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Misspecification-Averse Estimation
Researchers in economics are often concerned that their models are wrong, and about the consequences for estimation and inference. A large literature considers the problem of estimation and inference under misspecification with different optimality criteria, including worst-case mean squared error Huber1964, BonhommeWeidner2022 and confidence interval length ArmstrongKolesar2020. Other work considers the related problem of optimal decision-making in settings with misspecification concerns HansenSargent2001, HansenSargent2008,DuchiNamkoong2021DRO,GaoKleywegt2023DRO, CerriaViolioetal24. These contributions adopt different objective functions, and the choice among criteria has substantive consequences for the resulting procedures. A natural question is thus what objective a researcher should use when their model (and in particular, the model-implied mapping from parameters to data distributions) may be wrong, and what different choices of criteria imply for the resulting ranking of decision rules.
The question of how to evaluate decisions in the presence of model uncertainty is a central concern of the microeconomic theory of choice under ambiguity. That literature provides axiomatic foundations for classes of preferences, linking axioms on preferences to functional representations of the decision-maker's objective GilboaSchmeidler1989, maccheroni2006ambiguity, CerriaViolioetal24. This paper applies these tools to the problem of choosing estimation criteria when the model may be misspecified, connecting axiomatic foundations for ambiguity-averse preferences to the design of statistical procedures.
Our first contribution is to introduce constrained multiplier preferences as an optimality criterion for estimation under misspecification. Under this criterion, a researcher evaluates an estimator by its worst-case expected loss over distributions that satisfy constraints encoding what is known about potential misspecification, penalized by a Kullback-Leibler divergence term that reflects the difference from the baseline model. We start from a broad class of misspecification-averse preferences axiomatized by CerriaViolioetal24, which nests many different misspecification-robust criteria. We then discuss additional axioms that select specific subclasses within this family: constraint preferences BonhommeWeidner2022, which are characterized by the certainty independence axiom of GilboaSchmeidler1989; multiplier preferences HansenSargent2001, characterized by the sure thing principle following strzalecki2011axiomatic; and the constrained multiplier class, which combines constraint and multiplier features and whose axiomatic characterization is new. Building on results from the generalized empirical likelihood literature, we further show that when the constraints can be expressed as moment conditions, the constrained multiplier preference has a computationally tractable dual representation. The moment constraints allow the researcher to express partial trust in the model: for instance, one may trust the model's implications for certain moments of the data, but not the full data distribution.
Our second contribution is to derive optimal estimators for this criterion. We prove a local asymptotic minimax theorem that extends the classical lower bound of Hajek1972 to the constrained multiplier objective: for convex loss, no sequence of estimators can achieve worst-case risk below our bound. Obtaining this bound requires deriving a novel limit experiment that incorporates both misspecification concerns and moment information. We then characterize estimators that attain this bound. They take the form of Bayes rules under an exponentially tilted likelihood that incorporates the moment constraints. Under squared error loss, the structure simplifies: when no moment constraints are imposed, or when only the mean of a vector of sample average moments is constrained, the maximum likelihood estimator is optimal; when higher moments of the sample average moment vector are constrained, the optimal rule linearly adjusts the MLE using the sample average moments.\footnote{Contemporaneous and independent work by adusumilli2026 derives related asymptotic results for worst-case Bayes risk in the case with no constraints on misspecification, likewise showing that the rules which are asymptotically optimal under the model remain optimal under misspecification concern. His results cover parametric and semiparametric estimation with general symmetric and quasi-convex loss functions, as well as treatment assignment problems.} We further show that plug-in finite-sample analogs of the optimal rules in the limit experiment are asymptotically optimal under regularity conditions. Our results imply that the common practice in economics of specifying a full parametric model but conducting estimation based solely on a subset of model-implied moments due to misspecification concern andrews2025purpose is suboptimal absent extreme misspecification aversion.
Optimal estimation requires the researcher to specify their (scalar) degree of misspecification aversion since, absent such concern, conventional optimality results apply. Since it may sometimes be difficult for researchers to specify this preference parameter, following ArmstrongKlineSun2025 we close by exploring the scope for adapting to the degree of misspecification concern. Focusing on settings with a scalar parameter, a scalar moment condition, and squared error loss, we find that (as in ArmstrongKlineSun2025) two simple classes of decision rules attain nearly the same degree of adaptation as do much more flexible rules. The resulting simple adaptive rules may be attractive to researchers interested in applying our results but uncertain how to specify the misspecification-aversion parameter.
Section (ref) introduces the decision-theoretic framework and the axiomatization of constrained multiplier preferences. Section (ref) states the asymptotic risk bound. Section (ref) characterizes optimal estimators in the limit experiment and their feasible analogs, and Section (ref) studies adaptive rules. Appendix (ref) provides further detail on the axiomatic foundations of our approach, while the remaining proofs appear in Appendix (ref).
Consider a researcher who will observe data $X$ from a sample space $\mathcal{X}$. For an unknown parameter $\theta$ in a parameter space $\Theta$, this researcher must choose an action $a$ from a set of feasible actions $\mathcal{A}$. The researcher's objective is specified by a loss function $l:\mathcal{A}\times\Theta\to\mathbb{R}$, where $l(a,\theta)$ describes the loss from taking action $a$ when the true parameter is $\theta$.
Since the researcher does not observe $\theta$ directly, they must choose an action based on the data. A (randomized) decision rule $\delta:\mathcal{X}\to\Delta(\mathcal{A})$ maps data realizations to distributions over actions. A decision rule $\delta$ thus induces an expected loss that depends on $(\theta,X)$, \[L_\delta(\theta,X)=\int l(a,\theta)d\delta(a;X).\] We will take the induced loss functions $L:\Theta\times\mathcal{X}\to\mathbb{R}$ as our starting point and rank decision rules based on our preference over their induced losses $L_\delta$. Our approach is thus similar in spirit to Stoye2012, who takes risk (expected loss integrating over the distribution of the data) functions as a primitive to study the choice of decision rules. We work with loss rather than risk since risk depends on the distribution of the data, and we are interested in settings where this distribution may differ from that assumed by the model.
\paragraph{Example: Average Treatment Effects}
Suppose the researcher observes a sample of $n$ observations $X=(X_1,...,X_n)$ for $X_i=(Y_i,D_i)$, where $D_i\in\{0,1\}$ is a binary treatment and $Y_i\in\{0,1\}$ is a binary outcome. The unknown parameter $\theta=(\mu_0,\mu_1)$ collects the mean potential outcomes $\mu_d=E[Y_i(d)]$ in a population of policy interest (e.g. where the researcher is considering rolling out the treatment), and the target parameter is the average treatment effect (ATE) $\kappa(\theta)=\mu_1-\mu_0$. We consider loss $l(a,\theta)=(a-\kappa(\theta))^2$ and action space $\mathcal{A}=[-1,1]$.
