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Beyond Linearity: Semiparametric Solutions to Contamination Bias with Multi-valued Treatments
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\title{{\LARGE Beyond Linearity: Semiparametric Solutions to Contamination Bias with Multi-valued Treatments}}
\author{
Michel Csillag Finger\thanks{São Paulo School of Economics - FGV and Institute of Mathematics and Statistics - USP. Email: [email removed]} \and Vitor Possebom\thanks{São Paulo School of Economics - FGV. Email: [email removed]. This study was financed, in part, by the São Paulo Research Foundation (FAPESP), Brazil. Process Number \#2025/04857-0.}}
\date{}
\maketitle
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\begin{center}
First Draft: September 2026; This Draft: \monthyeardate\today
\href{https://sites.google.com/site/vitorapossebom/working-papers}{Please click here for the most recent version}
\
\large{\textbf{Abstract}}
\end{center}
We examine semiparametric solutions to contamination bias for nonbinary treatments. Deepening the discussion by \cite{gphk2024contamination}, we detail how spline functions approximate conditional expectation and propensity score functions under weak functional-form assumptions. Reanalyzing 18 regressions across 11 studies, we compare standard linear regressions against parametric and semiparametric versions of three contamination-robust estimators. We document large point-estimate discrepancies between parametric and semiparametric approaches and find that adopting flexible semiparametric solutions may not increase statistical uncertainty substantially. We recommend that researchers verify the robustness of their conclusions to the use of semiparametric tools that address contamination bias.
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\textbf{Keywords:} Contamination Bias, Semiparametric Estimation, Multi-valued Discrete Treatment, Police Use-of-force, Police Discrimination.
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\textbf{JEL Codes:} C21, C31, C51, D63, J15.
\newpage
\doublespacing
\section{Introduction}\label{sec:Intro}
Multi-valued discrete treatments are widespread in observations studies and randomized controlled trials. For example, \cite{weisburst2019police} focuses on police racial bias in use of force and considers four racial categories: white, black, Hispanic, and other. \cite{brownback2020instruction} study the impact of four types of performance-based incentives for community college instructors and students. These settings are typically analyzed with regressions of an outcome variable on indicator variables for each treatment arm and control variables. However, \citet{gphk2024contamination} show that these regression coefficients may present contamination bias, i.e., they are functions of the average effects of the treatment of interest and all other treatment arms. To avoid these issues, they propose three identification strategies that avoid contamination bias. Although they briefly discuss nonparametric implementation, they focus their estimation and empirical exercises on linear parametric estimators.
Because these estimators rely on strong linearity assumptions, we deepen their discussion about avoiding contamination bias through the usage of semiparametric smoothing functions. In particular, we detail how to use spline functions to approximate the necessary propensity score and conditional expectation functions. By doing so, we are able to easily implement semiparametric versions of the three estimators proposed by \cite{gphk2024contamination} using the statistical package provided by \cite{multe2024}. Since these semiparametric versions rely on weaker functional-form restrictions, they increase the credibility of the three solutions proposed by \cite{gphk2024contamination} for addressing contamination bias.
Next, we compare our proposed spline estimators against the parametric solutions implemented by \cite{gphk2024contamination} and against the usual linear regression estimator in relevant empirical settings. To do so, we replicate articles from top Economics journals. The replicated articles include six studies with continuous covariates previously analyzed by \citet{gphk2024contamination}, plus five additional articles from American Economic Review, Econometrica, and the Journal of Political Economy. As a result, we substantially increase the number of regressions used to compare seven estimators: standard linear regression with a full set of treatment-arm indicators, the parametric versions of the three solutions proposed by \cite{gphk2024contamination}, and our spline versions of these three solutions. Specifically, we analyze 18 regressions from 11 articles, for a total of 141 parametric estimates and 141 semiparametric estimates.
We present three sets of results concerning contamination bias in standard linear regressions, differences in the point estimates of parametric and semiparametric approaches, and differences in the statistical uncertainty behind the seven analyzed estimators. First, similarly to \cite{gphk2024contamination}, we document large variations in point-estimates when we compare the standard linear regression estimator against the strategies proposed by \cite{gphk2024contamination}. These discrepancies suggest that contamination bias might be a relevant problem in many empirical settings with nonbinary treatments. Second, we find substantial differences between the point estimates of the linear parametric estimators implemented by \cite{gphk2024contamination} and the semiparametric spline estimators discussed in this article. Third, when we compare the estimated standard errors of the seven analyzed estimators, we find that adopting approaches that address contamination bias may not lead to increases in statistical uncertainty and that adopting flexible semiparametric estimators may not lead to increased statistical uncertainty.
