **Stabilized weights.** Instead of $1/\hat e(\mathbf{X}_i)$, use $\hat p \,/\, \hat e(\mathbf{X}_i)$ for the treated and $(1-\hat p)\,/\,(1 - \hat e(\mathbf{X}_i))$ for controls, where $\hat p = \Pr(D=1)$ is the overall (marginal) treatment share. Multiplying by the marginal probability shrinks the variance of the weights without changing what they estimate — the giant weights get tamed by a constant factor — and it is essentially free. Most modern IPW uses stabilized weights by default. **Just diagnose the weights.** Always print the distribution of your weights: the max, the 99th percentile, and the **effective sample size** $\big(\sum_i w_i\big)^2 / \sum_i w_i^2$, which tells you how many "real" observations your weighted sample is worth. If your 600-student sample has an effective sample size of 40 because three units carry half the weight, no amount of trimming will save you — your data simply do not contain a credible counterfactual, and you should say so. The honest summary: IPW is beautiful in theory and treacherous in practice, and the treachery is entirely about extreme weights born of thin overlap. That treachery is precisely what the next method was invented to avoid. --- ## 2. Entropy balancing: solve for the weights directly ### 2.1 The complaint that motivates it Step back and notice something odd about the whole propensity-score pipeline, IPW and matching alike. We do not actually *care* about the propensity score. It is a means to an end: a *correctly specified* propensity model would, if we reweight by it, balance the covariates. But balancing the covariates is the *goal*; the propensity model is just an indirect route to it. We fit a logit, hope it is the right functional form, convert its predictions into weights, and then — this is the embarrassing part — *check afterward* whether the covariates actually came out balanced (the balance diagnostics of Ch 3.2). If they did not, we go back and add interactions and polynomials to the propensity model and try again, fishing for a specification that produces balance. Jens Hainmueller's **entropy balancing** (2012) asks the obvious question: if balance is what we want, why not just *solve for it directly*?[^hainmueller] Skip the propensity model. State the balance conditions you want — "the reweighted control group must have the same mean income, mean credit score, and mean age as the treated group" — as exact mathematical constraints, and find weights on the controls that satisfy them. There will be many such weight vectors (the constraints are far fewer than the number of control units), so among all weight vectors that achieve exact balance, pick the one that is **closest to uniform** — that disturbs the data the least. "Closest to uniform" is made precise by maximizing **entropy**, which is where the name comes from. [^hainmueller]: Hainmueller, J. (2012). Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies. *Political Analysis*, 20(1), 25–46. ### 2.2 The optimization, in plain terms Set up to estimate the ATT (the most common use). We will reweight the **control** units to look like the treated group, then compare. Let $w_i$ be the weight on control unit $i$. The problem is $$ \min_{\{w_i\}} \; \sum_{i \in \text{control}} w_i \log\!\frac{w_i}{q_i} \quad\text{subject to balance and normalization constraints,} $$ where $q_i$ is a *base weight* (uniform, $q_i = 1/N_c$, unless you have a reason otherwise). That objective $\sum w_i \log(w_i / q_i)$ is the **Kullback–Leibler divergence** of the weights from uniform — a standard measure of how far one distribution is from another. Minimizing it is the same as *maximizing entropy*: it keeps the weights as flat and as close to the base weights as possible. Flat weights are good weights — they mean no single unit dominates, which is exactly the explosion problem IPW could not control. The constraints are where you say what balance means. For each covariate $X_k$ you want to match, you impose $$ \sum_{i \in \text{control}} w_i\, X_{ik} \;=\; \bar X_{k}^{\text{treated}}, $$ i.e. the *weighted mean* of covariate $k$ among controls must **exactly equal** the treated-group mean of that covariate. You can add a second moment to match variances — $\sum_i w_i (X_{ik} - \bar X_k^{\text{treated}})^2 = $ treated variance — or cross-moments to match covariances. Plus the normalization $\sum_i w_i = 1$ and non-negativity $w_i \ge 0$. Two things make this both powerful and unusual. First, balance is **exact, not approximate**: the income means do not come out "close," they come out *equal*, by construction, to as many decimals as you like. There is no post-hoc balance check that can fail, because balance is the constraint, not the hope. Second, you choose *which moments* to balance — means only, or means and variances, or higher moments — making it transparent and tunable in a way the black-box logit is not. You are stating your balance requirements as an explicit, inspectable list. The solution has a tidy closed form. Because of the entropy objective, the optimal weights turn out to be exponential in the covariates, $w_i \propto q_i \exp\!\big(-\sum_k \lambda_k X_{ik}\big)$, where the $\lambda_k$ are Lagrange multipliers chosen to satisfy the constraints — found quickly by a low-dimensional convex optimization (one $\lambda$ per balance condition, not one per unit). You do not need to implement this by hand; the point is that it is a small, well-behaved problem with a unique answer, not a fishing expedition. ### 2.3 Why it is usually more stable than PSM and IPW Here is the punchline, and the reason entropy balancing has become a workhorse in empirical finance. Plain IPW gets balance *only if the propensity model is correctly specified*, and even then only *approximately* in any finite sample — and it pays for misspecification with exploding weights. Entropy balancing **guarantees exact balance on the moments you chose, in every sample, by construction**, while *simultaneously* keeping the weights as close to uniform as the constraints allow. It gives you the thing IPW only promises (balance) without the thing IPW inflicts (wild weights). There is even a deep connection: entropy balancing is *equivalent* to a particular logistic propensity model — but one fit by a loss function that targets covariate balance directly rather than predictive likelihood. So it is not magic from nowhere; it is what you get when you stop asking the propensity model to predict treatment well and start asking it to balance covariates well, which is what you actually needed all along. The practical contrast with matching (Ch 3.2) is just as sharp. Matching *discards* units (the unmatched controls, the ones outside the caliper) — Maya lost a third of her control group. Entropy balancing *keeps every unit* and simply gives some of them small weights; nothing is thrown away, so you retain more information and more precision. And it sidesteps the specification-search problem: there is no "which matching algorithm, which caliper, with or without replacement" menu to agonize over and to invite reviewer suspicion that you picked the spec that gave the prettiest answer. ### 2.4 Maya, reweighted Back to Maya. She wants the ATT: the effect of enrolling, for the kind of student who enrolled. She reweights her **non-enrollees** so that their weighted means of income, starting credit score, age, and prior-account count *exactly* match the enrollees' means, while keeping the weights as flat as possible. After solving, her balance table is perfect — every covariate difference is mechanically zero — and her effective sample size among controls is, say, 410 out of 600, a mild and honest cost. The reweighted control mean rate is $9.1$; the (unweighted, since this is ATT) enrollee mean is $8.4$; her entropy-balanced ATT is $8.4 - 9.1 = -0.7$ percentage points. Crucially, when she re-runs adding *variance* matching, the number barely moves — a reassurance the IPW wobble never gave her, because the estimate no longer rides on a fragile functional form. This is the kind of robustness that turns "I have a result" into "I have a result I trust." A Priya variant to show it generalizes beyond Maya's loans. Priya studies **climate insurance**: she wants to know whether buying parametric drought insurance changed a smallholder farm's investment in irrigation. Insured and uninsured farms differ on rainfall variability, farm size, distance to a bank branch, and prior losses — classic selection. Rather than match or fit a logit, Priya entropy-balances the uninsured farms to share the insured farms' means (and variances) of those four covariates, then compares irrigation spending. The weights stay tame even though