One decision rule the researcher could consider is the (non-randomized) difference-in-means estimator $\hat\kappa_{DM}=\bar{Y}_1-\bar{Y}_0=\frac{\sum_i D_i Y_i}{\sum_i D_i }-\frac{\sum_i (1-D_i) Y_i}{\sum_i (1-D_i) }$, which induces loss $L_{DM}(\theta,X)=(\hat\kappa_{DM}-\kappa(\theta))^2$.\footnote{If either the treatment or control group is empty, define $\hat\kappa_{DM}=0$.} Another decision rule is the (randomized) half-sample difference-in-means $\hat\kappa_{DM,\frac{1}{2}}$, which drops half of the treatment and control observations, selected at random, and computes the difference-in-means over the remaining observations. This induces loss $L_{DM,\frac{1}{2}}(\theta,X)=E\left[\left(\hat\kappa_{DM,\frac{1}{2}}-\kappa(\theta)\right)^2|X\right]$ where the expectation is over the randomness induced by dropping observations. $\triangle$
Standard decision-theoretic analysis proceeds by first characterizing preferences over a large menu of options (e.g. loss functions) including ones which may be infeasible, and then applying these preferences to select from the feasible set for a particular problem. This approach is useful for characterizing preferences because the feasible set in a given setting is often quite restrictive, and it is easier to characterize preferences when working with a larger menu of choices. For instance, constant loss functions that assign the same loss regardless of the state are used in many decision theory results, but will often be infeasible in estimation problems: if we consider an estimation problem under squared error loss $l(a,\theta)=(a-\kappa(\theta))^2,$ $L_\delta(\theta,X)$ will generally vary with $\theta$ so long as $\kappa(\theta)$ does.\footnote{By contrast, breza2025generalizability consider an estimation problem where the researcher may decline to report an estimate at a constant cost, in which case certain constant loss functions are feasible.}
We assume the researcher has a statistical model $\mathcal{Q}$, which they may not fully trust. Formally, let $\mathcal{Q}=\{Q_\theta:\theta\in\Theta\}$ denote the model, where each $Q_\theta\in\Delta(\mathcal{X})$ is a distribution for the data.\footnote{For simplicity, we further assume that $\mathcal{X}$ is Polish, that all distributions are defined on the Borel $\sigma$-algebra, and that functions discussed are Borel-measurable. Going forward, we suppress discussion of measure-theoretic details where possible.} Let $\mathcal{L}$ denote the set of all bounded loss functions $L:\Theta\times\mathcal{X}\to\mathbb{R}$.\footnote{As is standard in the decision theory literature, proofs of the sufficiency direction of our representation theorems (the direction where the axioms imply the representation) will proceed on the domain of simple (finite-valued) loss functions. Under an appropriate continuity axiom, such a representation admits a unique extension to bounded loss functions. See Appendix (ref) for details.} We take as primitive a family of preferences $\{\succsim_\Theta\},\{\succsim_\theta:\theta\in\Theta\}$ over loss functions in $\mathcal{L}$. Throughout, $L\succsim L'$ means that $\succsim$ weakly prefers $L$ to $L'$. The preference $\succsim_\Theta$ is the researcher's overall ranking of loss functions, and is the object we ultimately seek to characterize. The auxiliary preferences $\{\succsim_\theta\}_{\theta\in\Theta}$ represent the researcher's evaluations given parameter value $\theta$; we axiomatize representations for these first and then impose additional axioms to derive the desired representation for $\succsim_\Theta$. Intuitively, $\succsim_\theta$ represents the researcher's preference when they are certain that $\theta$ is the true parameter value but remain concerned that the data distribution $Q_\theta$ implied by their model may be misspecified. We therefore refer to $\{\succsim_\theta\}_{\theta \in \Theta}$ as conditional preferences. While we take these as primitive in the main text for ease of exposition, in Appendix (ref) we show how they may be derived from the researcher's actual preference over loss functions $\succsim_\Theta$.
By misspecification, we will mean that the data $X$ are distributed according to $P$ which differs from the distribution $Q_\theta$ implied by the true $\theta$. Importantly, we assume the true value of $\theta$ remains well-defined even when the statistical model is wrong. This is natural for parameters like causal effects and counterfactuals which remain well-posed even when the model the researcher uses to estimate them is incorrect, but is less natural for e.g. parameters in a parametric utility, for which it is difficult to define a “true value” absent correct specification.\footnote{We could weaken this assumption to instead require only that some function $\kappa(\theta)$ that enters the loss, $l(a,\theta)=\tilde{l}(a,\kappa(\theta)),$ have a model-agnostic definition, where the model implies a set of data distributions $\mathcal{Q}(\kappa^*)=\{Q_\theta:\theta\in\Theta,\kappa(\theta)=\kappa^*\}$ compatible with a given value $\kappa^*$. In particular, observe that for any function $f(Q_\theta)$, we have $\sup_{\theta\in\Theta}f(Q_\theta)=\sup_{\kappa^*}\sup_{Q\in\mathcal{Q}(\kappa^*)}f(Q),$ which may be used to reformulate our results in this format. While this is a weaker assumption on the interpretation of $\theta,$ it leads to substantially heavier exposition for some of our results, so we impose the stronger condition that the full vector $\theta$ has a model-free interpretation, or equivalently that $\kappa$ does and that $\kappa(\cdot)$ is invertible.} It also rules out the case where $\theta$ is defined as a statistical functional $\theta:\mathcal{Q}\to\mathbb{R}^p,$ since in this case the model is either well-specified (when $P\in\mathcal{Q}$) or $\theta$ is undefined (when $P\not\in\mathcal{Q}$). In the terminology of andrews2025purpose, we thus consider the problem of misspecification in the context of an econometric model described by the pair $(\theta,P),$ rather than a statistical model described solely by the distribution $P$ of the data.
\paragraph{Example: Average Treatment Effects, Continued}
Suppose the researcher's model posits that the data are generated by (i) drawing a random sample from a population of interest which (ii) satisfies the potential outcomes model with each unit's outcome depending only on their own treatment and (iii) running a randomized trial where treatment is independently assigned to each unit with probability $\frac{1}{2}$. Under these assumptions $Y_i=Y_i(D_i)$, where $D_i$ is independent of the potential outcomes $(Y_i(0),Y_i(1))$. Thus, under the researcher's model the observations $X_i=(Y_i,D_i)$ are i.i.d. draws from a multinomial distribution supported on $\{0,1\}^2$ with $E_{Q_\theta}[D_i]=\frac{1}{2},$ $E_{Q_\theta}[Y_i|D_i=0]=\mu_0,$ and $E_{Q_\theta}[Y_i|D_i=1]=\mu_1.$
There are many ways in which the researcher's model could be wrong. For instance, the population from which experimental participants are sampled could differ from the population of policy interest (in which case the marginal distribution of potential outcomes $Y_i(d)$ in the trial could differ from that implied by $\theta$). Alternatively, the experimental sample might not be drawn i.i.d., e.g. oversampling the friends of early experimental participants due to recruitment through social networks (in which case the outcomes would not be independent across units). Finally, treatment might not be assigned as prescribed by the protocol, e.g. treating participants with lower baseline outcomes with higher probability (in which case treatment would not be independent of potential outcomes). If we combine these possibilities, any value of $\theta=(\mu_0,\mu_1)$ could in principle be compatible with any distribution $P\in\Delta(\{0,1\}^{2n})$ for the observable data $X$.
For the forms of misspecification discussed above, $\theta$ remains well-defined as the average potential outcomes in the target population, though it will not be identified absent restrictions on the possible misspecification. There are other forms of misspecification one could contemplate, e.g. spillovers across units, where the definition of $\theta$ becomes more delicate. If spillovers are present, for instance, does $\mu_0$ correspond to the average outcome when no unit is treated, or when a given unit is untreated while others are treated i.i.d. with probability $\frac{1}{2}$? For our analysis, we presume the researcher has adopted a definition for the “true” $\theta$ which remains well-posed under the forms of misspecification they contemplate, and wishes to choose among decision rules in a way which is robust to this misspecification concern. $\triangle$
Results by maccheroni2006ambiguity and CerriaViolioetal24 imply axioms on the preference relations $\{\succsim_\Theta\},\{\succsim_\theta:\theta\in\Theta\}$ which hold if and only if these preferences are represented by $V_\Theta$ and $\left\{V_\theta:\theta\in\Theta\right\}$ respectively, for
in the sense that \[ L\succsim_\Theta L' \iff V_\Theta(L)\le V_\Theta(L')\] \[ L\succsim_\theta L'\iff V_\theta(L)\le V_\theta(L'). \] In both representations, $c_\theta:\Delta(\mathcal{X})\to[0,\infty]$ is a convex, lower-semicontinuous function with $c_\theta(Q_\theta)=0$ for each $\theta\in\Theta$. Intuitively, the preference $\succsim_\theta$ evaluates loss functions $L$ by (i) focusing on their behavior at parameter value $\theta$ and (ii) considering the penalized worst-case risk, which averages over $P$ but then subtracts off $c_\theta(P)$, effectively penalizing expected loss under data distributions $P$ that the researcher finds less plausible. We further assume that the researcher finds the model-implied distribution $Q_\theta$ at least as plausible as any other. The preference $\succsim_\Theta$ does the same for each $\theta,$ but further takes the worst case over all $\theta$. As an immediate consequence, $V_\Theta$ implies the (penalized) worst-case risk bound \[ \int L(\theta,x)dP(x) \le V_\Theta(L)+c_\theta(P) ~~ \forall \theta\in\Theta,P\in\Delta(\mathcal{X}) \] which controls, uniformly over $(\theta,P)$, how large the risk may be. Choosing a loss function which minimizes $V_\Theta(L)$ is thus the same as minimizing an upper bound on the risk.