We contribute to two strands of the economics literature. First, we extend the discussion about how to account for treatment effect heterogeneity. The mismatch between nonparametric, fully heterogeneous identification strategies and parametric estimators has been discussed by \cite{GoodmanBacon2021, dechaisemartin2020twowayFE, sun2021dynamic, callawaysantanna2021did} in the differences-in-differences setting, \cite{Sloczynski2022} in the binary treatment context, \cite{słoczyński2024} in the instrumental variable framework, \cite{Gelman2019} in the regression discontinuity setting, and \cite{gphk2024contamination} in the nonbinary treatment context. We contribute especially to the last setting by deeply discussing the semiparametric solutions suggested by \cite{gphk2024contamination}.
Second, we contribute to the empirical practice literature. Understanding how different estimators behave in relevant empirical settings has been a fruitful area of research. In instrumental variable contexts, \cite{Lal2024} discuss different ways to measure statistical uncertainty and instrument strength. In difference-in-differences settings, \cite{Bertrand2004} compares different approaches to measuring statistical uncertainty, while \citet{chiu2025analysis} discuss different estimators that account for heterogeneous effects and staggered treatment adoption. In the multi-valued discrete treatment setting, \cite{gphk2024contamination} compares three parametric solutions to contamination bias using eight standard regressions, for a total of 90 parametric estimates. We contribute especially to this last setting by analyzing semiparametric versions of these estimators and increasing the number of regressions analyzed to 18, for a total of 282 estimates.
The paper is organized as follows. Section \ref{sec:theory} explains the contamination bias problem in standard linear regression with multi-valued discrete treatment and presents three parametric and three semiparametric solutions to this issue. Section \ref{sec:method} explains how we selected the replicated articles and regressions used to assess the relevance of semiparametric estimators in a setting with nonbinary treatments. Section \ref{sec:results} discusses our main empirical findings, while Section \ref{sec:conclusion} concludes.
\section{Theoretical Background}\label{sec:theory}
We explain the identification issues related to contamination bias in Section \ref{sec:bias} and discuss six solutions in Section \ref{sec:solutions}. Although \citet{gphk2024contamination} discussed half of these solutions, we contribute by analyzing semi-parametric solutions in detail. We keep the exposition succinct for brevity.
\subsection{Understanding Contamination Bias} \label{sec:bias}
Suppose a linear regression model with an outcome variable $Y_i$, a $K$-dimensional treatment vector $X_i$, and covariates $W_i$. Specifically, each component $k \in \left\lbrace 1, \ldots, K \right\rbrace$ of $X_i$ is a binary variable $X_{k,i}$ indicating that the individual $i$ received treatment arm $k$, where all $K$ treatment arms are mutually exclusive. The covariates enter the regression through a function $g \in \mathcal{G}$, where $\mathcal{G}$ is a linear space of functions. Mathematically, this regression model is represented by
\begin{equation}
\label{eq:gen-model}
Y_i = X_i^{\prime} \beta + g(W_i) + U_i,
\end{equation}
where $U_i$ is the residual and $\beta$ and $g$ are defined by
\begin{equation}
\label{eq:gen-b-g}
(\beta, g) \coloneqq \mathop{\arg \min}_{\tilde{\beta} \in \mathbb{R}^K, \tilde{g} \in \mathcal{G}} \mathbb{E}[(Y_i - X_i^{\prime} \tilde{\beta} -\tilde{g}(W_i))^2].
\end{equation}
Let $Y_i (k)$ denote the potential outcome of unit $i$ under treatment $k$. Define the vector of treatment effects $\tau_i$ with elements $\tau_{ik} = Y_i (k) - Y_i (0)$. The vector $\tau (W_i) = \mathbb{E} [\tau_i | W_i]$ captures the conditional average treatment effects (CATE), where $\tau_k(W_i)$ corresponds to the CATE of treatment $k$. Define also the vector of propensity scores $p(W_i) = \mathbb{E}[X_i | W_i]$, where $p_k (W_i) = \mathbb{P} \left[X_{k,i} = k | W_i\right]$ is the conditional probability of receiving treatment arm $k$.
Following \citet{gphk2024contamination}, we make two assumptions:
\begin{assumption} \label{ass:mean-independence}
Given covariates, potential outcomes are mean-independent of the treatment, i.e., $\mathbb{E}[Y_i (k) | X_i, W_i] = \mathbb{E}[Y_i (k) | W_i]$ for all $k$.
\end{assumption}
\begin{assumption} \label{ass:g-specified}
The linear space $\mathcal{G}$ is flexible enough to contain either the true propensity score functions or the true conditional expectation of the untreated potential outcome. Formally, we have that either $p_k \in \mathcal{G}$ for all $k$ or $\mu_0 \in \mathcal{G}$, where $\mu_0 (w) := \mathbb{E}[Y_i (0) | W_i = w]$.