a few insured farms sit in unusual rainfall zones (the units that would have blown up an IPW estimator), precisely because the entropy objective refuses to let any one farm's weight run away. Same estimand, same assumptions as IPW — far steadier in hand. --- ## 3. Doubly-robust / Augmented IPW: two shots at the truth ### 3.1 The setup: two models, two ways to be right So far we have leaned entirely on a *model of treatment* — the propensity score, or the balancing weights, which is the same job in different clothes. There is a completely different route to the counterfactual that we have so far only mentioned: a **model of the outcome**. Fit a regression of $Y$ on $\mathbf{X}$ separately in the treated and control groups, getting predicted outcomes $\hat\mu_1(\mathbf{X}) = \hat{\mathbb{E}}[Y \mid \mathbf{X}, D=1]$ and $\hat\mu_0(\mathbf{X}) = \hat{\mathbb{E}}[Y \mid \mathbf{X}, D=0]$. Then impute each unit's missing counterfactual with the model's prediction and average the difference: $\hat\tau_{\text{reg}} = \frac{1}{N}\sum_i [\hat\mu_1(\mathbf{X}_i) - \hat\mu_0(\mathbf{X}_i)]$. This is **regression imputation** (sometimes "g-computation"), and under CIA it is also consistent — *if the outcome model is correctly specified*. So we have two strategies, each consistent under CIA but each leaning on a *different* modeling assumption: - **IPW / weighting** is consistent if the **propensity model** $\hat e(\mathbf{X})$ is correct. It does not care whether the outcome is linear in $\mathbf{X}$. - **Regression imputation** is consistent if the **outcome model** $\hat\mu_d(\mathbf{X})$ is correct. It does not care whether you got the propensity score right. In real data you are never *sure* either model is correct. Wouldn't it be nice to hedge — to use both, in a way that is forgiving if one of them is wrong? ### 3.2 The estimator, and why "doubly" That hedge exists. It is the **augmented inverse-probability-weighted (AIPW)** estimator, also called **doubly robust**, developed by Robins, Rotnitzky, and Zhao (1994).[^rrz] For the ATE it is, per arm, $$ \hat\mu_1^{\text{DR}} = \frac{1}{N}\sum_{i=1}^N \left[ \hat\mu_1(\mathbf{X}_i) + \frac{D_i\,\big(Y_i - \hat\mu_1(\mathbf{X}_i)\big)}{\hat e(\mathbf{X}_i)} \right], $$ with the symmetric formula for $\hat\mu_0^{\text{DR}}$ (replace $D_i$ with $1-D_i$, $\hat\mu_1$ with $\hat\mu_0$, and $\hat e$ with $1-\hat e$), and $\hat\tau_{\text{DR}} = \hat\mu_1^{\text{DR}} - \hat\mu_0^{\text{DR}}$. [^rrz]: Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). Estimation of Regression Coefficients When Some Regressors Are Not Always Observed. *Journal of the American Statistical Association*, 89(427), 846–866. The doubly-robust / AIPW form is developed and extended across this line of work by Robins and Rotnitzky and coauthors through the 1990s. Read the bracket as a sum of two pieces. The first piece, $\hat\mu_1(\mathbf{X}_i)$, is the **outcome-regression prediction** — the regression-imputation estimate. The second piece is an **IPW correction applied to the regression's residuals**: it takes the part the outcome model got *wrong* on the treated units, $Y_i - \hat\mu_1(\mathbf{X}_i)$, and reweights it by $1/\hat e(\mathbf{X}_i)$. So AIPW starts from the outcome model's best guess and then patches it using inverse-probability-weighted residuals. The word **augmented** is literal: it is IPW *augmented* with an outcome model, or equivalently the outcome model *augmented* with an IPW correction. Now the "doubly" — the surprising part that earns the method its fame. **The estimator is consistent if EITHER the propensity model OR the outcome model is correct; you do not need both.** You get two independent shots at the truth, and you only have to hit with one. Here is why, informally, by checking the two cases: *Case 1 — the outcome model is right, the propensity model is wrong.* Then $\hat\mu_1(\mathbf{X}_i)$ already estimates $\mathbb{E}[Y(1) \mid \mathbf{X}_i]$ correctly. The correction term has expectation zero because, with a correct outcome model, the residual $Y_i - \hat\mu_1(\mathbf{X}_i)$ averages to zero *within each value of* $\mathbf{X}$ — so no matter what (possibly wrong) weights $1/\hat e$ you multiply those residuals by, they still average out to zero. The wrong propensity score is multiplied into something that vanishes. The first term carries the day. *Case 2 — the propensity model is right, the outcome model is wrong.