\paragraph{Example: Average Treatment Effects, Continued}
Consider a researcher choosing between the Horvitz-Thompson estimator $\hat\kappa_{HT}=\frac{1}{n}\sum_i 2(2D_i-1)Y_i$ and the difference-in-means estimator $\hat\kappa_{DM}=\bar{Y}_1-\bar{Y}_0$. Provided the researcher's preferences are represented by (ref), they weakly prefer the difference-in-means estimator if and only if \[ V_\Theta(L_{DM})=\sup_{\theta\in[0,1]^2}\sup_{P\in\Delta(\{0,1\}^{2n})}\int \left(\hat\kappa_{DM}-\kappa(\theta)\right)^2dP(x)-c_\theta(P)\le\] \[ \sup_{\theta\in[0,1]^2}\sup_{P\in\Delta(\{0,1\}^{2n})}\int \left(\hat\kappa_{HT}-\kappa(\theta)\right)^2dP(x)-c_\theta(P)=V_\Theta(L_{HT}). \] Moreover, the optimized value $V_\Theta(L_{DM})$ of the left-hand side provides a bound on how quickly the performance of the difference in means estimator moves away from that prescribed by the researcher's model. $\triangle$
The representations (ref) and (ref) follow from arguments in maccheroni2006ambiguity and CerriaViolioetal24, but our framing of the problem (e.g. choice domain, preferences) differs from theirs. For completeness, and to aid interpretation for readers more used to working with risk and loss functions than the conventional decision-theory setup, we thus provide an axiomatization for our choice domain, along with proofs, in Appendix (ref).
The axiomatization discussed in Appendix (ref) leaves the form of the penalty function $c$ unspecified. Different choices of $c$ correspond to different attitudes toward misspecification, and restricting these attitudes narrows the class of penalties. We next discuss three important cases: constraint preferences, multiplier preferences, and a novel class of constrained multiplier preferences that combines the other two. In each case, we discuss the additional axioms that distinguish these preferences from the broader class characterized (ref) and (ref).
A particularly simple penalty arises when the researcher requires that the true distribution $P$ lie in some set but imposes no further penalties. Consider a closed, convex set $\mathcal{P}_\theta\subseteq\Delta(\mathcal{X})$ of distributions (the “ambiguity set”) that the researcher treats as plausible under parameter value $\theta$. We require that $Q_\theta \in \mathcal{P}_\theta$ so that the researcher finds their model plausible. Preferences of the form
\[ V_\Theta(L)=\sup_{\theta\in\Theta}\sup_{P\in\mathcal{P}_\theta}\int L(\theta,x)dP(x) \]
evaluate loss functions based on their worst-case performance over both parameters $\theta$ and distributions $P$ in the ($\theta$-specific) ambiguity set. Criteria of this form have a long history, and were discussed by e.g. Huber1964 in the context of contamination neighborhoods, and more recently by BonhommeWeidner2022 for estimation of misspecified economic models, while similar misspecification-aware setups are considered for confidence set construction by ArmstrongKolesar2020 and christensen2023sensitivity.\footnote{Note that while the class $\mathcal{P}_\theta$ considered in e.g. BonhommeWeidner2022 need not be convex, $V_\Theta(L)$ and $V_\theta(L)$ are unchanged if we replace all non-convex $\mathcal{P}_\theta$ by their convex hull.} Much of the large and active literature on distributionally robust optimization DuchiNamkoong2021DRO,GaoKleywegt2023DRO,MontielOleaetal2026 can also be cast in this form.\footnote{For instance, the problem of DuchiNamkoong2021DRO can be cast into our setting by taking $X=(X_1,...,X_n)$ to represent $n$ i.i.d. draws $X_i\in \mathcal{X}_0$, $\Theta\subseteq\Delta(\mathcal{X}_0),$ $Q_\theta=\times_{i=1}^n\theta=\theta^n$, $l(a,\theta)=E_\theta[l^*(a,X_i)]$ for some loss function $l^*,$ and $\mathcal{P}_\theta$ the convex hull of $\left\{P_0^n:P_0\in\Delta(\mathcal{X}_0),D_f(\theta|P_0)\le\rho\right\}$ for $D_f$ an $f$-divergence.}
\paragraph{Example: Average Treatment Effects, Continued}
Now suppose the experiment is conducted by two teams, with unit $i$ randomly assigned to team $C_i\in\{1,2\}$, where $\pi_j=P\{C_i=j\}$. The researcher's model maintains that outcomes do not depend on team assignment, so $E[Y_i(d)\mid C_i=j]=\mu_d$ for all $j$, and the likelihood pools data from both teams. However, the researcher has greater confidence in team 1's execution than in team 2's: for instance, team 2 may have deviated from the treatment assignment protocol, implemented the treatment less carefully, or measured outcomes differently. We may express this using constraint preferences.
Let $\mathcal{P}_\theta$ be the set of distributions which arise by (i) drawing $C_i$ i.i.d. with $P\{C_i=j\}=\pi_j$ (ii) drawing $(Y_i,D_i)|C_i=1$ as described by the researcher's model and (iii) drawing the remaining outcomes $\{(Y_i,D_i)|i\in\{1,...,n\}\text{ such that }C_i=2\}$ from some other distribution. This case lies strictly between full trust in the model ($\mathcal{P}_\theta=\{Q_\theta\}$) and complete agnosticism ($\mathcal{P}_\theta=\Delta(\mathcal{X})$): it requires that team 1's observations follow the model, while placing no restrictions (including independence) on team 2's observations. $\triangle$
The preference (ref) is a special case of (ref) with the penalty function
These preferences, which can be viewed as a version of GilboaSchmeidler1989 min-max preferences, are distinguished from the more general class of variational preferences by the Certainty Independence axiom. Certainty independence is stated using a constant loss, by which we mean a loss function $r\in\mathcal{L}$ that takes the same value $r\in\mathbb{R}$ for all $(\theta,x)$, and we slightly abuse notation by identifying constant functions with their value.
Intuitively, certainty independence requires that our preference over estimators not change if, with some probability independent of the data and parameter, we will be randomly switched to instead use an estimator with constant loss.
\paragraph{Example: Average Treatment Effects, Continued}
Again consider a researcher choosing between Horvitz-Thompson and difference-in-means. Now suppose that for each data realization this researcher is, with probability $1-\alpha$, randomized into instead using a noisy oracle $\hat\kappa_O=\kappa(\theta)+\varepsilon,$ where $\varepsilon\sim N(0,\sigma^2)$. This results in randomized procedures with induced losses $\alpha L_{HT}+(1-\alpha)\sigma^2$ and $\alpha L_{DM}+(1-\alpha)\sigma^2$. Certainty Independence requires that, for all $\alpha\in(0,1)$ and $\sigma^2$, the preference between these two randomized procedures be the same as the preference between the pure estimators $\hat\kappa_{HT}$ and $\hat\kappa_{DM}$. $\triangle$
While Proposition (ref) characterizes when the researcher's preferences have an (unpenalized) min-max form, the class of possible ambiguity sets $\mathcal{P}_\theta,$ and thus preferences, remains quite large. GhirardatoMarinacci2002 show that one can go further and infer the ambiguity sets $\mathcal{P}_\theta$ directly from $\succsim_\theta$.
To state this result, we must introduce some additional notation. For $(\theta,P)\in\Theta\times\Delta(\mathcal{X})$, define the subjective expected utility (SEU) preference $\succsim_{\theta,P}^{SEU}$ by \[L\succsim_{\theta,P}^{SEU} L' \iff \int L(\theta,x)dP(x)\le\int L'(\theta,x)dP(x).\] This is the preference of a decision maker who ranks decision rules based on their expected loss under $(\theta,P)$. Following GhirardatoMarinacci2002, $\succsim_\theta$ is more ambiguity averse than $\succsim_{\theta,P}^{SEU}$ if for all $L\in\mathcal{L}$ and $r\in\mathbb{R}$, \[L\succsim_\theta r \Rightarrow L\succsim_{\theta,P}^{SEU} r.\] In words, any time the preference $\succsim_\theta$ ranks a state-dependent loss $L$ as weakly better than a constant loss $r$, $\succsim_{\theta,P}^{SEU}$ must do so as well. Equivalently, $\succsim_\theta$ is “more cautious” than expected loss under $(\theta,P)$: it has a (weakly) higher bar for preferring uncertain losses to certain ones. GhirardatoMarinacci2002 establish that such comparisons identify $\mathcal{P}_\theta$.
