propensity score | Chen Xing

Balancing Weights for Causal Inference

Wed, 22 Oct 2025 00:00:00 +0000

TL;DR

Cohn et al. (2023) introduces the balancing approach to weighting for causal inference in observational studies. Unlike traditional methods that model the propensity score directly, balancing weights are estimated by solving an optimization problem that directly targets covariate balance between treatment groups. The authors demonstrate that this approach offers protection against model misspecification, connects naturally to bias-variance trade-offs, and can be augmented with outcome modeling for improved performance. Applied to the classic LaLonde job training data, balancing methods achieve better covariate balance than standard propensity score approaches while maintaining reasonable effective sample sizes.

What is this paper about?

Covariate balance is fundamental to causal inference: randomized experiments achieve it by design, while observational studies must adjust for it. This chapter addresses a key challenge in observational causal inference—how to construct weights that remove confounding by balancing observed covariates between treated and control groups. The traditional modeling approach estimates propensity scores (the probability of treatment given covariates) and inverts them to create weights, but this relies heavily on correct model specification. When the propensity score model is wrong, the resulting weights may fail to balance covariates in the sample, leading to biased treatment effect estimates. The chapter explores an alternative: directly finding weights that achieve balance in the observed data, rather than first modeling the propensity score.

What do the authors do?

The authors formalize the balancing approach as an optimization problem that jointly minimizes covariate imbalance and weight dispersion (variance). They show how different choices of the “model class” M—the set of functions of covariates to balance—correspond to different assumptions about the outcome model and lead to different optimization formulations. Using the LaLonde dataset (a constructed observational study where the true treatment effect is known), they compare three designs: balancing main covariate terms only, balancing up to three-way interactions, and balancing an infinite-dimensional reproducing kernel Hilbert space (RKHS). For each design, they evaluate covariate balance using standardized mean differences, examine the effective sample size (a measure of weight dispersion), and explore the bias-variance trade-off by varying regularization parameters. The authors also demonstrate how balancing weights can be augmented with outcome regression to further reduce bias, and they establish asymptotic normality results for inference.

Why is this important?

This work matters because most observational studies include covariates in their analysis, yet practitioners often don’t carefully consider whether their weighting method actually achieves balance on the relevant covariate functions. The balancing approach makes covariate balance a first-order design criterion rather than a post-hoc diagnostic check. It reveals the implicit bias-variance trade-offs in weighting methods and shows that different assumptions about the outcome model (linear, interactive, nonparametric) lead to fundamentally different weighting strategies. The framework unifies many existing methods (entropy balancing, stable weights, kernel balancing) under one optimization structure and clarifies the connection between the balancing and modeling approaches through Lagrangian duality. Importantly, the chapter provides practical guidance on design choices—what to balance, how much dispersion to tolerate, whether to allow negative weights—that applied researchers face but often lack principled ways to resolve.

Who should care?

Applied researchers in economics, epidemiology, public policy, education, and medicine who use inverse propensity weighting or other covariate adjustment methods in observational studies. Methodologists working on causal inference, especially those developing new weighting estimators or studying properties of existing ones. Graduate students learning causal inference who need to understand the trade-offs between different adjustment strategies and the implicit assumptions behind common practices. Policy evaluators who must justify their modeling choices and demonstrate that their treatment effect estimates are robust to covariate imbalance. Anyone who has struggled with poor covariate balance after propensity score weighting or wondered how to choose between competing adjustment methods would benefit from this framework.

Do we have code?

The chapter does not provide standalone replication code or software packages. However, the authors note that many of the specific balancing methods discussed are available in existing R packages: entropy balancing in the ebal package, covariate balancing propensity scores (CBPS) in the CBPS package, and stable balancing weights in the sbw package. The kernel balancing approach can be implemented using standard kernel methods in R or Python. The LaLonde dataset used throughout the examples is publicly available and widely used in the causal inference literature, making it straightforward to reproduce the analyses with these existing tools.

In summary, this chapter reframes propensity score weighting as a balance-optimization problem rather than a pure modeling exercise. By directly targeting the balancing property of inverse propensity weights, the approach offers robustness to propensity score misspecification while making explicit the bias-variance trade-offs inherent in any covariate adjustment. The LaLonde application demonstrates that balancing weights can substantially reduce covariate imbalance compared to standard methods, though at the cost of reduced effective sample size. The framework provides both theoretical insight (connecting balancing to dual regression, establishing asymptotic properties) and practical guidance (how to choose what to balance, when to augment with outcome modeling, whether to allow extrapolation) that fills an important gap in applied causal inference.

Reference

Cohn, Eric R., Eli Ben-Michael, Avi Feller, and José R. Zubizarreta (2023), “Balancing Weights for Causal Inference,” in Handbook of Matching and Weighting Adjustments for Causal Inference, Chapman and Hall/CRC. https://www.taylorfrancis.com/chapters/edit/10.1201/9781003102670-16/balancing-weights-causal-inference-eric-cohn-eli-ben-michael-avi-feller-josé-zubizarreta

Notes on Propensity Score Methods

Tue, 03 Jun 2025 00:00:00 +0000

Introduction

Here are my notes on propensity scores, mainly from Prof. Ding’s textbook (2024).

The traditional propensity score analysis workflow is shown in the image below, which I will not cover in detail. Instead, I will summarize the key theorems and results from Ding’s textbook.

Figure 1: Traditional propensity score analysis workflow

I will also provide some connections with Riesz Representer (RR).

Why connect with the Riesz Representer (RR)? The connection provides a powerful generalization of the foundational Rosenbaum-Rubin (1983) result.
Rosenbaum and Rubin showed that conditioning on the propensity score is sufficient for removing confounding bias when estimating causal effects. The Riesz representer extends this principle: it suffices to regress on the Riesz representer to obtain unbiased estimates of the average treatment effect.
The key insight is that the Riesz representer, like the propensity score, serves as a sufficient statistic – it captures all the confounding information necessary for unbiased estimation of your target causal parameter.