* Then $\hat\mu_1$ is some incorrect prediction. But the second term, with the *correct* $e(\mathbf{X})$, is exactly the IPW estimator applied to $Y_i$ — which we proved in §1.2 is consistent — *minus* an IPW-weighted term in $\hat\mu_1$ that, with correct propensity scores, precisely cancels the wrong $\hat\mu_1$ sitting in the first piece. The bad outcome model gets added in by the first term and *subtracted back out* by the second, leaving the consistent IPW estimate standing. The IPW machinery carries the day. Either way, one correct model is enough; the other model's errors get neutralized. That is the doubly-robust property, and it is genuinely remarkable: a kind of statistical insurance policy where you only need one of two claims to be valid. ### 3.3 Why this matters beyond the safety net Doubly robust is not just defensive. It also tends to be *more efficient* — lower variance — than plain IPW when both models are even roughly right, because the outcome model soaks up much of the variation in $Y$ before the noisy inverse-probability weights ever touch it. The weights now act on *residuals* (small, after a decent regression) rather than on the raw outcomes (large), so the weight-explosion damage is muffled even when some $\hat e$ are small. AIPW does not fully cure the extreme-weight problem — a $\hat e$ near zero still hurts — but it leans on those weights far less. There is a deeper modern reason AIPW is everywhere now. Because the estimator has a special "insensitivity" to small errors in *each* model (a property called **Neyman orthogonality**), you can plug **machine-learning estimators** — random forests, gradient boosting, neural nets — into both the propensity model and the outcome model, use sample-splitting (fit the models on one half, evaluate on the other) to avoid overfitting, and still get valid standard errors and confidence intervals for the treatment effect. This is the engine behind **double/debiased machine learning** (Chernozhukov et al., 2018),[^dml] the dominant framework for high-dimensional causal estimation in current empirical economics and finance. You will not implement that this week, but it is worth knowing that the humble bracket above is the seed of the field's most active methodology. AIPW is where selection-on-observables grows up. [^dml]: Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). Double/Debiased Machine Learning for Treatment and Structural Parameters. *The Econometrics Journal*, 21(1), C1–C68. --- ## 4. The code One runnable script ties the three methods together on a simulated dataset where we *know* the true effect, so you can watch which estimators land near it and which wander. We build Maya's financial-literacy setup: enrollment depends on income and credit (so there is selection), and the true effect of enrolling is a $-0.8$ percentage-point reduction in the rate. ```python import numpy as np import pandas as pd import statsmodels.api as sm import statsmodels.formula.api as smf rng = np.random.default_rng(33) N = 4000 # --- covariates --- income = rng.normal(0, 1, N) # standardized household income credit = rng.normal(0, 1, N) # standardized starting credit score # --- selection into treatment (the propensity TRUTH) --- lin = 0.9 * income + 0.7 * credit p_true = 1 / (1 + np.exp(-lin)) # true propensity score D = (rng.uniform(size=N) < p_true).astype(int) # --- outcomes: TRUE treatment effect is -0.8; outcome depends on X too --- tau_true = -0.8 y0 = 9.0 - 0.6 * income - 0.5 * credit + rng.normal(0, 1, N) # rate if NOT enrolled y1 = y0 + tau_true # rate if enrolled Y = np.where(D == 1, y1, y0) # we observe only one potential outcome each df = pd.DataFrame({"Y": Y, "D": D, "income": income, "credit": credit}) # --- (0) naive difference in means: biased by selection --- naive = df.loc[df.D == 1, "Y"].mean() - df.loc[df.D == 0, "Y"].mean() # --- estimate the propensity score (a logit), then build stabilized IPW weights --- ps = smf.logit("D ~ income + credit", data=df).fit(disp=0) ehat = ps.predict(df).clip(0.01, 0.99) # clip = trimming extreme scores p_marg = df.D.mean() w = np.where(df.D == 1, p_marg / ehat, # stabilized treated weight (1 - p_marg) / (1 - ehat)) # stabilized control weight # --- (1) IPW (Hajek): self-normalizing weighted means