In particular, since we have assumed that $Q_\theta \in \mathcal{P}_\theta$, Proposition (ref) implies that $\succsim_\theta$ is more ambiguity averse than $\succsim_{\theta,Q_\theta}^{SEU}$.
\paragraph{Example: Average Treatment Effects, Continued}
Consider a researcher choosing between the difference-in-means estimator $\hat\kappa_{DM}=\bar{Y}_1-\bar{Y}_0$ and the noisy oracle $\hat\kappa_O=\kappa(\theta)+\varepsilon,$ $\varepsilon\sim N(0,\sigma^2)$. If the researcher's conditional preferences $\succsim_\theta$ are represented by (ref) and satisfy Certainty Independence, and thus have a constraint representation with $Q_\theta \in \mathcal{P}_\theta$ for each $\theta \in \Theta$, it follows that $L_O\succsim_\theta L_{DM}$ whenever $\sigma^2\le E_{Q_\theta}[(\hat\kappa_{DM}-\kappa(\theta))^2].$ $\triangle$
Constraint preferences compute the worst-case expected loss over $\mathcal{P}_\theta$, and thus do not privilege any particular distribution in this set. Since we have assumed the researcher has a model $\mathcal{Q}$, however, it also seems natural that under parameter value $\theta$ they might find distributions “close” to $Q_\theta$ more plausible than those “far” from $Q_\theta$, and thus wish to penalize some notion of distance from the base model. As has previously been observed many times HansenSargent2001, HansenSargent2008, Kullback-Leibler (KL) divergence is an especially convenient penalty for many purposes, and leads to \[ V_\Theta(L)=\sup_{\theta\in\Theta}\sup_{P\in\Delta(\mathcal{X})}\left\{\int L(\theta,x)dP(x)-\lambda\cdot \operatorname{KL}(P\|Q_\theta)\right\}, \]
where $\lambda>0$ and \[\operatorname{KL}(P\|Q)=\int\log\left(\frac{dP}{dQ}(x)\right)dP(x)\] when $P\ll Q$ (i.e. all events which are probability zero under $Q$ are also probability zero under $P$) and $\operatorname{KL}(P\|Q)=\infty$ otherwise. The parameter $\lambda$ controls the degree of concern for misspecification: as $\lambda\to\infty$, the preference converges to expected loss under $Q_\theta$, while as $\lambda\to 0$, the preference places more weight on worst-case scenarios. The multiplier risk index $V_\Theta$ is equivalent to the worst-case Bayes risk used by adusumilli2026 for asymptotic analysis of point estimation and treatment assignment problems.
Results in the literature again characterize this penalty relative to the broad class $c_\theta(\cdot)$. strzalecki2011axiomatic shows a tight connection between the KL divergence and Savage's Sure Thing Principle. To state the Sure Thing Principle, for all $L,L'\in \mathcal{L}$ define a spliced loss equal to $L$ on an event $\mathcal{E}$ and $L'$ otherwise, \[ L_\mathcal{E}L'(\theta,X)=
. \]
The Sure Thing Principle requires that preferences between loss functions that agree on some set ($\mathcal{E}^c$, in this case) depend only on their values elsewhere. In our setting, this axiom has a natural connection to pretesting.
\paragraph{Example: Average Treatment Effects, Continued}
The experimental design implies $E_{Q_\theta}[D_i]=\frac{1}{2}$, which the researcher could test in order to check implementation fidelity. Suppose that, conditional on not rejecting the null $H_0:E_P[D_i-\frac{1}{2}]=0$, the researcher will choose between the Horvitz-Thompson estimator $\hat\kappa_{HT}$ and the difference-in-means estimator $\hat\kappa_{DM}$, while if the null is rejected they will use some alternative procedure. The Sure Thing Principle requires that the ranking between $\hat\kappa_{HT}$ and $\hat\kappa_{DM}$ under $\succsim_\theta$, conditional on the test not rejecting, not vary depending on what procedure is used when the test rejects. $\triangle$
The results of strzalecki2011axiomatic imply that the Sure Thing Principle holds if and only if $c_\theta(\cdot)$ is KL divergence relative to some centering distribution. Our requirement that $c_\theta(Q_\theta)=0$ further ensures that this centering distribution is $Q_\theta$ (see Appendix (ref) for an axiomatic justification). We further impose an axiom, adapted from lanzani2025supplement, which enforces that the misspecification-aversion parameter $\lambda$ is constant across $\theta$.
\paragraph{Example: Average Treatment Effects, Continued} Suppose the researcher is considering an estimator $\hat{\kappa}$ whose induced distribution over squared estimation errors $ (\hat{\kappa}-\kappa(\theta))^2$ does not depend on $\theta$. For example, consider the (infeasible) estimator $\hat{\kappa}$ which yields an estimation error of $1/2$ if the sample fraction of treated individuals exceeds $1/2$ and $0$ otherwise.\footnote{Note that in the limit experiment discussed in the next section, there do exist estimators whose model-implied distribution does not depend on the parameter.} Since the law of the sample fraction of treated individuals is $n^{-1}\text{Bin}(n,1/2)$ under $Q_\theta$ for all $\theta$, $\hat{\kappa}$ satisfies the property (ref) for all $\theta,\theta'\in\Theta$. $\triangle$
Loosely speaking, Monotone Continuity ensures that the representation only cares about countably additive probabilities. When $\mathcal{X}$ is finite, it has no bite.
Finally, say that an event $\mathcal{E} \subseteq \mathcal{X}$ is nonnull under $\succsim_\theta$ if there exist $L,L',M \in \mathcal{L}$ such that $L_{\mathcal{E}} M \succ_\theta L'_{\mathcal{E}} M$. To state our representation theorem, we make the mild assumption that $\mathcal{X}$ has at least three disjoint nonnull events for each $\succsim_\theta$, and that there exists $q\in(0,1)$ such that for each $\theta,$ there exists an event $\mathcal{E}_\theta\subseteq \mathcal{X}$ with $Q_\theta(\mathcal{E}_\theta)=q$.
Multiplier preferences treat all forms of misspecification symmetrically: deviations from the model are penalized solely based on their KL divergence. This is natural when the researcher has no view about which aspects of the model are more likely to fail. In many economic applications, however, researchers appear to have more confidence in some model predictions than in others. For such researchers, it is natural to combine both elements: a hard constraint ruling out certain forms of misspecification a priori, and a KL penalty governing concern about remaining misspecification within the ambiguity set.
To capture the resulting preferences, we introduce a novel class of constrained multiplier preferences, which rule out some DGPs a priori and then continuously penalize deviations within the ambiguity set using Kullback-Leibler divergence:
Mathematically, (ref) and (ref) correspond to special cases of (ref) and (ref), respectively, which take $c_\theta(\cdot)$ equal to the sum of the convex indicator for the set $\mathcal{P}_\theta$, as in constraint preferences, and the KL divergence from the model distribution $Q_\theta$, as in multiplier preferences.
\paragraph{Example: Average Treatment Effects, Continued}
In our discussion of constraint preferences we considered the case where $\mathcal{P}_\theta$ exactly pins down the distribution of the data collected by team 1 while imposing no constraints on the data from team 2. This represents extreme distrust of the second team, for instance treating it as equally plausible that this team adhered faithfully to the experimental protocol and that they fabricated the data wholesale. Constrained multiplier preferences accommodate the intermediate case where the researcher thinks the data from the second team may have a distribution different than that predicted by the model (in principle allowing any $P\in\mathcal{P}_\theta$ with $\operatorname{KL}(P\|Q_\theta)<\infty$) but continuously discounts distributions $P$ which are further from $Q_\theta$ as measured by KL divergence. $\triangle$
Our axiomatization of constrained multiplier preferences will build on the constraint and multiplier preference axiomatizations. Specifically, let $\succsim_{\theta}^C$ denote the preference the researcher would have if they (i) took as given that the true parameter is $\theta$ and (ii) were certain the true distribution lay in $\mathcal{P}_\theta$ but did not privilege any distribution in this set. Formally, we assume the preferences $\left\{\succsim_{\theta}^C:\theta\in\Theta\right\}$ satisfy the conditions of Propositions (ref) and (ref), and thus are constraint preferences with ambiguity set $\mathcal{P}_\theta$.