Setting & Notation

Binary treatment $Z$
Potential outcomes $\{Y(0), Y(1)\}$
Propensity score: $\P(Z = 1 \mid X)$, where $X$ represents covariates

Two approaches learning causal relationships:

Outcome process (via outcome regression)
Treatment assignment mechanism (via propensity score)

The following summarizes the key theorems and results related to propensity scores from Prof. Ding’s textbook.

1. The propensity score as a dimension reduction tool

Theorem 1 (pscore as dimension reduction tool).

$$ \text { If } Z \indep \{Y(1), Y(0)\} \mid X, \text { then } Z \indep \{Y(1), Y(0)\} \mid e(X) . $$

Covariates $X$ can be high dimensional, but the propensity score, $e(X) \in \R$, is a 1-dimensional scalar
We can view the propensity score as a dimensional reduction tool

2. Propensity score stratification

Idea: Discretize the estimated propensity score by its $K$ quantiles:
$$ Z \indep \{Y(1), Y(0)\} \mid \hat{e}^{\prime}(X)=e_k \quad(k=1, \ldots, K) . $$
Estimate ATE within each subclass and then average by the block size
Advantage: The propensity score stratification estimator only requires the correct ordering of the estimated propensity scores rather than their exact values, which makes it relatively robust compared with other methods

3. Propensity score weighting

Theorem 2 (Invese propensity score weighting (IPW)).

If $Z \indep \{Y(1), Y(0)\} \mid X$ and $ 0 < e(X) < 1$, then $$E\{Y(1)\}=E\left\{\frac{Z Y}{e(X)}\right\}, \quad E\{Y(0)\}=E\left\{\frac{(1-Z) Y}{1-e(X)}\right\}$$ and $$ \begin{aligned} \tau &=E\{Y(1)-Y(0)\}\\ &=E\left\{\frac{Z Y}{e(X)}-\frac{(1-Z) Y}{1-e(X)}\right\} \\ &=E\left\{HY \right\} \end{aligned} $$ where $H := \left[\frac{Z}{e(X)}-\frac{(1-Z) }{1-e(X)}\right]$ is called the Horvitz-Thompson transform.

Connection the Riesz Representer (RR)

Remark 1 (RR in the case of ATE).

In the case of ATE, the Riesz Representer, $\alpha(Z, X)$, has the same form as above Horvitz-Thompson transform, $$ \alpha(Z, X) = \left[\frac{Z}{e(X)}-\frac{(1-Z) }{1-e(X)}\right] $$

3.1 Estimation

The sample version of IPW is called the Horvitz–Thompson (HT) estimator,
$$ \hat{\tau}^{\mathrm{ht}}=\frac{1}{n} \sum_{i=1}^n \frac{Z_i Y_i}{\hat{e}\left(X_i\right)}-\frac{1}{n} \sum_{i=1}^n \frac{\left(1-Z_i\right) Y_i}{1-\hat{e}\left(X_i\right)} $$
HT estimator $\hat{\tau}^{\mathrm{ht}}$ has many problems
Problem: lack of invariance, i.e. if we replace $Y_i$ by $Y_i + c$, $\hat{\tau}^{\mathrm{ht}}$ changed because it depends on $c$. This is not reasonable.
Solution: normalizing the weights
$$ \hat{\tau}^{\text {hajek }}=\frac{\sum_{i=1}^n \frac{Z_i Y_i}{\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{Z_i}{\hat{e}\left(X_i\right)}}-\frac{\sum_{i=1}^n \frac{\left(1-Z_i\right) Y_i}{1-\hat{e}\left(X_i\right)}}{\sum_{i=1}^n \frac{1-Z_i}{1-\hat{e}\left(X_i\right)}} . $$
Hajek estimator is invariant to the location transformation

3.2 Strong overlap condition

Many asymptotic analyses require a strong overlap condition,

$$ 0<\alpha_{\mathrm{L}} \leq e(X) \leq \alpha_{\mathrm{U}}<1 $$ In practice,

Crump et al. (2009) suggested $α_L = 0.1$ and $α_U = 0.9$
Kurth et al. (2005) suggested $α_L = 0.05$ and $α_U = 0.95$

4. Balancing property

Theorem 3 (balancing property).

Conditional on $e(X)$, the treatment and the covariates are independent
Within the same level of the propensity score, the covariate distributions are balanced across the treatment and control groups
Useful implication: we can check whether the propensity score model is specified well enough to ensure the covariate balance in the data

4.1 Propensity score is a balancing score

Definition 1.

Theorem 4 (Propensity score is a balancing score).

This is relevant in subgroup analysis
The conditional independence in (11.5) ensures unconfoundedness holds given the propensity score, within each level of $X_1$. Therefore, we can perform the same analysis based on the propensity score, within each level of $X_1$, yielding estimates for two subgroup effects

5. Doubly Robust or AIPW

The following Theorem is summarized from Prof. Wager’s lecture notes (2024).

Theorem 5 (strong double robustness of AIPW estimator).

Define the outcome regression as $$ \mu_{(z)}(x)=\mathbb{E}\left[Y_i(z) \mid X_i=x\right], $$ Define AIPW estimator as $$ \begin{aligned} \hat{\tau}_{A I P W} & =\underbrace{\frac{1}{n} \sum_{i=1}\left(\hat{\mu}_{(1)}\left(X_i\right)-\hat{\mu}_{(0)}\left(X_i\right)\right)}_{\text { outcome regression estimator }} \\ & +\underbrace{\frac{1}{n} \sum_{i=1}^n\left(\frac{Z_i}{\hat{e}\left(X_i\right)}\left(Y_i-\hat{\mu}_{(1)}\left(X_i\right)\right)-\frac{1-Z_i}{1-\hat{e}\left(X_i\right)}\left(Y_i-\hat{\mu}_{(0)}\left(X_i\right)\right)\right)}_{\text { applying IPW to the regression residuals }} \end{aligned} $$ If we use estimators $\hat{\mu}_{(z)}(x)$ and $\hat{e}(x)$ that are both consistent with root-mean squared error (RMSE) decaying faster than $n^{-\alpha_\mu}$ and $n^{-\alpha_e}$ respectively, and if furthermore $\alpha_\mu+\alpha_e \geq 1 / 2$, then $$ \begin{aligned} & \sqrt{n}\left(\hat{\tau}_{A I P W}-\tau\right) \Rightarrow \mathcal{N}\left(0, V_{A I P W}\right) \\ & V_{A I P W}=\operatorname{Var}\left[\tau\left(X_i\right)\right]+\mathbb{E}\left[\frac{\sigma_0^2\left(X_i\right)}{1-e\left(X_i\right)}\right]+\mathbb{E}\left[\frac{\sigma_1^2\left(X_i\right)}{e\left(X_i\right)}\right] \end{aligned} $$ where $$ \sigma_{(z)}^2(x)=\operatorname{Var}\left[Y_i(z) \mid X_i=x\right] $$