\end{assumption}
\citet[Proposition 1]{gphk2024contamination} shows that the regression coefficient $\beta_{k}$ of Equation \eqref{eq:gen-model} is a sum of two terms, where the first term is a convex combination of the $k$-th treatment arm's CATEs and the second term is a possibly non-zero linear combination of the CATEs of the other treatment arms. Formally, \citet[Proposition 1]{gphk2024contamination} derives the following result:
\begin{proposition} \label{prop:general-bias}
Under Assumptions \ref{ass:mean-independence} and \ref{ass:g-specified}, the treatment coefficients in Equation \eqref{eq:gen-model} identify
\begin{equation} \label{eq:general-bias}
\beta_k = \mathbb{E}[\lambda_{kk}(W_i) \tau_{k}(W_i)] + \sum_{\ell \neq k} \mathbb{E}[\lambda_{k\ell}(W_i) \tau_{\ell}(W_i)],
\end{equation}
where $\mathbb{E}[\lambda_{kk}(W_i)] = 1$ and $\mathbb{E}[\lambda_{k\ell}(W_i)] = 0$. Moreover, contamination weights $\lambda_{k\ell}(W_i)$ will be non-zero unless treatment assignment is unconditionally random (i.e., $p_k (W_i) = \mathbb{P} \left[X_{k,i} = 1 | W_i\right]$ is a constant function for every $k$).
\end{proposition}
In settings where treatment assignment depends on covariates, Proposition \ref{prop:general-bias} implies that the coefficient $\beta_k$ does not identify any sort of average of the corresponding $k$-th treatment arm's CATEs. It shows that this regression coefficient identifies a linear combination of \textit{all} CATEs due to the contamination bias term in Equation \eqref{eq:general-bias}.
\subsection{Avoiding Contamination Bias} \label{sec:solutions}
\citet{gphk2024contamination} propose three solutions to identify interpretable convex combinations of CATEs in settings with multi-valued discrete treatments. As pointed out in \citet[p.~4031]{gphk2024contamination}, a key assumption in all these strategies is that the conditional expectation function of each potential treatment outcome is in the space of functions $\mathcal{G}$. For this reason, \citet[e.g., Equation (16)]{gphk2024contamination} suggest using nonparametric smoothing functions (e.g., splines) to approximate the relevant conditional expectation function and propensity scores. Despite this suggestion, in their empirical analysis, \citet[Sections III.C and IV]{gphk2024contamination} implement only linear estimators that restrict $\mathcal{G}$ to be the space of linear functions of the covariates.
Our main contribution is to follow their suggestion of using nonlinear methods to approximate the relevant conditional expectation functions and propensity scores. We empirically implement their three solutions using splines as a semiparametric approximation to these functions. Beyond reanalyzing all the articles that were discussed by \cite{gphk2024contamination} and contain continuous covariates, we include additional articles published in the Journal of Political Economy, Econometrica, and the American Economic Review. As a consequence, we substantially increase the number of regressions that are used to compare seven estimators: standard linear regression (Equation \eqref{eq:ew-model}), the parametric versions of the three solutions proposed by \cite{gphk2024contamination} and the spline version of these three solutions.
To understand these seven estimators, we need to build our notation. Define the expected conditional potential outcome as $\mu_k (w) = \mathbb{E}[Y_i (k) | W_i = w]$ for $k = 0, \dots, K$, implying that $\tau_k (w) = \mu_k (w) - \mu_0 (w)$. Define a transformation of the covariates $q_k (w)$ such that $q_k (\cdot) \in \mathcal{G}$. For example, we can choose $\mathcal{G}$ to contain infinite sequences of semiparametric smooth basis functions $\{ b_j (W_i) \}_{j = 0}^{\infty}$.
As previously mentioned, we focus on splines as these semiparametric smooth basis functions. Specifically, we use the definition of univariate splines by \citet[p.~5571]{chen2007}: for a spline of order $r$ with $J$ knots for a continuous variable $W_{i}$, we use
\begin{equation} \label{eq:spline-def}
\text{Spl}(r, J; W_i) = \sum_{\ell=1}^{r-1} W_i^\ell + \sum_{j=1}^{J} [\max \{ W_i - t_j, 0 \} ]^{r-1},
\end{equation}
where the knots are given by $t_j \text{ with } j = 0, \dots, J + 1$ such that $\min \{ W_i \} = t_0 < t_1 < t_2 < \dots < t_{J-1} < t_{J} < t_{J+1} = \max \{ W_i \}$.