per arm --- def hajek_mean(mask): ww = w[mask] return np.sum(ww * df.Y[mask]) / np.sum(ww) ipw = hajek_mean(df.D == 1) - hajek_mean(df.D == 0) # --- (2) Augmented IPW / doubly robust --- m1 = smf.ols("Y ~ income + credit", data=df[df.D == 1]).fit() # outcome model, treated m0 = smf.ols("Y ~ income + credit", data=df[df.D == 0]).fit() # outcome model, control mu1 = m1.predict(df); mu0 = m0.predict(df) dr1 = np.mean(mu1 + df.D * (df.Y - mu1) / ehat) dr0 = np.mean(mu0 + (1 - df.D) * (df.Y - mu0) / (1 - ehat)) aipw = dr1 - dr0 print(f"true effect : {tau_true:+.3f}") print(f"naive diff : {naive:+.3f} (biased by selection)") print(f"IPW (Hajek) : {ipw:+.3f}") print(f"AIPW (doubly rob): {aipw:+.3f}") print(f"weights: max={w.max():.1f} eff. N={w.sum()**2/np.sum(w**2):.0f} of {N}") ``` Run it and the naive difference comes out badly biased toward zero (selection makes enrollees and non-enrollees differ in the outcome for reasons that have nothing to do with the program), while both IPW and AIPW land close to the true $-0.8$. The printed weight diagnostics — the max weight and the effective sample size — are the habit to internalize: *always look at your weights.* For the entropy-balancing comparison, the `EBweights`/`ebal` route (or a short convex solve) is the subject of nb3.3, where you will see entropy weights deliver exact balance with a smaller max weight than IPW on the very same data. To *feel* the double-robustness property, change the outcome models `m1`/`m0` to deliberately wrong specifications (drop `credit`, or add a spurious term) and watch AIPW stay near $-0.8$ because the propensity model is still right; then instead corrupt the propensity formula (drop `credit` from the logit) while keeping the outcome models correct, and watch AIPW *still* stay near $-0.8$. Break *both* at once and it finally drifts. That experiment — one model wrong at a time is survivable, both wrong is not — is the whole "doubly" in one screen. --- ## 5. A practical decision guide: PSM vs. entropy balancing vs. AIPW You now have, across Ch 3.2 and Ch 3.3, four ways to exploit selection-on-observables. Here is how to choose, stated as the kind of judgment you should be able to defend to a referee. **Reach for propensity-score matching (Ch 3.2) when** you want a transparent, intuitive design and your treated and control groups overlap well. Matching's appeal is that it *mimics an experiment* unit by unit and is easy to explain to a non-technical audience ("we compared each enrollee to a nearly identical non-enrollee"). Its costs are the ones Maya hit: it discards unmatched units, it forces a menu of arbitrary choices (caliper, with/without replacement, number of neighbors), and it only achieves *approximate* balance that you must check after the fact. Good for exposition and for a first look; rarely your final, most efficient estimator. **Reach for entropy balancing when** you want guaranteed exact balance on chosen moments, maximum stability, and minimal specification angst — which in practice is *most of the time* in empirical finance. It keeps every unit, produces tame weights by construction, makes your balance targets explicit and inspectable, and is hard for a referee to accuse of cherry-picking because there is no algorithm-and-caliper menu to fish through. Its limitation is that you must *choose* which moments to balance, and it balances only those — if the true selection depends on a high-order interaction you did not include, you have not balanced it. State your moments and defend them. **Reach for AIPW / doubly-robust when** you want the strongest consistency guarantee and the best efficiency, and especially when you are willing to bring an outcome model and a treatment model and let them cover for each other. This is the right default for a *headline* causal estimate you intend to publish: it is consistent if either model is right, it is more efficient than plain IPW, and it is the on-ramp to machine-learning-based estimation if your covariate set is large. Its cost is conceptual overhead (two models, not one) and that it still suffers — though less — from extreme weights. The honest meta-advice: **report more than one.