Check my previous post: Intuition for Doubly Robust Estimator
AIPW provides a natural starting point for understanding Double Machine Learning
Key insight of RR in DML framework: Leverage the Riesz Representer, a “generalized version of propensity score” to “correct the bias”

More general, Li et al. (2018a) gave a unified discussion of the causal estimands in observational studies.

Theorem 6 (Ding (2024), Section 13.4).

Summary Table of common estimands:

This table provides us a good way to understand and remember IPW estimator for ATT
How to remember $\tau^h$? Apply IPW on “pseudo outcome” $Yh(X)$ then divide by $E(h(X))$
When the parameter of interest is ATT, then $$E(h(X)) = E(e(X)) = E(E(Z \mid X)) = E(Z) = \P(Z = 1) = e$$
Use it to better understand IPW for ATT

7. Propensity Score in Regression

PS as a covariate

Theorem 7 (regression with pscore as a covariate).

Under unconfoundedness, the coefficient of $Z$ in the population OLS fit of $$ Y \sim 1+Z + e(X) $$ equals $\tau_{\mathrm{O}}$, $$ \tau_{\mathrm{O}}=\frac{E[e(X)\{1-e(X)\} \tau(X)]}{E[e(X)\{1-e(X)\}]} ,$$ which is the overlap-weighted average treatment effect

Based on above Theorem, we also have:

Corollary 1.

Under unconfoundedness,
1. the coefficient of $Z$ in the population OLS fit of $$ Y \sim 1+Z + e(X) + X $$ also equals $\tau_{\mathrm{O}}$,

2. the coefficient of $Z-e(X)$ in the population OLS fit of $$ Y \sim [Z - e(X)] \quad \text{or} \quad Y \sim 1 + [Z - e(X)] $$ also equals $\tau_{\mathrm{O}}$

PS as a weight

There is a convenient way to obtain $\hat{\tau}^{\text{hajek}}$ based on WLS.

Proposition 1 (convenient to obtain $\hat{\tau}^{\text{hajek}}$ based on WLS).

Need to use bootstrap for standard error
Why does the WLS give a consistent estimator for $\tau$ ?
In RCT with a constant propensity score, we can simply use the coefficient of $Z_i$ in the OLS fit of $Y_i$ on ( $1, Z_i$ ) to estimate $\tau$
In observational studies, we need to deal with the selection bias. The key idea is:
- If we weight the treated units by $\frac{1}{e(X_i)}$ and the control units by $\frac{1}{1-e(X_i)}$, then both treated and control groups can represent the whole population
- Thus, by weighting, we effectively have a pseudo-randomized experiment
- Remark 2 (IPCW).
  
  Inverse Probability of Censoring Weighting (IPCW) follows the same idea — it adjusts for censoring bias by reweighting observations based on their probability of being uncensored.
Consequently, the difference between the weighted means is consistent for $\tau$. The numerical equivalence of $\hat{\tau}^{\text {hajek }}$ and WLS is not only a fun numerical fact itself but also useful for motivating more complex estimators with covariate adjustment

Reference

Ding, Peng (2024), A First Course in Causal Inference, CRC Press.

Wager, S. (2024). Causal inference: A statistical learning approach. https://web.stanford.edu/~swager/causal_inf_book.pdf

Notes on causal inference with No Overlap – Regression Discontinuity

Sat, 31 May 2025 00:00:00 +0000

Here is my notes on regression discontinuity from Prof. Ding’s textbook (2024) and Prof. Imai’s lecture notes.

Motivation

We often cannot run a randomized experiment and have to use/design observational studies to find a setting where credible causal inference is possible.

The key is the knowledge of treatment assignment mechanism.

Regression discontinuity design (RD Design):

RD Design is a simple and widely used quasi-experimental design. The term “quasi experimental” is to emphasize that these approaches are still framed using concepts from randomized experiments but require econometric innovations to compensate for the lack of random treatment assignment.

Sharp RD Design: treatment assignment is based on a deterministic rule (i.e. we have full knowledge of how treatment is assigned)
Fuzzy RD Design: encouragement to receive treatment is based on a deterministic rule

Setting

Binary treatment $Z\in \{0,1\}$
Potential outcomes $\{Y(0), Y(1)\}$
There is a running variable $X \in \R$ such that $Z=I\left(X \geq x_0\right)$, where $x_0$ is a pre-determined threshold. Note that, the treatment assignment is deterministic
The unconfoundedness assumption holds automatically $$ Z \indep \{Y(1), Y(0)\} \mid X $$
The overlap assumption does not hold

$$ e(X)=\operatorname{pr}(Z=1 \mid X)=1\left(X \geq x_0\right) = \text{1 or 0} $$

Identification

RD can identify a local average causal effect at the cutoff point $x_0$
Estimand:

$$ \tau\left(x_0\right)=E\left\{Y(1)-Y(0) \mid X=x_0\right\} . $$

Assumption 1 (continuity assumption).