It is important to note that the spline definition in Equation \eqref{eq:spline-def} only applies for continuous covariates.
To include categorical and binary covariates, we use a full set of indicator variables. Specifically, we define $S(W_i) \in \mathcal{G}$ such that
\begin{equation} \label{eq:qk-spline}
S (W_i) =
\begin{cases}
\text{Spl} (r, J; W_{ij}), & \text{if } W_{ij} \text{ is a continuous variable} \\
W_{ij}, & \text{if } W_{ij} \text{ is a binary variable} \\
\sum_{\ell = 2}^{L}\mathbbm{1}\{ W_{ij} = \ell \} , & \text{if } W_{ij} \text{ is a categorical variable, with } L \text{ categories}
\end{cases}
\end{equation}
as one possible transformation of the covariates, which we will use in the next sections.
In the next three sections, we present the three solutions proposed by \citet{gphk2024contamination} with their respective spline extensions.
Each solution should be compared to a ``simple'' regression, which is the ``canonical'' method of estimating $\hat{\beta}$, with a coefficient for each treatment and for each covariate.
It is equivalent to choose $\mathcal{G} = \{\alpha + w'\gamma : [\alpha, \gamma']' \in \mathbb{R}^{1+\text{dim}(W_i)}\}$.
Explicitly, this model may suffer from contamination bias and is given by Equation \eqref{eq:simple-reg}.
\begin{equation} \label{eq:simple-reg}
Y_i = \alpha_{0} + \sum_{k=1}^{K} X_{ik} \beta_k + W_i' \gamma + U_i
\end{equation}
\subsubsection{Estimating Average Treatment Effects (ATE)}
\citet{gphk2024contamination} extend the ATE estimator proposed for binary treatments by \citet[Subsection~5.3]{imbens2009developments} and use the following regression model:
\begin{equation} \label{eq:ate}
Y_i = \dot{\alpha}_0 + \sum_{k = 1}^{K} X_{ik} \dot{\beta}_k + W_i' \alpha_{W,0} + \sum_{k = 1}^{K} X_{ik} (W_i - \overline{W})' \gamma_{W, k} + \dot{U}_i
\end{equation}
where $\overline{W} = \frac{1}{N} \Sigma_{i=1}^{N} W_i$ and $N$ is the number of observations. There are two main differences from the standard regression in Equation \eqref{eq:simple-reg}. First, we add interaction terms to ensure each treatment coefficient depends only on the outcome of interest and the untreated outcome, avoiding contamination bias if the conditional expectation functions of the potential outcomes are linear in the covariates. Second, covariates $W_i$ are demeaned to ensure that the coefficients appropriately identify the ATE if the conditional expectation functions are linear in the covariates.
If the relevant conditional expectation functions are not linear, we follow the suggestion by \citet{gphk2024contamination} and implement a regression specification with splines:
\begin{equation} \label{eq:spline}
Y_i = \dot{\alpha}_{S,0} + \sum_{k = 1}^{K} X_{ik} \dot{\beta}_{S,k} + S (W_i)^{\prime} \alpha_{S, 0} + \sum_{k=1}^{K} X_{ik} \left(S (W_i) - \overline{S} (W_i) \right)^{\prime} \gamma_{S, k} + \dot{U}_{S,i}
\end{equation}
where $\overline{S} (W_i) = \frac{1}{N} \sum_{i=1}^{N} S (W_i)$. Note that this spline estimator achieves the semiparametric efficiency bound under the strong overlap \citep{chen2007} and that its standard errors can be estimated using the usual parametric formulas \citep{Ackerberg2012}.
\subsubsection{Easiest-to-Estimate Averages of Treatment Effects (EW)}\label{SecEW}
Instead of interacting the covariates with the treatment-arm indicator variables, \citet[Equation (24)]{gphk2024contamination} also propose running separate regressions for each treatment arm, i.e.,
\begin{equation}
\label{eq:ew-model} Y_i = \ddot{\alpha}_k + X_{ik} \ddot{\beta}_k + W_i' \ddot{\gamma}_k + \ddot{U}_{ik},
\end{equation}
where we use a sample containing only units assigned to either treatment arm $k$ or the control group. According to \citet{gphk2024contamination}, coefficient $\ddot{\beta}_k$ captures a weighted average of treatment arm $k$'s CATEs if the conditional expectation functions of the potential outcomes are linear in the covariates.
If the relevant conditional expectation functions are not linear, we follow the suggestion by \citet{gphk2024contamination} and implement a regression specification with splines:
\begin{equation}
\label{eq:ew-spline} Y_i = \ddot{\alpha}_{S,k} + X_{ik} \ddot{\beta}_{S,k} + S_k (W_i)' \ddot{\gamma}_{S,k} + \ddot{U}_{S,ik},
\end{equation}
where we use a sample containing only units assigned to either treatment arm $k$ or the control group.