Similarly, let $\succsim_\theta^M$ denote the preference the researcher would have if they (i) took as given that the true parameter is $\theta$ and (ii) were more concerned with performance “close” to the model-implied distribution $Q_\theta$ but did not have hard constraints on the class of possible data distributions and (iii) satisfied the Sure Thing Principle. Formally, we assume the preferences $\left\{\succsim_{\theta}^M:\theta\in\Theta\right\}$ satisfy conditions of Proposition (ref), and thus are multiplier preferences with centering distribution $Q_\theta$.
We consider a researcher who both believes the constraints on misspecification imposed by $\succsim_{\theta}^C$ and is more concerned with DGPs close to $Q_\theta$ as in $\succsim_\theta^M$. The next axiom captures how their conditional preference given $\theta$, $\succsim_\theta$, combines these two elements.
The Indirect Pareto axiom requires that the conditional preference $\succsim_\theta$ strictly prefers the loss function $L$ to a constant loss $r$ if and only if $L$ can be decomposed into two parts, corresponding to the constraint and multiplier preferences respectively, each of which is strictly preferred to its share of the constant under the respective component preference. We show that this property characterizes the constrained multiplier preference.
One practically appealing feature of multiplier preferences is that they imply highly tractable dual representations.
This result shows that the ranking over losses implied by multiplier preferences is precisely the same as the ranking one would obtain by assuming the researcher's model $Q_\theta$ is correct but using the exponentiated loss $\exp\left(\frac{1}{\lambda}L\right)$. Consequently, one can compute optimal decision rules under multiplier preferences by applying standard arguments for the correctly-specified case to the transformed loss function.
Parallel convex duality arguments also imply a tractable dual for constrained multiplier preferences in some contexts. In particular, we focus on the case where the ambiguity set $\mathcal{P}_\theta$ can be written as the set of distributions satisfying a collection of moment equalities. For ambiguity sets of this form, convex duality arguments similar to those in the generalized empirical likelihood literature NeweySmith2004,Kitamura2009 imply a finite-dimensional dual for the constrained multiplier problem, even when $L$ may be unbounded.
The representation ((ref)) introduces Lagrange multipliers $\beta$ that enforce the moment constraints. The infimum over $\beta$ is the dual to the original problem of maximizing over $\mathcal{P}_\theta$. While this optimization problem does not in general have a closed form solution, it is convex and sufficiently tractable to enable both computation and theoretical analysis.
\paragraph{Example: Average Treatment Effects, Continued}
Consider the two-team ATE setting introduced above. The ambiguity set $\mathcal{P}_\theta$ restricts the joint distribution of the team-1 observations. Fully expressing these constraints using moment equalities would require per-observation restrictions (e.g. $E_P[(Y_i-\mu_0)(1-D_i)1\{C_i=1\}]=0$ for each $i$) and cross-observation restrictions (e.g. zero covariance between $(D_i-\frac{1}{2})1\{C_i=1\}$ and $(D_j-\frac{1}{2})1\{C_j=1\}$ for $i\ne j$), with the total number of moment equalities growing with the sample size.
To obtain a more parsimonious (but also more permissive) set of moment conditions, we may instead constrain a sample average moment function. Let
and define $\varphi(\theta,X)=\frac{1}{n}\sum_{i=1}^n\psi(\theta,X_i)$. The first two components capture the conditional mean outcomes among treated and control units in team 1, while the third and fourth elements restrict treatment and team assignment, respectively. The constraint $E_P[\varphi(\theta,X)]=0$ requires that the moment conditions hold on average across units, but does not constrain the marginal for a given unit and thus leads to a weakly higher worst-case risk. As we discuss in the next section we can strengthen the constraint by including higher moments of the sample average.
With the moment-equality constraint set $\mathcal{P}_\theta=\{P:E_P[\varphi(\theta,X)]=0\}$, Proposition (ref) applies, and the researcher's preference $\succsim_\Theta$ ranks estimators $\hat\kappa_\delta$ based on \[ \sup_{\theta\in\Theta}\inf_{\beta\in\mathbb{R}^4}\lambda\cdot\log\left(E_{Q_{n,\theta}}\left[\exp\left(\frac{1}{\lambda}(\hat\kappa_\delta-\kappa(\theta))^2-\beta'\varphi(\theta,X)\right)\right]\right),\] where we have used that the ranking is unchanged by monotone transformations of $V_\Theta$. $\triangle$
Before moving on, we briefly note that the analytic tractability of constrained multiplier preferences extends to settings with moment inequality, rather than equality, constraints. Specifically, if $\mathcal{P}_\theta=\{P:E_P[\varphi(\theta,X)]\le 0\}$, one can extend Proposition (ref) to show that \[ V_\theta(L)=\inf_{\beta\in\mathbb{R}^b}\sup_{\eta\in\mathbb{R}^b_+}\lambda\cdot\log\left(E_{Q_\theta}\left[\exp\left(\frac{1}{\lambda}L(\theta,X)-\beta'(\varphi(\theta,X)+\eta)\right)\right]\right)=\] \[ \inf_{\beta\in\mathbb{R}^b_+}\lambda\cdot\log\left(E_{Q_\theta}\left[\exp\left(\frac{1}{\lambda}L(\theta,X)-\beta'\varphi(\theta,X)\right)\right]\right).\]
While we derived constrained multiplier preferences in a finite-sample setting, in most interesting economic models finite-sample performance is analytically intractable. Following a foundational analytic approach for models without misspecification concern, we thus study local asymptotic performance instead. This section develops a local asymptotic minimax (LAM) theorem for constrained multiplier preferences, paralleling a classical result for the correctly specified case. We begin by introducing the asymptotic framework and reviewing the classical LAM theorem as a benchmark, then state our LAM result and discuss its implications.
Consider a sequence of estimation problems indexed by the sample size $n$. For sample size $n$ the researcher observes data $X^n=(X_1,\ldots,X_n)$, which under their model is drawn from a distribution in $\mathcal{Q}_n=\{Q_{n,\theta}:\theta\in\Theta\}$, where $\Theta\subseteq\mathbb{R}^p$.\footnote{While we focus on parametric models to tractably accommodate time-series applications, for i.i.d. data we expect that our analysis below, like that of adusumilli2026, will extend to semiparametric models.} For instance, if the researcher's model implies the data are i.i.d. then $Q_{n,\theta}=\times_{i=1}^n Q_{1,\theta}$ for $Q_{1,\theta}$ the distribution of a single observation, though the framework allows for more general dependence structures. We study performance when the true parameter is local to a base value $\theta_0$, taking the form $\theta_{n,h}=\theta_0+h/\sqrt{n}$ for a local parameter $h\in H=\mathbb{R}^p$, and we shorthand $Q_{n,\theta_{n,h}}=Q_{n,h}$. The loss function in the sample of size $n$ is \[ l_n(a,\theta)=\ell(\sqrt{n}(a-\kappa(\theta))), \] where $\kappa:\Theta\to\mathbb{R}^d$ is the target parameter and $\ell:\mathbb{R}^d\to\mathbb{R}$ is a fixed loss function. We are interested in the worst-case asymptotic performance of decision rule sequences $\delta_n$ over the local parameter space $H$.
The notion of local asymptotic normality, a foundational tool for characterizing asymptotic performance, formalizes a sense in which regular statistical models are asymptotically equivalent to Gaussian location experiments.
The LAN condition says that, in a local neighborhood of $\theta_0$, the log-likelihood ratio is asymptotically quadratic in the local parameter $h$ with Hessian $-I_0$, and thus resembles the log-likelihood of a normal model with Fisher information (and inverse variance) $I_0$. For i.i.d. data, LAN follows from standard differentiability conditions on the single-observation density van_der_vaart_asymptotic_1998.
