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Aggregating many estimators using estimated weights
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{\LARGE\bf Aggregating many estimators using estimated weights}
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\begin{abstract}
Consider an increasing number of consistent estimators to be averaged when only estimated weights are available. The underlying parameter of interest can be identical across estimators (homogeneity) or not (heterogeneity). The contribution of the paper is threefold. First, it is shown that the interaction of the estimated weights with the estimators can generate specific bias terms. This constrains the number of estimators that can be aggregated when weight estimation is ignored in inference. Second, the paper proposes estimated adaptive weights, which allow for standard Gaussian inference in a uniform manner and are asymptotically optimal both under homogeneity and heterogeneity. Third, conditions ensuring the validity of the \citet{cochran1937} Q test of homogeneity with estimated variance are given.
\end{abstract}
\noindent
{\it Keywords:} distributed estimation, meta-analysis, common effect, random effects
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\section{Introduction}
\label{sec:introduction}
Models with many agents, goods, choices, or interactions may involve very large amounts of data for each observation; see \citet{varian2014}, \citet{einav2014}, and \citet{ng2017} for general discussions of the resulting applications, challenges, and methods. Optimising, or even evaluating, a likelihood or another objective function can then become computationally difficult. We consider settings in which the initial sample can be divided into meaningful subsamples, such as geographic or institutional units, within which the parameter of interest can be estimated feasibly. The considered class of subsample estimators is broad and includes standard econometric estimators. Final estimates are obtained by aggregating the subsample estimates using estimated weights. However, usual optimal weights are generally infeasible in full generality, especially when the estimator variance depends on the parameter in an unknown way which can vary across subsamples.
We propose a feasible adaptive weighting scheme, which is asymptotically optimal whether the underlying parameter is constant (homogeneous) or not (heterogeneous) across subsamples. Under suitable conditions linking the number of estimators to subsample size, the procedure allows for optimal inference across this class of aggregators. We further propose a test of parameter homogeneity, a property in line with external validity. Under homogeneity, the consistency rate of our aggregation procedure is given by the square-root of the total sample size, and falls down to the square-root of the number of subsamples under heterogeneity.
These results connect to several strands of the computer science, econometrics, and statistics literatures, particularly distributed estimation, meta-analysis, and panel data.
The distributed estimation literature has developed various methods which can be useful for econometrics in a rich data environment or with complex models. In this approach, several computers can work jointly to compute a gradient and a Hessian over subsamples, which are then summed to produce full sample counterparts used in the next iteration of descent or Newton-Raphson optimisation algorithms \citep{tsitsiklis1986,chu2006,duan2022}. However, the partial gradient and Hessian must be centralized at each iteration, which can be costly if there are many subsamples or if many iterations are needed to achieve convergence. It is also ineffective in the case of subsample-specific parameters, such as in the location choice analysis of \citet{wang2025}. In this case, it may be faster to run estimation on each computer and to aggregate the results to produce an average estimation. In the homogeneous context, this simple distributed method has been considered by \citet{mcdonald2010}, \citet{zhang2013,zhang2015}, \citet{rosenblatt2016}, and \citet{volgushev2019}, among others. While this branch of the literature typically assumes that the observations are i.i.d., \citet{gu2023} consider aggregation of estimators whose variance can vary across subsamples. They use optimal inverse-variance weights and a (sub)sample-splitting procedure to ensure that the weights are independent of the aggregated estimators.
However, the computer science literature typically assumes parameter homogeneity, i.e., that the parameter of interest remains constant across subsamples. This concern is addressed in the meta-analysis literature. Many meta-analyses from the economic literature rely on a Bayesian analysis of the \citet{rubin1981} hierarchical model which assumes that the heterogeneous parameter of interest is drawn from a normal distribution. See the review section of \citet{vivalt2020} who relates heterogeneity with the important issue of external validity. Some papers such as \citet{meager2019,meager2022} and \citet{bandiera2021} consider a small number of studies. Other works are closer to the rich-data environment of interest here. For instance, \citet{brown2024} and \citet{wolfson2019} consider between 500 and 700 estimates, while \citet{xue2020} analyse more than 12,000 ones. While papers consider the possibility of publication bias (see \citealp{andrews2019} and the references therein), the baseline model assumes that the estimators are unbiased. The estimator variance is regarded as known, justified by the belief that the subsample sizes are large enough ``to guarantee a good estimate of the true within-study variance, with little or no impact on the results'' as noted in \citet{bellio2016}. Among the few exceptions is \citet{liu2015} who consider estimated inverse-variance weights under parameter homogeneity and for a finite number of estimators, a framework inappropriate for distributed estimation but for which variance estimation does not affect limit distributions. By contrast, \citet{cochran1937} and \citet{cochran1953} consider a Gaussian framework where weight estimations affect finite-sample distribution.