\subsubsection{Easiest-to-Estimate Common Weighting (CW) Scheme}\label{SecCW}
Although the EW solution in Section \ref{SecEW} is easy to estimate and has interesting efficiency properties according to \citet[Corollary 1]{gphk2024contamination}, it only allows comparisons between treatment $k$ and the control group. To facilitate the comparison between any treatment arms while putting more weight on covariate values that satisfy the overlap condition, \cite{gphk2024contamination} propose a weighted least squares estimator that regresses the outcome variable $Y_i$ on treatment vector $X_i$ and a constant, where unit $i$'s weight is given by
\begin{equation}
\label{eq:cw}
\left( \sum_{k=0}^{K} \frac{\pi_k (1 - \pi_k)}{p_k (W_i)} \right)^{-1} \cdot \left(p_{D_{i}} (W_i)\right)^{-1},
\end{equation}
$\pi_k$ is the unconditional probability of treatment arm $k$, and $D_{i} = k$ if and only if $X_{ik} = 1$. To feasibly implement this regression, the unit weights use $\hat{\pi}_k = \sfrac{\sum_{i=1}^{N} X_{ik}}{N}$ for $\pi_k$ and parametric multinomial logit estimators $\hat{p}_k (W_i)$ for the propensity score functions $p_k (W_i)$. The k-coefficient in this weighted regression captures a weighted average of treatment arm $k$'s CATEs if the linearity assumptions behind the propensity score estimators hold.
If these linearity assumptions do not hold, we follow the suggestion by \citet{gphk2024contamination} and implement a regression specification with splines for the propensity score estimators. To do so, we apply the spline transformation on the covariates and use these transformed variables in the multinomial logit estimator.
\section{Comparing Solutions to Contamination Bias in Empirical Settings}\label{sec:method}
To understand the importance of semiparametric solutions to contamination bias, it is paramount to compare our proposed spline estimators against the parametric solutions implemented by \cite{gphk2024contamination} and against the usual linear regression estimator (Equation \eqref{eq:simple-reg}) in relevant empirical settings. To do so, we replicate articles from top Economics journals. The replicated articles include six studies with continuous covariates previously analyzed by \citet{gphk2024contamination}, plus five additional articles from American Economic Review, Econometrica, and the Journal of Political Economy.
To select the articles in our replication exercise, we follow closely the method used by \citet[Appendix C.1]{gphk2024contamination}. However, these authors included only articles published by the American Economic Association while we also searched for articles published in Econometrica, the Journal of Political Economy, the Quarterly Journal of Economics, and the Review of Economic Studies. Specifically, articles from these journals were filtered for containing in the title, abstract, or body any word related to ``experiments'' (\textit{stratified, random, RCT, experiment}) or to racial disparities (\textit{racial/ethnic differences, discrimination, disparities, gaps}). The resulting articles were manually analyzed to verify (1) the presence of multi-valued discrete treatments, and (2) the public availability of a replication package.
Using this procedure, we found eight experimental articles and three observations studies. Table \ref{tab:articles} lists those articles and the replicated columns of each of their published tables. Unlike \cite{gphk2024contamination}, we reanalyze more than one regression per article, implying that we analyze a total of 18 standard regressions, 141 parametric estimates, and 141 semiparametric estimates. In comparison, \cite{gphk2024contamination} analyze 9 standard regressions, 30 treatment effects, and 90 parametric estimates.
After checking that the published tables are appropriately generated by the replication files, we use the \texttt{MulTE} package \citep{multe2024} to implement the parametric and semiparametric solutions discussed in Section \ref{sec:solutions}. To implement our proposed semiparametric approach, we generate each spline term separately and include it in the regression specification as additional covariates. For brevity, we focus our attention on spline specifications (Equation \eqref{eq:spline-def}) with order $r = 3$ and $J = 2$ knots, where the knots are the 33rd and 67th percentile of each covariate.
\section{Results}\label{sec:results}
Section \ref{sec:specific-analysis} reports the detailed results regarding the work of \cite{weisburst2019police} while Section \ref{sec:general-analysis} summarizes the results of all replicated regressions. Appendix \ref{ap:tables} reports the detailed results of all replicated articles.