1. $E\{Y(1) \mid X=x\}$ is continuous from the right at $x_0$
2. $E\{Y(0) \mid X=x\}$ is continuous from the left at $x_0$

We have $$ \begin{align} E\left\{Y(1) \mid X=x_0\right\} & =\lim _{\varepsilon \rightarrow 0+} E\left\{Y(1) \mid X=x_0+\varepsilon\right\} \tag{continuity} \\ & =\lim _{\varepsilon \rightarrow 0+} E\left\{Y(1) \mid Z=1, X=x_0+\varepsilon\right\} \tag{def of Z}\\ & =\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=1, X=x_0+\varepsilon\right), \end{align} $$ Similarly, $$ E\left\{Y(0) \mid X=x_0\right\}=\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=0, X=x_0-\varepsilon\right) $$ So the local average causal effect at $x_0$ can be identified by the difference of the two limits
Advantage: internal validity
Disadvantage: external validity

Key Theorem

Theorem 1.

Assume that the treatment is determined by $Z=I\left(X \geq x_0\right)$ where $x_0$ is a predetermined threshold. Assume that $E\{Y(1) \mid X=x\}$ is continuous from the right at $x_0$ and $E\{Y(0) \mid X=x\}$ is continuous from the left at $x_0$. Then the local average treatment effect at $X=x_0$ is identified by $$ \tau\left(x_0\right)=\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=1, X=x_0+\varepsilon\right)-\lim _{\varepsilon \rightarrow 0+} E\left(Y \mid Z=0, X=x_0-\varepsilon\right) $$

$\tau\left(x_0\right)$ is nonparametrically identified.

Reference

Ding, P. (2024). A First Course in Causal Inference. CRC Press.

Lecture notes: Regression Discontinuity Design

Notes on Callaway & Sant’Anna (2021) – Staggered Adoption DiD

Tue, 27 May 2025 00:00:00 +0000

0. Motivation

Staggered‐adoption policies break the canonical two-period / two-group DiD model. It has been shown that the traditional two-way fixed-effects (TWFE) regression can assign negative weights to treatment effects, thereby obscuring their dynamic and heterogeneous patterns.

Callaway & Sant’Anna (2021) propose a divide-and-conquer strategy:

Divide the messy staggered panel into many honest $2\times2$ DiDs
Conquer by estimating each “little” DiD under familiar assumptions, then combine them with user-chosen weights to answer specific questions

The key takeaway:

Use $\operatorname{ATT}(g, t)$ as a building block so we can transparently see how things are constructed
Many different aggregation schemes are possible: they deliver different parameters
Can allow for covariates via regressions adjustments, IPW, and DR.

1. Setup

Data structure: Panel of units $i$ over time $t = 1,\dots,T$
Let $D_{it}$ be a binary variable. $D_{it} = 1$ if unit $i$ is treated in period $t$; $D_{it} = 0$ otherwise
Cohorts
- Define $G$ as the time period when a unit first becomes treated. For all units that eventually get treated, $G$ defines which “group” they belong to
- Define $G_{g}$ as a binary variable. $G_{g} = 1$ if a unit is first treated in period $g$ (i.e. $G_{i,g} = \1\{G_i = g\}$ )
- Define $G=\infty$ as “never treated” group.
Potential outcomes
- $Y_{it}(g)$: outcome at time $t$ if first treated in period $g$.
- $Y_{it}(\infty)$: outcome if never treated.
Cohort-time ATT

Assume $\mathrm{iid}$, drop unit index $i$. A parameter of interest that has clear interpretation is the $\operatorname{ATT}(g, t)$: $$ \operatorname{ATT}(g, t)=\mathbb{E}\left[Y_t(g)-Y_t(\infty) \mid G_g=1\right], \text { for } t \geq g . $$ This defines one “clean” $2\times2$ DiD for each pair $(g,t)$.

Remark 1.

Why focus on time after treatment starting period, $t \ge g$? Because we need the no anticipation assumption. Before treatment taking place, $t < g$, there is no treatment effect.

2. Key Assumptions

Given that we never observe $Y(\infty)$ in post-treatment periods among units that have been treated, we need to make assumptions to identify $\operatorname{ATT}(g, t)$.

Assumption 1 (No anticipation).

$$ Y_{i t}(g)=Y_{i t}\left(g^{\prime}\right) \quad \forall \ i \text{, } t<\min \left\{g, g^{\prime}\right\} $$ Treatment cannot affect pre-treatment outcomes.

The no anticipation assumption has exactly the same content as in the $2\times2$ case.

Assumption 2 (Parallel trends based on 'never-treated' control).

For each $t \in\{2, \ldots, T\}, g \in \mathcal{G}$ such that $\textcolor{red}{t \geq g}$, $$ \mathbb{E}\left[Y_t(\infty)-Y_{t-1}(\infty) \mid G_g=1\right]=\mathbb{E}\left[Y_t(\infty)-Y_{t-1}(\infty) \mid G=\infty\right], \tag{1} $$

Note that, (1) is equivalent to the following: for $t \in\{2, \ldots, T\}, g \in \mathcal{G}, t \geq g$ ,

$$ \mathbb{E}\left[Y_t(\infty)-Y_{g-1}(\infty) \mid G_g=1\right]=\mathbb{E}\left[Y_t(\infty)-Y_{g-1}(\infty) \mid G=\infty\right], \tag{1’} $$

Assumption 3 (Parallel trends based on 'Not-Yet-Treated' groups).

For each $(s, t) \in\{2, \ldots, T\} \times\{2, \ldots, T\}, g \in \mathcal{G}$ such that $\textcolor{red}{t \geq g, s \geq t}$ $$ \mathbb{E}\left[Y_t(\infty)-Y_{t-1}(\infty) \mid G_g=1\right]=\mathbb{E}\left[Y_t(\infty)-Y_{t-1}(\infty) \mid D_s=0, G_g=0\right], \tag{2} $$

Similarly, (2) is equivalent to (2’), that is, changing $Y_{t-1}(\infty)$ to $Y_{g-1(\infty)}$ in (2).