This model is also estimated in a frequentist way in the statistical or medical literature. However, \citet{kulinskaya2014} point to a lack of well-grounded knowledge, noting that the ``[ubiquitous related Maximum Likelihood procedure] has been applied to thousands of data sets, despite the lack of theoretical or simulation studies to confirm when it does work, if ever''. While \citet{zeng2015} provide a theoretical study of the MLE based on an increasing number of estimators with known distribution up to the parameter of interest and the heterogeneity parameter variance, they restrict to the case where the parameter heterogeneity variance decreases with the average subsample size.
A third related area is panel-data econometrics. Estimation of nonlinear panel-data models with individual and time effects has attracted considerable interest, see e.g., \citet{fernandezval2016} and the references therein among many others. For linear panel data models, \citet{lu2023}
propose a Bootstrap approach for average individual and time parameters which is uniformly valid under both homogeneity and heterogeneity. For a nonlinear regression distribution specification, \citet{fernandezval2026} consider the standard mean of individual parameters, which are bootstrapped to obtain uniformly valid confidence intervals under both homogeneity and heterogeneity, without weights that would capture estimator variance heterogeneity.
As noted in \citet{vivalt2020} among others, parameter heterogeneity poses a threat to external validity. Testing for homogeneity was first considered by \citet{cochran1937}, in a Gaussian framework limited to a finite number of univariate estimators. \citet{swamy1970} applied a similar approach for testing homogeneity of a panel data linear regression slope with a fixed number of individuals and growing time dimension, while \citet{pesaran2008} allow both to grow. See also \citet{ritz2008} for related results.
Our results extend the literature in several directions. First, many references are tied to specific models and estimation methods, mostly related to linear regression. Our framework, presented in Section \ref{Frmwrk}, allows for general estimation methods which can be biased, thereby ensuring wide applicability in a large data framework. Second, potential heterogeneity in both the parameter and its variance is considered, calling for estimated weights when the variance is unknown. Indeed, averaging estimators without weighting becomes inefficient under homogeneity. The paper introduces \emph{adaptive}
weights in \eqref{Hatwstar}, which are asymptotically optimal under homogeneity and heterogeneity. Theorem \ref{Opt} establishes that these weights allow for standard Gaussian inference which is uniformly valid, producing optimal confidence intervals in the class of weighted estimators without resorting to the Bootstrap. This requires using a parameter-heterogeneity variance which vanishes fast enough under homogeneity. The proposed one \eqref{Checksig} extends \citet{cochran1954}, \citet{swamy1970} and \citet{hedges1985} to our general setup. Third, Theorem \ref{Homtest} shows the validity of the proposed homogeneity test. Fourth, all our results, starting with Proposition \ref{Expansion}, account for the interaction bias terms created by weight estimation, as well as a correlation bias term caused by the dependence of the population weights on the subsample parameters, see Proposition \ref{Randweightshet}.
The rest of the paper is organized as follows. Section~\ref{sec:framework} introduces the framework. Section~\ref{sec:main-assumptions} groups and discusses our main Assumptions while Section~\ref{sec:main-results} gathers our main results. Section~\ref{sec:simulation-application} reports the results of simulation experiments for instrument regression and discrete choice specifications together with an application to racial differences in mortgage denial across U.S. metropolitan areas. Section~\ref{sec:concluding-remarks} concludes. Appendix~\ref{app:additional-simulation-results} provides additional simulation results and Appendix~\ref{app:proof-section} gathers the proofs of our main results.
\section{Heterogeneity, estimation and weights \label{Frmwrk}}
\label{sec:framework}
\paragraph{Sample and Subsamples.}
Consider a sample $X_i$, $i=1,\ldots,N$, which is divided into disjoint groups of observations, $\left\{X_i,i \text{ in }\mathcal{G}_g\right\}$, indexed by $g$, $g=1,\ldots,G_N$, where the number of groups $G_N$ increases with the total sample size $N$. Each group $\mathcal{G}_g$ gathers $n_g$ observations, with $\sum_{g=1}^{G_N} n_g = N$.
This division into subsamples can be arbitrarily decided by the researcher, as in the standard distributed estimation scenario where the total sample size $N$ is large and using the full sample leads to computational difficulties when estimating a model of interest. Alternatively, the subsample structure can come from the data collection process. Each subsample may correspond to a different data source. For example, the subsample index $g$ may denote a geographic location, such as a city or a country.