For LAN models, the classical local asymptotic minimax theorem gives a lower bound on the worst-case local asymptotic risk of any sequence of estimators.
The right-hand side is the minimax risk in the Gaussian limit experiment where one observes $X\sim N(h,I_0^{-1})$ and wishes to estimate $Kh$ under loss $\ell$. The bound says no sequence of estimators can achieve lower worst-case local asymptotic risk than the finite-sample risk in this Gaussian problem. Under squared error loss, the bound reduces to $KI_0^{-1}K'$, achieved by any efficient estimator, including the maximum likelihood estimator. The classical LAM theorem thus provides a theoretical foundation for familiar efficiency claims.
\paragraph{Example: Average Treatment Effects, Continued}
Under the model described in Section (ref), the family $\{Q_{n,\theta}\}$ is LAN at any interior $\theta_0=(\mu_0,\mu_1)$ with Fisher information $I_0=\frac{1}{2}\operatorname{diag}(1/\sigma_0^2,1/\sigma_1^2)$ for $\sigma_d^2=\mu_d(1-\mu_d)$. Since $K=(-1,1)$, the classical LAM bound under squared error loss is $KI_0^{-1}K'=2(\sigma_0^2+\sigma_1^2)$, achieved by the difference-in-means estimator. $\triangle$
We now develop an analogous result for constrained multiplier preferences. The key additional ingredient is a moment function $\psi:\Theta\times\mathcal{X}_0\to\mathbb{R}^k$ satisfying $E_{Q_{n,\theta}}[\psi(\theta,X_i)]=0$ for all $n$ and $\theta$, where $\mathcal{X}_0$ denotes the sample space for a single observation $X_i$. This moment function encodes the researcher's beliefs about which aspects of the model are correctly specified. We assume that the researcher believes misspecification does not affect the first $M$ moments of the scaled sample average of $\psi$ evaluated at the true value of $\theta$. Including higher moments is potentially important, since the forms of misspecification allowed by constrained multiplier preferences include ones which change the dependence structure of the data. Thus, even if misspecification does not affect the marginal distribution of each observation, the distribution of sample averages could still change. Constraining the moments of the sample average up to order $M$ restricts such possibilities. We emphasize, however, that the moment conditions $\psi$ need not point-identify $\theta$ (e.g. the dimension of $\psi$ might be lower than that of $\theta$) so our results apply even in settings where there are too few moment functions to allow moment-based estimation.
Formally, let \[ Y_{n,h}=\frac{1}{\sqrt{n}}\sum_{i=1}^n\psi(\theta_{n,h},X_i), \] and for $m=(m_1,\ldots,m_k)\in\mathbb{N}_0^k$ with $1\le \sum_{s=1}^k m_s\le M$, define $\tilde W^m_{n,h}=\prod_{s=1}^k Y_{n,h,s}^{m_s}.$ Let $\tilde W_{M,n,h}$ collect $\tilde W^m_{n,h}$ over all such $m$, and define \[ W_{M,n,h}=\tilde W_{M,n,h}-E_{Q_{n,h}}[\tilde W_{M,n,h}]. \] The constraint set \[ \mathcal{P}^M_{n,h}=\left\{P\in\Delta(\mathcal{X}_0^n):E_P[W_{M,n,h}]=0\right\} \] consists of all data distributions that preserve the first $M$ moments of $Y_{n,h}$. This takes the form assumed in Proposition (ref) with $\varphi(\theta_{n,h},X^n)=W_{M,n,h}$, so the duality result applies.
\paragraph{Example: Average Treatment Effects, Continued} Returning to the two-team ATE example, recall the per-observation moment function (ref). The constraint set $\mathcal{P}^M_{n,h}$ enforces that misspecification not affect the first $M$ moments of $Y_{n,h}=\frac{1}{\sqrt{n}}\sum_i\psi(\theta_{n,h},X_i)\in\mathbb{R}^4$. For $M=1$, as discussed above this constrains the mean of $Y_{n,h}$, requiring that $\sum_{i=1}^nE_P[\psi(\theta_{n,h},X_i)]=0$. For $M\ge 2$, the constraints additionally involve moments of $Y_{n,h}$ that depend on pairwise covariances of $\psi(\theta,X_i)$ across observations, restricting the impact of misspecification of the cross-observation dependence within team 1's data. $\triangle$
To derive our LAM theorem we impose two assumptions. The first collects regularity conditions on the model and the moment function.
These conditions are standard: stationarity, a central limit theorem for the scaled moments, and smoothness of the moment function. We also restrict the loss.
Relative to the classical LAM theorem, Assumption (ref) strengthens quasi-convexity to full convexity, but allows asymmetric loss. Convexity ensures that randomized estimators cannot improve on deterministic ones, which simplifies our asymptotic results. Note that while the results of Section (ref) consider the case of bounded loss to simplify the axiomatic derivations, here we apply the resulting preferences to unbounded loss functions.
By Proposition (ref), in the sample of size ${n}$ the worst-case constrained multiplier risk of an estimator $\delta_n$ over a set of local parameters $I \subset H$ is \[ \sup_{h\in I}\sup_{P\in\mathcal{P}^M_{n,h}}\left\{\mathbb{E}_P\left[l_n(\delta_n(X^n),\theta_{n,h})\right]-\lambda \operatorname{KL}(P\|Q_{n,h})\right\}= \] \[ \sup_{h\in I}\inf_\beta\lambda\cdot\log\left(E_{Q_{n,h}}\left[\ell^*\left(\sqrt{n}\left(\delta_n(X^n)-\kappa(\theta_{n,h})\right)\right)\exp\left(\beta'W_{M,n,h}\right)\right]\right) \] for $\ell^*(u)=\exp\left(\frac{1}{\lambda}\ell(u)\right)$. Under the conditions above, if we consider the liminf as $n\to\infty$ and take the worst case over $I$, we obtain the following local asymptotic minimax bound.
Like the classical LAM theorem, Theorem (ref) shows that the local asymptotic risk, now considering the constrained multiplier risk, is lower bounded by the risk in a Gaussian limit experiment. The limit experiment now involves two statistics: $X$, which plays the same role as in the standard LAM theorem and corresponds to the asymptotic analog of the maximum likelihood estimator, and $Y$, which is the limit of the scaled sample average $Y_{n,0}$ and captures the information in the moment conditions. The finite-sample constraint set $\mathcal{P}^M_{n,h}$, which requires that the first $M$ moments of $Y_{n,h}\approx Y_{n,0}+\Psi h$ be preserved, maps to the constraint $E_P[W_{M,h}]=0$ in the limit experiment.
Our use of a fixed, sample-size independent $\lambda$ in Theorem (ref) is important for the result.\footnote{Formally, our use of a fixed $\lambda$ corresponds to using a single, sample-size independent constrained multiplier preference with state space $H\times\mathcal{X}$ where $\mathcal{X}=\mathcal{X}_0^\infty$. Our asymptotic results then concern choice from the sequence of loss menus $\mathcal{L}_n=\{l_n(\delta_n(X),\theta_{n,h}):h\in H,X\in\mathcal{X},\delta_n\in\mathcal{D}_n\}$ where the rules in $\mathcal{D}_n$ are constrained to depend on $X=(X_1,X_2,...)$ only through the first $n$ observations. Work in progress by Ricky Li explores the axiomatic implications of the other aspects of the LAM criterion, in particular the focus on the liminf and the use of finite index sets $I$.} Note, in particular, that if $Q_{n,\theta}$ and $P$ are both product measures (corresponding to i.i.d. sampling), then $\operatorname{KL}(P\|Q_{n,\theta})$ is equal to $n$ times the KL divergence for a single draw from each distribution. Thus, to attain a fixed KL divergence as $n\to\infty,$ the per-observation KL divergence must shrink at a $\frac{1}{n}$ rate. For many decision rules $\delta_n$ the worst-case distributions $P$ which attain or approximate the constrained multiplier risk will not be i.i.d., but as the above calculation suggests they will nevertheless often correspond to “local” misspecification. Thus, while the preferences we consider do not directly impose an assumption of local misspecification, a focus on the locally misspecified cases emerges naturally.