3. Identification: Long-Difference Estimands

Under no anticipation and one of the parallel-trends assumptions, each $\mathrm{ATT}(g,t)$ equals a simple long-difference DID:

Using never-treated

$$ \operatorname{ATT}^{\text {never}}(g, t)=\mathbb{E}\left[Y_{t}-Y_{g-1} \mid G_g=1 \right]-\mathbb{E}\left[Y_{ t}-Y_{g-1} \mid G=\infty\right] $$

Using not-yet-treated

$$ \operatorname{ATT}^{\text {not-yet}}(g, t)=\mathbb{E}\left[Y_{t}-Y_{g-1} \mid G_g=1 \right]-\mathbb{E}\left[Y_{ t}-Y_{g-1} \mid D_t = 0, G=\infty\right] $$

Why it’s called "long difference"? Longer Time Span!

Remark 2.

Q: Why it’s called "long difference"?

A: Longer Time Span! Check my plot above. The "long difference" refers to the fact that the comparison often spans from a pre-treatment period (i.e. $g-1$) to a later period (post-treatment, $t \ge g$), potentially covering multiple time periods. This contrasts with "short differences," which might involve comparing outcomes in consecutive periods or shorter time windows.

For each treated cohort, the method computes the difference in outcomes between the pre-treatment period and a specific post-treatment period, potentially far apart in time. This extended gap emphasizes the "long" aspect, as it captures the cumulative effect of the treatment over time.

Moreover, with covariates, one can form doubly-robust estimators that combine generalized propensity scores $p_g(X_i)$ and outcome models $m_{g,t}(X_i)$. At high level, the form of this estimator is identical to AIPW–ATT in cross–section, but we need to replace “levels” (e.g. $Y$) with “changes” (e.g. $\Delta Y$).

For more details, check Theorem 1 in the paper.

4. Aggregation of $\mathrm{ATT}(g,t)$

Any overall summary $\theta$ is a weighted average of the cell-specific ATTs:

$$ \theta=\sum_{g=2}^T \sum_{t=g}^T w_{g, t} \operatorname{ATT}(g, t), \quad \sum_{g, t} w_{g, t}=1 . $$

Common choices:

Cohort-heterogeneity: Average effect of participating in the treatment that units in group $g$ experienced,

$$ \theta_S(g)=\frac{1}{T-g+1} \sum_{t=2}^T 1\{g \leq t\} \mathrm{ATT}(g,t) $$

Calendar time heterogeneity
Event-study / dynamic treatment effects

5. Limitation & Extension

Lee & Wooldridge (2023) argue that Callaway & Sant’Anna (2021) method is less efficient but more resilient to functional form of covariates. The following is from their working paper:

… CS (2021) method uses only the period just prior to the intervention in defining the control group, thereby discarding potentially useful information in earlier time periods.

In fact, Wooldridge (2021) shows that, under the standard “error components” structure on the error, with a homoskedastic time-constant component and homoskedastic and serially uncorrelated idiosyncratic errors, the POLS estimator is both best linear unbiased (BLUE) and asymptotically efficient. These theoretical results imply that the CS (2021) estimators are inefficient under a standard set of assumptions. The simulations in Wooldridge (2021) bear this out, showing the CS approach can be very inefficient. Balanced against the loss in precision is that the CS approach can be less biased when parallel trends are violated.

To improve the efficiency, instead of using long differences, Lee and Wooldridge (2023) use all suitable control observations in transforming the outcome variable. Specifically, rather than using the single period just prior to the treatment, $Y_{g-1}$, they use the pre-treatment average, $\frac{1}{g-1}\sum_{s = 1}^{g-1}Y_s$.

The rolling method (Lee and Wooldridge, 2023) transforms panel data with staggered interventions by subtracting each unit’s average outcome across all pre-treatment periods from their outcome in the current period of interest. This transformation, combined with no anticipation and parallel trends assumptions, makes the treatment assignment unconfounded for the transformed outcome in each cohort/time cross-section. With unconfoundedness holding, we can then apply standard treatment effects estimators, including doubly robust methods and matching, utilizing all not-yet-treated units as the valid control group for that specific cross-section.

6. R code Example

The following R code example is provided by Professor Scott Cunningham.

library(readstata13)
library(ggplot2)
library(did) # Callaway & Sant'Anna

castle <- data.frame(read.dta13('https://github.com/scunning1975/mixtape/raw/master/castle.dta'))
castle$effyear[is.na(castle$effyear)] <- 0 # untreated units have effective year of 0

# Estimating the effect on log(homicide)
atts <- att_gt(yname = "l_homicide", # LHS variable
               tname = "year", # time variable
               idname = "sid", # id variable
               gname = "effyear", # first treatment period variable
               data = castle, # data
               xformla = NULL, # no covariates
               #xformla = ~ l_police, # with covariates
               est_method = "dr", # "dr" is doubly robust. "ipw" is inverse probability weighting. "reg" is regression
               control_group = "nevertreated", # set the comparison group which is either "nevertreated" or "notyettreated" 
               bstrap = TRUE, # if TRUE compute bootstrapped SE
               biters = 1000, # number of bootstrap iterations
               print_details = FALSE, # if TRUE, print detailed results
               clustervars = "sid", # cluster level
               panel = TRUE) # whether the data is panel or repeated cross-sectional

# Aggregate ATT
agg_effects <- aggte(atts, type = "group")
summary(agg_effects)

# Group-time ATTs
summary(atts)

# Plot group-time ATTs
ggdid(atts)

# Event-study
agg_effects_es <- aggte(atts, type = "dynamic")
summary(agg_effects_es)

# Plot event-study coefficients
ggdid(agg_effects_es)

Reference

Callaway, Brantly and Pedro H. C. Sant’Anna (2021), “Difference-in-Differences with multiple time periods,” Journal of Econometrics, Themed Issue: Treatment Effect 1, 225 (2), 200–230.

Lee, S. J., & Wooldridge, J. M. (2023). A Simple Transformation Approach to Difference-in-Differences Estimation for Panel Data (SSRN Scholarly Paper No. 4516518). Social Science Research Network. https://doi.org/10.2139/ssrn.4516518

Sant’Anna, Pedro H. C. and Jun Zhao (2020), “Doubly robust difference-in-differences estimators,” Journal of Econometrics, 219 (1), 101–22.