The proposed framework also covers cases where subsamples are not necessarily observed. In meta-analysis, the subsample index $g$ can, for instance, identify a study based on the subsample $\left\{X_i,i \text{ in }\mathcal{G}_g\right\}$. Each study may report subsample estimation results, without disclosing all the underlying data.
The results in this paper are for the case where all the groups grow, i.e., $\inf_{1 \leq g \leq G_N} n_g$ diverges with $N$ in an (approximately) balanced way, with each subsample containing a similar proportion of the observations. That is, for some $C>0$ large enough and when $N$ grows,
\begin{align*}
\frac{1}{C} \leq \min_{1\leq g \leq G_N} \frac{n_g}{N/G_N} \leq \max_{1\leq g \leq G_N} \frac{n_g}{N/G_N} \leq C,
\end{align*}
meaning that all the $ n_g$'s are asymptotically of exact order $N/G_N$.
\paragraph{Parameter heterogeneity.}
The distribution of the observations can be identical or differ across groups. In the latter case, this may affect the multidimensional parameter of interest. As a consequence, the parameter of interest will be denoted from now on as $\theta_g$, $g=1,\ldots,G_N$, to allow for subsample variations. Our framework considers the following two scenarios:
\begin{equation}
\begin{array}{lll}
\theta_g = \theta & \text{for all $g=1,\ldots,G_N$,} & \text{(Homogeneity)} \\
\theta_g = \theta +\delta_g & \text{for all $g=1,\ldots,G_N$, } & \text{(Heterogeneity)} \\
\end{array}
\label{Par}
\end{equation}
where the deviations $\delta_g$ are centered, independent, and identically distributed random vectors with variance $\Sigma$. The homogeneity case is often referred to as the common-effect or fixed-effect model\footnote{With a singular ``effect'', as $\theta_g=\theta$ does not vary across studies.}, while the heterogeneity case corresponds to the random effects model in the meta-analysis literature. The homogeneity case is ubiquitous in distributed estimation papers, especially in those where numerical estimator computation proceeds by summing subsample gradients to compute a sample one. The heterogeneity scenario considered here is common in meta-analysis and panel-data econometrics. The parameter $\theta$ in \eqref{Par} then becomes the parameter of interest. Estimation of $\Sigma$ will also be considered.
\paragraph{Estimation of group parameters.}
Each $\theta_g$ can be estimated using an estimator $\widehat{\theta}_g$ based on the subsample
$\left\{X_i,i\in \mathcal{G}_g\right\}$ with $n_g$ observations.
We assume that $\widehat{\theta}_g$, $g=1,\ldots,G_N$ are independent given the $\theta_g$'s. However, we do not assume that the estimators are identical, as economic parameters can be estimated differently across studies in a meta-analysis.
The paper focuses on estimators that converge at the usual $\sqrt{n_g}$ parametric rates, assuming that the bias and variance of the $\widehat{\theta}_g$'s satisfy
\begin{align*}
\mathrm{Var} \left(\left. \widehat{\theta}_g \right| \theta_g \right)
=
\frac{\Omega_{n_{g},g} \left( \theta_g\right)}{n_g}
,
\qquad
\mathbb{E} \left[\left. \widehat{\theta}_g \right| \theta_g\right] = \theta_g + \frac{B_{n_g,g} \left(\theta_g\right)}{n_g^b}, \quad b>\frac{1}{2},
\end{align*}
where $\Omega_{n_{g},g} \left( \cdot \right)$ and $B_{n_g,g} \left(\cdot\right)$ respectively converge to bounded and non-vanishing $\Omega_{g} \left( \cdot \right)$ and $B_{g} \left(\cdot\right)$.
As a consequence, $\widehat{\theta}_g$ can be represented as
\begin{align}
\widehat{\theta}_g = \theta + \delta_g + \widehat{\varepsilon}_g + \frac{B_{n_g,g} \left(\theta_g\right)}{n_g^b},
\text{ with }
\left\{
\begin{array}{l}
\mathbb{E} \left[\delta_g\right] = 0 \text{ and } \mathrm{Var} \left(\delta_g\right) = \Sigma,
\\
\mathbb{E} \left[\widehat{\varepsilon}_g \left|\theta_g\right.\right] = 0 \text{ and } \mathrm{Var} \left[\widehat{\varepsilon}_g \left|\theta_g\right.\right] = \frac{\Omega_{n_{g},g} \left( \theta_g\right)}{n_g},
\\
\text{ implying in particular } \mathrm{Cov} \left(\delta_g,\widehat{\varepsilon}_g \right)=0,
\end{array}
\right.
\label{Representation}
\end{align}
where $\widehat{\varepsilon}_g = \widehat{\theta}_g - \theta_g - B_{n_g,g} \left(\theta_g\right)/n_g^b$, the pair $\left(\delta_g,\widehat{\varepsilon}_g\right)$ being independent across subsamples but not necessarily identically distributed.