\subsection{In-depth Analysis of Weisburst (2019)} \label{sec:specific-analysis}
To build intuition around the contamination bias present in Equation \eqref{eq:simple-reg} and the parametric and semiparametric solutions discussed in Section \ref{sec:solutions}, we start by discussing the estimates regarding Panel A in Table 2 of \citet{weisburst2019police}. This article analyzes police racial bias in use-of-force incidents in Dallas, Texas, between 2013 and 2016. To replicate this analysis, we implement the following standard regression:
\begin{equation} \label{eq:weisburst}
FR_{a} = \alpha + \beta_{B} \mathbbm{1}(\text{Black})_{a} + \beta_H \mathbbm{1}(\text{Hispanic})_{a} + \beta_O \mathbbm{1}(\text{Other})_{a} + X'_{a} \delta + \lambda_s + \lambda_b + \varepsilon_{a}
\end{equation}
where $a$ indexes an arrestee, the dependent variable indicates whether a use-of-force event occurred during the arrest, and the comparison group consists of white arrestees. Covariates $X_a$ include the arrestee's age and sex, average crime rates in the arrestee's police beat and semester (percentage of violent incidents, property crimes, drug use, prostitution arrests, traffic arrests, and disorderly arrests), and average characteristics of police officers in the arrestee's police beat and semester (age, sex, trainee status, experience, experience squared, black, hispanic, or other race). Regression \eqref{eq:weisburst} also includes fixed effects for both beat and semester ($\lambda_b$ and $\lambda_s$). Additionally, we implement the parametric and spline approaches discussed in Section \ref{sec:solutions}.
Table \ref{tab:weisburst} presents our main results for the setting discussed by \cite{weisburst2019police}. Column (1) corresponds to the published estimated coefficients, which are estimated based on Equation \eqref{eq:simple-reg} with the controls and fixed effects described in Equation \eqref{eq:weisburst}. Columns (2), (4) and (6) report the estimates for the linear parametric versions of the ATE solution (Equation \eqref{eq:ate}), the EW solution (Equation \eqref{eq:ew-model}) and the CW solution (Section \ref{SecCW}). Columns (3), (5) and (7) report the estimates for the semiparametric spline versions of the ATE solution (Equation \eqref{eq:spline}), the EW solution (Equation \eqref{eq:ew-spline}) and the CW solution (Section \ref{SecCW}). We report standard errors clustered at the police beat level in parentheses.
\begin{table}
\begin{center}
\caption{Full results for \citet{weisburst2019police}} \label{tab:weisburst}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{@{}l @{}cSSSSS SSSSSS @{}}
\toprule \hline
& \multicolumn{7}{c}{Full sample} \\
\cmidrule(lr){2-8}
Estimates & \multirow{2}{*}{$\hat{\beta}$} & \multicolumn{2}{c}{ATE} & \multicolumn{2}{c}{EW} & \multicolumn{2}{c}{CW}\\
\cmidrule(lr){3-4} \cmidrule(lr){5-6} \cmidrule(lr){7-8}
& & {Linear} & {Spline} & {Linear} & {Spline} & {Linear} & {Spline}\\
& \text{(1)} & \text{(2)} & \text{(3)} & \text{(4)} & \text{(5)} & \text{(6)} & \text{(7)} \\ \hline
Black & 0.172 & 0.342 & 0.540 & 0.109 & 0.061 & 0.246 & 0.660\\
& (0.274) & (0.396) & (0.952) & (0.267) & (0.289) & (0.292) & (2.281)\\
Hispanic & 0.043 & -0.330 & -0.413 & -0.496 & -0.265 & -0.466 & -0.378\\
& (0.394) & (0.395) & (0.499) & (0.341) & (0.341) & (0.289) & (1.418)\\
Other & 1.130 & 0.223 & -6.452 & 1.244 & -0.054 & 0.106 & -0.368\\
& (0.652) & (0.622) & (2.837) & (0.679) & (0.766) & (0.712) & (1.435)\\
\hline
Number of controls & & \multicolumn{1}{r}{256} & \multicolumn{1}{r}{310} & & & & \\
Sample size & & \multicolumn{1}{r}{7,488} & \multicolumn{1}{r}{7,488} & & & &
\\
\bottomrule
\end{tabular}
\end{center}
\emph{Notes:} This table reports the coefficients for \citet{weisburst2019police} for all estimators described in Section \ref{sec:solutions}. This table starts by replicating the coefficients from Panel A in Table 2 of the cited article. This article is an observational study. We report standard errors clustered at the police beat level in parentheses.
\end{table}
First, we interpret the original point estimates of Regression \eqref{eq:weisburst}, which are already measured in percentage points. In Column (1), we find that the use-of-force rate is 0.172 higher for black arrestees, 0.043 higher for Hispanic arrestees, and 1.130 higher for ``other'' arrestees in comparison to white arrestees.