How does doubly robust DiD estimator works? Check this: Lecture 5: How Covariates can make your DiD More Plausible

did R package 📦

Intuition for Doubly Robust Estimator

Sun, 18 May 2025 00:00:00 +0000

Introduction

To estimate ATE, we can either use outcome regression or inverse propensity weighting (IPW). While each approach has merits, combining them offers significant advantage – double robustness. In this post, I summarize the intuition for doubly robust estimator from Professor Ding’s textbook (Ding 2024), and connects this framework to debiased machine learning (DML) through Riesz representation theory. By understanding these connections, we can gain some insights into how modern causal inference methods effectively correct for bias in treatment effect estimation.

Two characterizations of the ATE

Assumption 1 (Basic setting).

SUTVA, unconfoundedness and overlap

Let $Y(0), Y(1)$ be potential outcomes and $D$ be a binary treatment variable. Consider ATE.

First, we can use the outcome regression,

$$ \tau = \E\{\mu_1(X) - \mu_2(X) \}, $$ where $$ \mu_1 = \E\{Y(1) \mid X \} = \E\{Y \mid D = 1, X \}, $$ $$ \mu_0 = \E\{Y(0) \mid X \} = \E\{Y \mid D = 0, X \} $$

Second, we can use the inverse propensity score weighting (IPW) approach,

$$\tau = \E\left\{\frac{DY}{e(X)} \right\} - \E\left\{\frac{(1-D)Y}{1-e(X)} \right\},$$

where $e(X) = \P(D = 1 \mid X)$ is the propensity score.

It completely ignores the outcome model. However, if there exist covariates $X$ that are predictive of $Y$, then even when the outcome model is misspecified, including it can reduce the variance compared to using IPW alone.

Key Insights

This motivates combining the two approaches to:

1. Reduce the variance of the IPW estimator

2. Reduce the bias of the outcome regression

Reducing the Variance

$$ \mu_1=E\{Y(1)\}=E\left\{Y(1)-\mu_1\left(X, \beta_1\right)\right\}+E\left\{\mu_1\left(X, \beta_1\right)\right\} . $$

Idea:

View $Y-\mu_1\left(X, \beta_1\right)$ as a “pseudo potential outcome”
Then apply IPW to it:

$$ \begin{aligned} \mu_1 & = E\left\{\frac{D\left\{Y-\mu_1\left(X, \beta_1\right)\right\}}{e(X)}\right\}+E\left\{\mu_1\left(X, \beta_1\right)\right\} \\ & =E\left\{\frac{D\left\{Y-\mu_1\left(X, \beta_1\right)\right\}}{e(X)}+\mu_1\left(X, \beta_1\right)\right\}, \end{aligned} $$

Similarly,

$$ \begin{aligned} \mu_0 & =E\left\{\frac{(1-D)\left\{Y-\mu_0\left(X, \beta_0\right)\right\}}{1-e(X)}\right\}+E\left\{\mu_0\left(X, \beta_0\right)\right\} \\ & =E\left\{\frac{(1-D)\left\{Y-\mu_0\left(X, \beta_0\right)\right\}}{1-e(X)}+\mu_0\left(X, \beta_0\right)\right\}, \end{aligned} $$

Notice that,

$$ \begin{aligned} \mu_1 - \mu_0 & = E\left\{\frac{D\left\{Y-\mu_1\left(X, \beta_1\right)\right\}}{e(X)}+\mu_1\left(X, \beta_1\right)\right\} \\ &\quad - E\left\{\frac{(1-D)\left\{Y-\mu_0\left(X, \beta_0\right)\right\}}{1-e(X)}+\mu_0\left(X, \beta_0\right)\right\} \end{aligned} $$

$\mu_1 - \mu_2$ gives us the AIPW estimator.

Reducing the Bias

$$ \mu_1 =E\left\{\frac{Z\left\{Y-\mu_1\left(X, \beta_1\right)\right\}}{e(X)}\right\}+E\left\{\mu_1\left(X, \beta_1\right)\right\} $$

Idea: We can view $Y-\mu_1\left(X, \beta_1\right)$ as the regression residuals, from which we apply IPW to extract useful signals. Alternatively, we can view $\mu_1\left(X, \beta_1\right) - Y$ as the bias, which we then use IPW to correct the bias.

Connecting to DDML: Riesz Representation for Bias Correction

In the generic debiased framework (Chernozhukov, Newey, and Singh 2022), we leverage the Riesz representer to correct for bias, similar to the approach described above. This method parallels our use of IPW to extract signals from residuals, but specifically employs the Riesz representer for bias correction.

Data is $Z = \{Y, D, X\}$
Let $g$ be outcome regression, $g(D, X)=E[Y \mid D, X]$
Suppose moment is of the form: for some moment $m()$ that is linear in $g$

$$\theta-E[m(Z ; g)]=0,$$ $$\theta = E[m(Z ; g)] :=E[g(1, X)-g(0, X)]$$

Then debiased version of the moment is of the form:

$$ \theta-E[m(Z ; g)+a(X) \cdot(Y-g(X))]=0, $$

where

$a(X)$ is the Riesz Representer of the linear functional $L(g) := E[m(Z ; g)]$. The existence of $a(X)$ is guaranteed by the Riesz representation theorem, for all square integrable $g(\cdot)$, we have

$$ E[m(Z ; g)]=E[a(X) \cdot g(X)]. $$

As we consider the ATE, the Riesz Representer are just some “inverse propensity score terms”,

$$ a(D, X):=\frac{D}{e(X)}-\frac{(1-D)}{1-e(X)}, $$

Consider the expression

$$ E[m(Z ; g)+a(X) \cdot(Y-g(X))] $$

The key intuition here is that $Y−g(X)$ represents the residual part, and we use the Riesz Representer $a(X)$ to correct this bias/residual term.