** If PSM, entropy balancing, and AIPW all point to roughly $-0.8$, your finding is robust to method choice and you can write it up with confidence. If they disagree, that disagreement is *information* — almost always a sign of thin overlap or a fragile specification — and the disagreement itself belongs in your paper, not swept under the rug of whichever method gave the prettiest number. This is the empirical-spec discipline from the Conventions applied to method choice: name the estimand (ATE or ATT), name the covariates you balanced on, name the overlap region you restricted to, and name the assumption — CIA — that the whole exercise rides on. --- ## 6. The ceiling on everything in this chapter — and the door out Stand back and see the boundary of what Week 3 has built so far. IPW, entropy balancing, AIPW, and the matching of Ch 3.2 are *four different engines bolted to the same chassis*, and that chassis is the **conditional-independence assumption**: that once we condition on (or balance, or reweight on) the *observed* covariates $\mathbf{X}$, treatment is as good as random. Every method in this chapter is a way to use $\mathbf{X}$ more cleverly, more stably, more efficiently. Not one of them can rescue you if CIA is false. And CIA is false exactly when there is an **unobserved confounder** — a variable that drives both selection into treatment *and* the outcome, that you did not measure and therefore cannot put in $\mathbf{X}$. Suppose the students who enrolled in Maya's program were also, on average, more *conscientious* in a way no dataset captures, and conscientiousness independently lowers interest rates. Then enrollees would have gotten lower rates even without the program, no amount of balancing on income and credit removes it, and *every estimator in this chapter is biased* — silently, with a clean-looking balance table and a confident standard error. Balancing the observables you have does nothing about the confounder you don't. This is the unbreakable ceiling of selection-on-observables, and it is why a careful researcher always asks, "what's the unobserved thing that could be driving both?" That question is the doorway to Ch 3.4. When you cannot defend CIA — when you fear an unobserved confounder you cannot measure — you need a fundamentally different source of identifying variation: not a richer set of controls, but an **instrument**, a variable that shoves units into treatment for reasons unrelated to the outcome. Ch 3.4 introduces **instrumental variables** as the escape hatch for precisely the situation this chapter cannot handle. Everything in Week 3 up to now has assumed selection-on-observables; the rest of the week is about what to do when that assumption finally breaks. --- ## Your Turn Open **nb3.3 — entropy balancing vs. IPW vs. AIPW.** You will take a single simulated dataset with a known treatment effect and selection on observables, and run all three estimators side by side: build stabilized IPW (Hájek) weights and inspect their distribution and effective sample size; solve for entropy-balancing weights that exactly match covariate means (then means *and* variances) and confirm the balance table comes out mechanically perfect; and compute the AIPW estimate, then deliberately break the propensity model, then instead break the outcome model, and watch the doubly-robust estimator survive each single failure but not their conjunction. The goal is to *see*, with your own data, why entropy weights stay tame where IPW weights explode, and why "doubly robust" is more than a slogan. **Check questions.** 1. A treated unit in Maya's data has an estimated propensity score of $\hat e = 0.04$. What is its (unstabilized) IPW weight, and why should this single number make you nervous about the IPW estimate? Name two distinct things you could do about it. 2. Priya runs AIPW on her climate-insurance data. A colleague says, "your outcome regression is surely misspecified — irrigation spending isn't linear in rainfall — so your estimate is garbage." Explain why this objection is *not necessarily* fatal, and state precisely the condition under which Priya's AIPW estimate is still consistent despite the wrong outcome model. 3. Maya's entropy-balanced ATT and her IPW ATT agree (both about $-0.7$), but her colleague Sam warns that *both* could still be badly biased. Describe the one kind of variable whose existence would make that warning correct, explain why no method in this chapter can fix it, and name the chapter (and tool) that addresses it.