Second, these estimates may suffer from contamination bias as explained in Section \ref{sec:bias}. The differences between Column (1)'s coefficients and the coefficients in all the other columns provide possible estimates for the contamination bias present in the standard regression. For example, when we compare the estimated coefficient for black arrestees in the linear specification for the ATE solution against the coefficient in Column (1), we estimate the contamination bias at $-0.170$ (i.e., $= 0.172 - 0.342$), implying that the bias might have a similar magnitude to the estimated coefficient.
We reach similarly concerning point estimates when we compare Column (1)'s coefficients and the coefficients in all the linear parametric specifications (Columns (2), (4), and (6)). However, these differences are small relative to the uncertainty around these estimated coefficients.
When we allow for nonlinear relationships between the potential outcomes and the covariates, the contamination bias problem is even more concerning. For example, for black arrestees in the ATE solution, we estimate the contamination bias at $-0.368$ (i.e., $= 0.172 - 0.540$). Moreover, the estimated contamination bias is usually larger in magnitude when we compare Column (1)'s coefficients and the coefficients in all the spline semiparametric specifications (Columns (3), (5), and (7)). However, these differences are still small relative to the uncertainty around these estimated coefficients.
Third, note that the spline specification substantially increases the number of control variables. While the linear specification uses 256 control variables, the more flexible spline specification uses 310. This extra flexibility comes at the cost of much larger estimated standard errors in Columns (3), (5), and (7) in comparison to Columns (2), (4), and (6).
The overall message of this replication exercise is that point estimates vary substantially across all specifications. Such large variation might reflect contamination bias in the standard regression approach, the extra flexibility that allows spline specification to capture nonlinear relationships, or the large statistical uncertainty in this empirical setting. To have a better understanding of these issues, we analyze all our replicated regressions in the next section.
\subsection{Analysis of all Selected Papers} \label{sec:general-analysis}
We now present the results for all regressions shown in Table \ref{tab:articles}\footnote{\label{FtBur}In our estimations, we removed three outliers. The estimates are the ATE estimates of Moral Incentives, Cash Rebate, and Religious Placebo in \citet{bursztyn2019moral}.
The estimates had either abnormally large normalized coefficients or standard error ratios.}. Specifically, we analyze a total of 18 standard regressions, 141 parametric estimates, and 141 semiparametric estimates. Given the large number of estimated coefficients, we provide a broad discussion of their point estimates and standard errors.
To allow comparison across papers and against the standard OLS regressions, we implement the normalization procedure proposed by \citet{gphk2024contamination}. We first multiply the estimated coefficient of our six solutions to contamination bias (Section \ref{sec:solutions}) by the sign of the original regression coefficient and then divide it by the standard error of the OLS coefficient. Mathematically, the normalized coefficient is given by $\hat{\beta}_{Norm} = \text{s}(\hat{\beta}_{OLS}) \cdot \hat{\beta}_{TE} / \text{SE} (\hat{\beta}_{OLS})$,
where $\hat{\beta}_{TE}$ is the estimate for one of our six solutions to contamination bias (Section \ref{sec:solutions}) and $\text{s}(\hat{\beta})$ is a function that returns 1 if $\hat{\beta} > 0$, $-1$ if $\hat{\beta} < 0$ and 0 if $\hat{\beta} = 0$.
After normalizing all estimated coefficients, we take the difference between the spline-based ones and the linear ones. Specifically, for each effect of interest, we compute $\hat{\beta}_{Norm Dif} = \hat{\beta}^{Spline}_{TE} - \hat{\beta}^{Linear}_{TE}$. Due to the previous normalization procedure, $\hat{\beta}_{Norm Dif}$ is comparable across all regressions and provides an easy-to-interpret measure of the distance between the two point estimates. Figure \ref{fig:histogram-full} contains a histogram of $\hat{\beta}_{Norm Dif}$ while Figure \ref{fig:histogram} shows a zoomed-in version of this histogram, which focuses on normalized differences between $-5$ and 5.