Double robust estimation of ATT

How about the double robust estimator for ATT? Can we also use the same idea to derive it? Yes!

Assumption 2 ( "one-sided" unconfoundedness and overlap).

$$ D \indep Y(0) \mid X \text { and } e(X)<1 . $$

Theorem 1.

Under the "one-sided" unconfoundedness and overlap assumption, $$ \E\{Y(0) \mid D=1\}=\frac{1}{e}\E\left\{\frac{e(X)}{1-e(X)} (1-D) Y\right\} \tag{1} $$ and $$ \tau_{\mathrm{T}}=\E(Y \mid D=1)-\E\left\{\frac{e(X)}{e} \frac{1-D}{1-e(X)} Y\right\}, $$ where $e=\P(D=1)$ is the marginal probability of the treatment.

We also have a doubly robust estimator for $\E\{Y(0) \mid D=1\}$ which combines the propensity score and the outcome models.

Theorem 2.

Define $$ \tilde{\mu}_{0 \mathrm{T}}^{\mathrm{dr}} := \frac{1}{e} \E\left[ \frac{e(X, \alpha)}{1 - e(X, \alpha)}(1-D)\left\{Y-\mu_0\left(X, \beta_0\right)\right\}+D \mu_0\left(X, \beta_0\right)\right] , $$ Under above Assumption, if either $e(X, \alpha)=e(X)$ or $\mu_0\left(X, \beta_0\right)=\mu_0(X)$, then $\tilde{\mu}_{0 \mathrm{T}}^{\mathrm{dr}}= \E\{Y(0) \mid D=1\}$.

How to come up with $\tilde{\mu}_{0 \mathrm{T}}^{\mathrm{dr}}$ ? Exactly the same idea as before!

$$ \E\{Y(0) \mid D=1\} = \E\{Y(0) - \mu_0\left(X, \beta_0\right) \mid D=1\} + \E\{\mu_0\left(X, \beta_0\right) \mid D=1\} $$

Now, we can view $Y(0) - \mu_0\left(X, \beta_0\right)$ as a “pseudo potential outcome” under the control and apply eqn (1) to weight it, then we can get the form of $\tilde{\mu}_{0 \mathrm{T}}^{\mathrm{dr}}$ .

Reference

Ding, Peng (2024) A first course in causal inference. Chapman & Hall.

Chernozhukov, Victor, Whitney K. Newey, and Rahul Singh (2022), “Automatic Debiased Machine Learning of Causal and Structural Effects,” Econometrica, 90 (3), 967–1027.

Adjust Censoring and Confounding Bias by IP Weighting

Tue, 01 Oct 2024 00:00:00 +0000

Introduction

In the context of causal inference, adjusting for both censoring bias and confounding bias is crucial, particularly in survival analysis where right-censored data often complicates causal effect estimation. Right censoring occurs when the outcome of interest (e.g., time to an event) is not observed within the study period for some subjects, making it challenging to correctly assess the causal effect of a treatment. Moreover, confounding bias arises when treatment assignment is influenced by pre-treatment covariates, potentially leading to biased estimates of treatment effects if not properly accounted for.

To address these issues, inverse probability weights (IPW) are commonly employed. IPW adjusts for confounding by reweighting observations based on their treatment probabilities given covariates. In addition, inverse probability of censoring weights (IPCW) further adjust for censoring by reweighting observations based on their probabilities of being uncensored. Together, these techniques allow us to estimate causal effects in the presence of both confounding and censoring biases, providing more accurate insights into the relationships between treatments and outcomes.

This post explains how to apply IP weights to adjust for these biases, with references to key resources.

Hernán and Robins’ Causal Inference: What If (2020), especially Chapter 8.5 and Chapter 12.6, covers censoring and missing data in detail
Zubizarreta et al.’s Handbook of Matching and Weighting Adjustments for Causal Inference (2023), Chapter 21, discusses treatment heterogeneity in survival outcomes. These resources form the basis for our explanation on using IP weights in survival analysis

IP Weighting

Imagine we want to estimate the causal effect of eco-friendly packaging on a product’s selling price. However, the selling price is right-censored — meaning we only observe the price for items that have been sold, while unsold items remain censored (i.e., their final selling price is unknown).

Setting:

Let $W \in \{0,1\}$ be a binary treatment variable (e.g., $W= 1$, if the item has a eco-friendly package; $W = 0$, otherwise)
Let $Y(w, s)$ be potential outcomes
Let $S = \textbf{1} \{C = 0\}$ be non-censoring indicator, where $C \in \{0,1\}$ is the censoring indicator
- e.g. $S = 1$, if the item is non-censored (in this case sold); $S = 0$, if the item is censored (in this case on-sale)
Let $X, L \in \mathcal{X}$ be two sets of covariates and $X \subset L$. Then there exist a function $f(\cdot)$ such that $X = f(L)$

Considering $Y(W = w, S = s)$, our analysis was necessarily restricted to uncensored individuals, i.e., those with $S = 1$, because those were the only ones with known values of the outcome $Y$. Thus, the causal effect of interest is:

$$ \tau = \mathbb{E}\{Y(w = 1, s = 1)\} - \mathbb{E}\{Y(w = 0, s= 1)\} $$

Assumptions:

Recall the three identification conditions in Chapter 8.5 (Hernán and Robins, 2020, p. 113-114). Note that, the book uses $Y, A, C, L$ to represent the outcome, treatment, censoring indicator and covariates respectively.

“First, the average outcome in the uncensored individuals must equal the unobserved average outcome in the censored individuals with the same values of $A$ and $L$. This provision will be satisfied … if the variables in $A$ and $L$ are sufficient to block all backdoor paths between $C$ and $Y$.”

“Second, IP weighting requires that all conditional probabilities of being uncensored given A and the variables in L must be greater than zero.”

“The third condition is consistency, including sufficiently well-deﬁned interventions. IP weighting is used to create a pseudo-population in which censoring $C$ has been abolished, and in which the effect of the treatment $A$ is the same as in the original population.”