\paragraph{Example: Average Treatment Effects, Continued}
In the limit experiment (ref), $X\in\mathbb{R}^2$ corresponds to the maximum likelihood estimator for $(\mu_0,\mu_1)$ pooling data from both teams, while $Y\in\mathbb{R}^4$ captures the team-1 specific moment information. The matrices governing the limit experiment are \[I_0^{-1}=2\operatorname{diag}(\sigma_0^2,\sigma_1^2),\quad \Psi=
,\quad \Omega=\operatorname{diag}\left(\frac{\pi_1\sigma_0^2}{2},\,\frac{\pi_1\sigma_1^2}{2},\,\frac{\pi_1}{4},\pi_1(1-\pi_1)\right).\] Note that the third and fourth rows of $\Psi$ are zero, so the corresponding elements of $Y$ are uninformative on their own, but narrow the (asymptotic analog of the) ambiguity set. $\triangle$
When $M=0$ there are no moment constraints, so the statistic $Y$ plays no role: the adversary is free to distort its distribution. If $\ell$ is additionally symmetric around zero, the result reduces to the classical LAM theorem applied to the loss $\ell^*$. Specifically, in this case $\ell^*(u)=\exp(\ell(u)/\lambda)$ is symmetric and (quasi-)convex, so the classical LAM theorem (Proposition (ref)) applies for each value of $\lambda$. More generally, the choice of $M$ reflects what restrictions the researcher places on the forms of misspecification they consider. When the researcher is uncertain what value of $M$ to impose, the $M\to\infty$ limit provides a natural benchmark.
Taking $M\to\infty$ requires that misspecification not distort any moment of $Y_{n,h}$. Since the normal distribution is determined by its moments, this forces $Y$ to remain normally distributed in the limit. The risk bound then simplifies because the KL divergence decomposes: the adversary can distort the conditional distribution of $X$ given $Y$, but not the marginal of $Y$. The resulting bound has an intuitive form, with an inner conditional expectation of the exponentiated loss over $X\mid Y$, inside a logarithm, integrated over the marginal of $Y$.
Theorem (ref) and Corollary (ref) provide lower bounds on the risk achievable by any sequence of estimators under constrained multiplier preferences. In the next section, we characterize estimators which attain these bounds.
To derive optimal estimators, we begin by exploiting the invariance structure of the limit experiment to show that we can limit attention to (asymptotically) equivariant decision rules. We then characterize optimal rules for important special cases and show that their finite-sample analogs, based on plugging the MLE and moments into the limit-experiment optimal rule, are asymptotically optimal. Practically, these results show how researchers who believe certain implications of their model (captured by $\psi$ and $M$) and have a given degree of misspecification concern (captured by $\lambda$) can optimally combine likelihood and moment information. One implication of our results is that the widespread current practice (documented and discussed in e.g. andrews2025purpose) of focusing solely on moment-based estimation in settings where misspecification is a concern is unlikely to be optimal outside cases of extreme concern (i.e. $\lambda\to 0$).
The limit experiment (ref) exhibits an important invariance structure. Consider the group $G=\mathbb{R}^p$ acting on the sample space by $g\circ(X,Y)=(X+g,Y-\Psi g)$, on the action space $\mathbb{R}^d$ by $g\circ a=a+Kg$, and on the parameter space by $g\circ h=h+g$. These transformations leave the loss unchanged: $\ell((g\circ a)-K(g\circ h))=\ell(a-Kh)$ for all $a,h,g$. Following LehmannCasella1998, we say that a decision rule $\delta$ is equivariant if $\delta(X+g,Y-\Psi g)=\delta(X,Y)+Kg$ for all $g\in\mathbb{R}^p$. Let $\mathcal{D}^E$ denote the class of equivariant decision rules in the limit experiment.
While the constrained multiplier objective is nonstandard, we extend the classical Hunt-Stein theorem to show that for minimax purposes, it is without loss to limit attention to equivariant decision rules.
A useful consequence of equivariance is that the constrained multiplier risk does not depend on $h$. Specifically, for any $\delta^E\in\mathcal{D}^E$, transitivity of the group action implies \[ \sup_{P\in\mathcal{P}_{h}^M}\left\{E_{P}\left[\ell\left(\delta^E\left(X,Y\right)-Kh\right)\right]-\lambda \operatorname{KL}(P\|Q_h)\right\}=\sup_{P\in\mathcal{P}_{0}^M}\left\{E_{P}\left[\ell\left(\delta^E\left(X,Y\right)\right)\right]-\lambda \operatorname{KL}(P\|Q_0)\right\}. \] The minimax problem over equivariant rules thus reduces to \[ \inf_{\delta^E\in\mathcal{D}^E}\sup_{P\in\mathcal{P}_{0}^M}\left\{E_{P}\left[\ell\left(\delta^E\left(X,Y\right)\right)\right]-\lambda \operatorname{KL}(P\|Q_0)\right\}. \]
\paragraph{Example: Average Treatment Effects, Continued}
In the two-team ATE example, the group $G=\mathbb{R}^2$ shifts $(X,Y)$ by $(g,-\Psi g)$ and shifts the action by $Kg=(-1,1)g$. Equivariance requires that the estimator respond to a location shift in the data $(X,Y)$ by a parallel shift in the estimated treatment effect. The MLE $KX=(-1,1)X$ is equivariant, as is the (limit experiment analog of the) efficient GMM estimator $-K(\Psi'\Omega^{-1}\Psi)^{-1}\Psi'\Omega^{-1}Y$. $\triangle$
We next characterize optimal equivariant rules, treating the $M<\infty$ and $M=\infty$ cases in turn. By the duality of Proposition (ref) and invariance, the minimax problem for equivariant rules with $M<\infty$ becomes
Since this is a joint infimum over $(\delta^E,\beta)$, the order of minimization does not matter. For each fixed $\beta$, however, the minimization over $\delta^E$ corresponds to finding the best equivariant decision rule under an exponentially tilted likelihood, since \[ \inf_{\delta^E\in\mathcal{D}^E}E_{Q_0}\left[\ell^*\left(\delta^E\left(X,Y\right)\right)\exp\left(\beta'W_{M,0}\right)\right]=\] \[ \inf_{\delta^E\in\mathcal{D}^E}E_{Q_0}\left[\ell^*\left(\delta^E\left(X,Y\right)\right)\frac{\exp\left(\beta'W_{M,0}\right)}{E_{Q_0}\left[\exp\left(\beta'W_{M,0}\right)\right]}\right]E_{Q_0}\left[\exp\left(\beta'W_{M,0}\right)\right]. \] Theorem 6.5 of Eaton1989 proves that the best equivariant estimator in a location problem is the Bayes decision rule under the flat prior, from which we immediately obtain the form of the optimal rule for our problem.
The previous result provides the form of the optimal estimator for the case with $M<\infty$. In the case of $M=\infty,$ the risk of an equivariant rule takes the form
We have not found a closed-form characterization of the optimal rule in the $M=\infty$ case under general loss. However, one can show that the risk (ref) is convex in $\delta^E$, so if we restrict to a linear class of rules $\delta_\Gamma(X,Y)=\Gamma\phi(X,Y)$ for a finite-dimensional vector of basis functions $\phi(X,Y)$, the problem of finding the optimal rule in the class is likewise convex.
Using this convexity, it is straightforward to solve numerically for optimal rules in a given linear class provided one can tractably compute expectations under $Q_0$. For the important special case of squared error loss, however, we are able to go further and exactly characterize the optimal decision rule for all $M$.
Part (b) implies that increasing the number of moment constraints $M$ beyond two does not improve the optimal risk under squared error loss. Intuitively, we show that the best equivariant decision rule for $M=\infty$ is linear in $(X,Y)$, but for such rules the worst-case distribution $P^*$ is Gaussian, and a Gaussian distribution that matches the first two moments of $Y$ automatically matches all higher moments as well.