\begin{figure}[!htb]
\begin{center}
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/hist_differences_rev5_EN.png}
\caption{Histogram of the Difference (Spline - Linear) of Normalized Treatment Effects - Full Sample}
\label{fig:histogram-full}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.48\textwidth}
\includegraphics[width=\textwidth]{Figures/hist_differences_zoom_rev5_EN.png}
\caption{Histogram of the Difference (Spline - Linear) of Normalized Treatment Effects - Zoomed In Version}
\label{fig:histogram}
\end{subfigure}
\end{center}
\caption{Histogram of the Difference (Spline - Linear) of Normalized Treatment Effects}\label{FigHist}
\textit{Notes:} This figure presents histograms of the difference of normalized coefficients, $\hat{\beta}_{NormDif}$, for each treatment effect estimator in Section \ref{sec:solutions}. The normalized coefficient is given by $\hat{\beta}_{Norm} = \text{s}(\hat{\beta}_{OLS}) \cdot \hat{\beta}_{TE} / \text{SE} (\hat{\beta}_{OLS})$, where $\text{s}(\hat{\beta})$ is the sign function and $\hat{\beta}_{TE}$ is one of the estimators. The difference between the normalized coefficients is defined as $\hat{\beta}_{NormDif} = \hat{\beta}^{Spline}_{TE} - \hat{\beta}^{Linear}_{TE}$. Figure \ref{fig:histogram} focuses on normalized coefficients between $-5$ and 5 while Figure \ref{fig:histogram-full} uses the full sample after excluding the outliers mentioned in Footnote \ref{FtBur}.
\end{figure}
Initially, note that linear parametric estimates and semiparametric spline estimates are not always similar to each other. If they were, the histograms in Figure \ref{FigHist} would be a spike at 0, since we divide both types of estimates by the same denominator in the normalization process. This result suggests that allowing for nonlinear relationships between the potential outcomes and the covariates may impact the estimated treatment effect in real empirical settings.
Next, we must ask whether these differences between linear parametric estimates and semiparametric spline estimates are substantial. To answer this question, we can benchmark our normalized differences against two possible thresholds. First, we measure these differences in standard errors of the original OLS estimate. For example, a difference greater than 0.2 might be regarded as large since it implies that the difference between the parametrically and semiparametrically estimated coefficients is the size of one-fifth of the original OLS standard error. Importantly, almost 30\% (42/141) of our estimated differences are above this threshold. Second, we can benchmark these differences against the standard normal distribution. For example, we find that 11 estimated differences exceed the 95th percentile of the standard normal, i.e., 7.8\% of our estimated differences are larger in absolute value than 1.645. These two benchmarks suggest that there are substantial differences between the linear parametric estimators proposed by \cite{gphk2024contamination} and the semiparametric spline estimators suggested in Section \ref{sec:solutions}.
Lastly, we must ask whether allowing nonlinear relationships in our spline specifications comes at the cost of increased statistical uncertainty. To answer this question, we analyze the ratio between the estimated standard error of all our six solutions to contamination bias (Section \ref{sec:solutions}) and the estimated standard error of the original OLS regression (Equation \eqref{eq:simple-reg}). Mathematically, we calculate $\sfrac{SE_{TE}}{SE_{OLS}}$. Table \ref{tab:quantiles} shows the quantiles of these six ratios. Each pair of rows is associated with one of the three solutions (ATE, EW, or CW) and, within each pair, the first row is associated with semiparametric spline estimators while the second row is associated with linear parametric estimators.
First, we compare the same solution using linear parametric estimators versus semiparametric spline estimators. Unsurprisingly, almost all of the percentiles in the spline rows are larger than the percentiles in the linear rows.
Second, and most interestingly, most estimated standard errors for the estimators that address contamination bias are not much larger than the estimated standard error of the usual regression in Equation \eqref{eq:simple-reg}. For example, the 50th percentile is around 1 for all specifications, implying that the estimated standard errors of half of the coefficients for all six solutions in Section \ref{sec:solutions} are smaller than the estimated standard errors for the usual regression in Equation \eqref{eq:simple-reg}. Additionally, the highest standard error of the EW solution is only 24\% bigger than the standard error of its comparable OLS estimator. These results imply (i) that adopting approaches that address contamination bias concerns may not lead to increases in statistical uncertainty and (ii) that allowing nonlinear relationships between potential outcomes and covariates may not lead to increased statistical uncertainty.
\section{Conclusion}\label{sec:conclusion}
Multi-valued discrete treatments are widespread in observations studies and randomized controlled trials. \cite{gphk2024contamination} shows that standard regression approaches do not estimate a proper weighted average of treatment effects, presenting contamination bias. They discuss three alternative estimators that avoid this bias and implement their parametric versions.
In this paper, we deepen their analysis by explaining and implementing three semiparametric versions of their solutions to contamination bias. We find that (i) contamination bias might be a relevant problem in many empirical settings, (ii) there exist substantial differences between the point estimates of linear parametric estimators and semiparametric spline estimators, and (iii) adopting flexible semiparametric solutions to contamination bias may not lead to increased statistical uncertainty. Because our proposed spline estimators are easy to implement using the statistical package provided by \cite{multe2024}, we recommend that empirical researchers verify the robustness of their conclusions to the use of semiparametric tools that address contamination bias.
As future avenues of research, it is important to explore how machine learning estimators \citep{Cetin2025} behave in empirical settings with multi-valued discrete treatments.
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