Following above three identification conditions, we make the following assumptions. Denote the propensity score as $e(\cdot)$ and censoring score as $c(\cdot)$,

$$ \begin{alignat}{2} &\text{1. (Unconfoundedness):} \quad && \{Y(w= 0, s = 1), Y(w= 1, s = 1)\} \perp \!\!\! \perp W \mid X \\[1.5em] &\text{2. (Overlap):} \quad && 0 < e(X) := \mathbb{P}(W = 1 \mid X) < 1 \\[1.5em] &\text{3. (Ignorable censoring):} \quad && \{Y(w= 0, s = 1), Y(w= 1, s = 1)\} \perp \!\!\! \perp S \mid (W, L) \\[1.5em] &\text{4. (Positivity):} \quad && c(W, L) := \mathbb{P}(S = 1 \mid (W, L)) > 0 \end{alignat} $$

Claim:

$$ \begin{align} \tau &= \mathbb{E}\left[Y(w= 1, s = 1) - Y(w= 0, s = 1)\right] \\[1em] &= \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)} - \frac{S (1-W) Y}{c(W, L) (1-e(X))}\right] \quad \tag{2} \end{align} $$

Proof:

First, note that the eqn (2) is well defined because of A2 and A4.

I only give the proof for, $$\mathbb{E}\left[Y(w= 1, s = 1)\right] = \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)}\right],$$ as the $\mathbb{E}\left[Y(w= 0, s = 1)\right]$ part follows the same logic.

Recall that,

By our setting, $X = f(L)$ for some known function $f$
$c(W, L) := \mathbb{P}(S = 1 \mid W, L) = \mathbb{E}\left[S \mid W,L \right]$

Then we have,

\begin{aligned} \mathbb{E}\left[\frac{S W Y}{c(W, L) e(X)}\right] &= \mathbb{E}\left\{\frac{W}{e(f(L))c(W,L))} \cdot \mathbb{E}\left[SY(w, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by LIE)} \\[1.5em] & = \mathbb{E}\left\{\frac{W}{e(f(L))c(W,L))} \cdot \mathbb{E}\left[S \,\middle|\, W,L \right] \cdot \mathbb{E}\left[Y(w, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by A3)} \\[1.5em] & = \mathbb{E}\left\{\frac{W}{e(f(L))} \cdot \mathbb{E}\left[Y(w, s=1) \,\middle|\, W,L \right] \right\} \\[1.5em] & = \mathbb{E}\left\{\mathbb{E}\left[\frac{W}{e(f(L))} \cdot Y(w=1, s=1) \,\middle|\, W,L \right] \right\} &&\text{(by SUTVA)} \\[1.5em] & = \mathbb{E}\left[\frac{W}{e(X)} \cdot Y(w=1, s=1) \right] \\[1.5em] & = \mathbb{E}\left[Y(w=1, s=1) \right] &&\text{(by A1 and IPW)} \end{aligned}

The last equation holds by the classical proof of the unbiasedness of inverse propensity score weighting (IPW) estimator. For more details, one can check Theorem 11.3 at page 158-159 in Peng Ding’s textbook (Ding, 2023).

Q.E.D.

Future Work

How to create a doubly robust estimator based on above setting? One related literature is the causal survival forest model (Cui et al., 2023), which estimates heterogeneous treatment effects in time-to-event setting and obtains doubly robustness property. But sometimes we are more interested in the downstream outcomes (e.g. selling price) after the event (e.g. being sold). This motivates us to create a new estimator that is doubly robust…

References

Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC.

Zubizarreta, J. R., Stuart, E. A., Small, D. S., & Rosenbaum, P. R. (2023). Handbook of Matching and Weighting Adjustments for Causal Inference. CRC Press.

Cheng, C., Li, F., Thomas, L. E., & Li, F. (Frank). (2022). Addressing Extreme Propensity Scores in Estimating Counterfactual Survival Functions via the Overlap Weights. American Journal of Epidemiology, 191(6), 1140–1151. https://doi.org/10.1093/aje/kwac043

YouTube tutorial about IPCW: “Survival Analysis, Censoring and Time Scales”

Cui, Y., Kosorok, M. R., Sverdrup, E., Wager, S., & Zhu, R. (2023). Estimating heterogeneous treatment effects with right-censored data via causal survival forests. Journal of the Royal Statistical Society Series B: Statistical Methodology, 85(2), 179–211. https://doi.org/10.1093/jrsssb/qkac001

Ding, Peng. “A First Course in Causal Inference.” arXiv, October 3, 2023. https://doi.org/10.48550/arXiv.2305.18793.

propensity score | Chen Xing

Balancing Weights for Causal Inference

TL;DR

What is this paper about?

What do the authors do?

Why is this important?

Who should care?

Do we have code?

Reference

Notes on Propensity Score Methods

Introduction

Setting & Notation

1. The propensity score as a dimension reduction tool

2. Propensity score stratification

3. Propensity score weighting

3.1 Estimation

3.2 Strong overlap condition

4. Balancing property

4.1 Propensity score is a balancing score

5. Doubly Robust or AIPW

6. Other Estimands related to IPW

7. Propensity Score in Regression

PS as a covariate

PS as a weight

Reference

Notes on causal inference with No Overlap – Regression Discontinuity

Motivation

Setting

Identification

Key Theorem

Reference

Notes on Callaway & Sant’Anna (2021) – Staggered Adoption DiD

0. Motivation

1. Setup

2. Key Assumptions

3. Identification: Long-Difference Estimands

4. Aggregation of $\mathrm{ATT}(g,t)$

5. Limitation & Extension

6. R code Example

Reference

Intuition for Doubly Robust Estimator

Introduction

Two characterizations of the ATE

Key Insights

Reducing the Variance

Reducing the Bias

Connecting to DDML: Riesz Representation for Bias Correction

Double robust estimation of ATT

Reference

Adjust Censoring and Confounding Bias by IP Weighting

Introduction

IP Weighting

Setting:

Assumptions:

Claim:

Proof:

Future Work

References