\paragraph{Example: Average Treatment Effects, Continued}
In the two-team ATE example, part (a) says the optimal rule when $M\le 1$ is $KX$, the MLE pooling data from both teams. Under $M\ge 2$, the researcher can exploit the team-1 moment conditions: the optimal rule linearly adjusts the MLE using $Z^I=Y+\Psi X$, which is the asymptotic analog to the moment conditions evaluated at the MLE. The optimal adjustment matrix $C^*$ can be computed numerically. $\triangle$
The results above characterize optimal decision rules in the limit experiment. In practice, however, the researcher observes only finite-sample data. We now show that plug-in finite-sample analogs of limit-experiment rules converge in both distribution and, under an integrability condition, risk.
The results in this subsection are local to the fixed base value $\theta_0$ used to define the limit experiment. Accordingly, the limit-experiment rule may depend on the derivative $K=\frac{\partial}{\partial\theta'}\kappa(\theta_0)\in\mathbb{R}^{d\times p}$ as well as on $\Sigma$. For a continuous decision rule $\delta^c(X,Y;K,\Sigma)$ in the limit experiment, equivariant in $(X,Y)$ for each fixed $(K,\Sigma)$, define a plug-in finite-sample analog as \[ \delta_n^c=\kappa(\hat\theta_n^{MLE})+\frac{1}{\sqrt{n}}\delta^c\left(0,\;\frac{1}{\sqrt{n}}\sum_{i=1}^n\psi(\hat\theta_n^{MLE},X_i);\;\hat K_n,\hat\Sigma_n\right), \] where $\hat\theta_n^{MLE}$ is the maximum likelihood estimator, $\hat K_n$ is a consistent estimator of $K$ (e.g. the derivative of $\kappa$ at $\hat\theta_n^{MLE}$), and $\hat\Sigma_n$ is a consistent estimator of $\Sigma$.
To obtain convergence of the (dual) constrained multiplier objective, we strengthen convergence in distribution to convergence of moments.
While the results in Section (ref) characterize optimal decision rules under constrained multiplier risk, the resulting rules depend on the misspecification-aversion parameter $\lambda$. While dependence of optimal rules on $\lambda$ is natural from a theoretical perspective (after all, a researcher entirely unconcerned with misspecification already knows the MLE is optimal), from a practical perspective it introduces a free parameter that a researcher interested in applying our methods must choose. In this section, we follow ArmstrongKlineSun2025 and examine, for a special case of our setting, whether there exist simple estimators that perform reasonably for many different values of $\lambda$.
In particular, we consider the special case of squared error loss $\ell=\|u\|^2$ where both the parameter $\theta$ and the moment condition $\psi$ are scalar, where we normalize $I_0=1$ and $\Psi=-1$, so the limit problem is fully described by $\Omega=\operatorname{Var}(Y)$.\footnote{As in ArmstrongKlineSun2025, one could more broadly interpret this setting as the limit problem when we are interested in a scalar target parameter and would like to combine an efficient but potentially misspecified estimator (represented by $X$) and a less efficient but more robust estimator (represented by $Y$). The distinction between our analysis in this section and that of ArmstrongKlineSun2025 then stems from their focus on a version of constraint preferences, vs. ours on constrained multiplier preferences.} We take $M=\infty,$ corresponding to the case where the researcher thinks misspecification does not affect the moments at all. Thus, the sole remaining decision for the researcher is the (unavoidable) choice of which moment function $\psi$ they think remains valid under misspecification.
To search for procedures which perform well across different values of $\lambda$, let $R_{\infty}^{\lambda}(\delta)$ denote the constrained multiplier risk of $\delta$ under misspecification-aversion parameter $\lambda,$ \[ R_{\infty}^{\lambda}(\delta)=\sup_{h\in\mathbb{R},\, P\in\mathcal{P}_{h}^\infty}\left\{E_{P}\left[\ell\left(\delta\left(X,Y\right)-Kh\right)\right]-\lambda \operatorname{KL}(P\|Q_h)\right\}. \] Following ArmstrongKlineSun2025 we consider the adaptive regret criterion \[ A_{\infty}(\delta) = \sup_{\lambda \in \Lambda} \frac{R_{\infty}^{\lambda}(\delta)}{\min_{\tilde{\delta}}R_{\infty}^{\lambda}(\tilde{\delta})}, \] which compares the performance of the rule $\delta,$ across values of $\lambda,$ to the performance of the family of $\lambda$-by-$\lambda$ optimal rules. By construction $A_{\infty}(\delta)\ge1,$ where if the adaptive regret is close to this lower bound it tells us that the rule $\delta$ is “nearly” optimal in a proportional sense uniformly across $\lambda$ values, and thus that if we opt to use $\delta$ rather than taking a stand on the “correct” $\lambda$ and using the resulting rule, the price of doing so (measured as the proportional increase in constrained multiplier risk) is not very high.
By Proposition (ref), we know that to solve for the $\lambda$-specific minimized risk $\min_{\tilde{\delta}}R_{\infty}^{\lambda}(\tilde{\delta}),$ it suffices to consider linear equivariant rules, greatly facilitating computation. Analogously, in the broader search for adaptive rules, we limit attention to equivariant rules $\delta^E\in\mathcal{D}^E$. Let us normalize $I_0=1$ and $\Psi=-1,$ and again define $Z^I=Y+\Psi X$ as the limit experiment analog of the sample average moments evaluated at the MLE, or (equivalently) as the difference between the GMM and ML estimators. One can show that $Z^I$ is a maximal invariant in the limit experiment, and thus that any equivariant decision rule can be written as \[ \delta^E(X,Y)=X+\gamma(Z^I). \] Thus, the problem of finding an equivariant estimator with a small adaptive risk is equivalent to picking a “good” $\gamma$.
We consider two simple parameterizations of $\gamma(\cdot)$ with a single tuning parameter, both of which ArmstrongKlineSun2025 find perform well according to their criterion. The first is the soft-thresholding estimator with \[ \gamma_{ST,\tau}(Z^I) = \max\left\{|Z^I| - \tau, 0\right\}\mathrm{sgn}(Z^I), \] which is closely related to the LASSO estimator Tibshirani1996. The second is the adaptive empirical risk minimization (ERM) estimator Magnus2002,deChaisemartin2020 with \[ \gamma_{ERM, \tau}(Z^I) = \frac{(Z^I)^2}{(Z^I)^2+\tau} Z^I. \] In each class, we choose the tuning parameter $\tau$ to minimize the adaptive risk $A_\infty$. To benchmark the performance of these simple parametric classes, we compare their adaptive risk to that of a flexible specification of $\gamma.$ In particular, motivated by Proposition (ref), we parameterize $\gamma$ as a linear spline (flattened for the extreme values of $Z^I$) and solve numerically for the optimal parameters via convex optimization.\footnote{Specifically, the maximization which defines $A_\infty$ preserves the convexity established by Proposition (ref), and we approximate the maximum over $\lambda$ by maximization over a finite, evenly spaced grid of $\log(\lambda)$ values from [-3,6].}
Figure (ref)(a) shows the resulting constrained multiplier risk functions for the $\lambda$-by-$\lambda$ optimal estimator, the adaptive linear spline estimator, and the two simple estimators, all for the case of $\Omega=2$. As expected, for small $\lambda$ the optimal risk is close to that of GMM (i.e. $\Omega=2$) while for large $\lambda$ it is close to that of MLE (i.e. $I_0=1$). In between, we see that the linear spline estimator has better performance than the simple estimators over an intermediate range of $\lambda$ values, but performs quite similarly for large and small $\lambda$. Figure (ref)(b) illustrates adaptive performance more directly, plotting the risk ratio $\frac{R_{\infty}^{\lambda}(\delta)}{\min_{\tilde{\delta}}R_{\infty}^{\lambda}(\tilde{\delta})}$ as a function of $\lambda$. Here we see that, consistent with the findings of ArmstrongKlineSun2025 for their optimality criterion, the simple soft-thresholding and ERM estimators perform nearly as well as the more complicated linear spline procedure, with an adaptive risk close to 1.5 for all procedures considered. Whether a 50% increase in constrained multiplier risk is an acceptable tradeoff for eliminating dependence on $\lambda$ seems to depend on one's priorities, but we find it encouraging that near-optimal adaptation is possible using simple combinations of the ML and GMM estimators.
Figure (ref) presents similar results when $\Omega = 6$. The simple estimators perform comparably to the linear spline. However, in the intermediate range of $\lambda$, the relative performance is less clear because we optimize only the maximum of the regret criterion.