For brevity, the bias rate exponent $b$ is common to all estimators. The restriction $b>1/2$ ensures that the estimator Mean Squared Errors (MSE hereafter) are dominated by the variance term when $n_g$ grows, as expected for asymptotically normal and centered $\widehat{\theta}_g$'s.
In addition to $\widehat{\theta}_g$, we assume that a consistent estimator $\widehat{\Omega}_g = \widehat{\Omega}_g \left(\widehat{\theta}_g\right)$ of $\Omega_{n_{g},g} (\theta_g)$ is available.
\paragraph{Weighted estimators.}
The weighted estimators studied in this paper are of the form:
\begin{align}
\widehat{\theta} \left(\widehat{W}\right)
=
\left[
\sum_{g=1}^{G_N} \widehat{W}_g
\right]^{-1}
\sum_{g=1}^{G_N} \widehat{W}_g \widehat{\theta}_g,
\quad
& \widehat{W}_g = \widehat{W}_g\left(\widehat{\theta}_g;X_i, 1 \leq i \leq N \right)= \widehat{W}_g\left(\widehat{\theta}_g\right), \quad g=1,\ldots G_N,
\label{Weighted}
\end{align}
where $\widehat{W}_g$ is a conformable square matrix estimating some population weight $W_g = W_g \left(\theta_g\right)$. Multiplying all $\widehat{W}_g$'s by an invertible matrix does not change the weighted estimator \eqref{Weighted}. Hence, a sequence of weights $\left\{\widehat{W}_g, 1\leq g \leq G_N \right\}$ can be identified with $\left\{\varPhi_N \cdot \widehat{W}_g, 1\leq g \leq G_N \right\}$, where $\left\{\varPhi_N,N\geq 1\right\}$ is a sequence of deterministic or stochastic invertible matrices. In particular, the constant weights $\widehat{W}_g = \widehat{W}$ correspond to equal weighting.
Note that \eqref{Representation} gives the decomposition:
\begin{align*}
\widehat{\theta} \left(\widehat{W}\right)
=
\theta +
\left[
\sum_{g=1}^{G_N} \widehat{W}_g
\right]^{-1}
\sum_{g=1}^{G_N} \widehat{W}_g \left(\delta_g + \widehat{\varepsilon}_g \right)
+
\left[
\sum_{g=1}^{G_N} \widehat{W}_g
\right]^{-1}
\sum_{g=1}^{G_N} \widehat{W}_g
\frac{B_{n_g,g} (\theta_g)}{n_g^b}.
\end{align*}
If $b$ is strictly larger than 1 or if the $B_{n_g,g} (\theta_g)$'s cannot be removed via a suitable bias correction, the weighted average of the bias components $B_{n_g,g} (\theta_g)/n_g^b$ can be viewed as a negligible remainder term with respect to the weighted average of the stochastic terms $\delta_g + \widehat{\varepsilon}_g$, at the price of some restrictions to be specified later on. Associated constant optimal weights are proportional to the inverse of the variance of $\delta_g + \widehat{\varepsilon}_g$, which is equal to
$
\Sigma
+
\mathbb{E} \left[\Omega_{n_{g},g} (\theta_g)\right]/n_g
$,
suggesting the optimal population weights:
$
W_g^{\sharp}=\left( \Sigma + n_g^{-1} \cdot \mathbb{E} \left[\Omega_{n_{g},g} \left(\theta_g \right)\right]\right)^{-1}
$.
The difficulty is that the expectation $\mathbb{E} \left[\Omega_{n_{g},g} (\theta_g)\right]$ can be difficult to estimate in full generality, so that implementing empirical versions of the weights is not feasible. For instance, if the i.i.d. $X_i$ of subsample $\mathcal{G}_g$ have the distribution $\mathcal{N} (\theta_g, \Omega_g (\theta_g))$ given $\theta_g$ where $\Omega_g (\cdot)$ is unknown and varies with $g$ due to unobserved heterogeneity, it is possible to estimate $\Omega_g (\theta_g)$ using the subsample variance. But recovering the full $\Omega_g (\cdot)$ as necessary to estimate $\mathbb{E} \left[\Omega_{n_{g},g} (\theta_g)\right]$ may be out of reach due to subsample dependence. Another example uses a discrete choice model $Y_i = \mathbb{I} \left(U_i \leq X_i \theta_g\right)$ for all $i$ in $\mathcal{G}_g$, where the $U_i$ are i.i.d. with a logistic distribution, independent of the $X_i$'s and $\theta_g$'s. In this model, the asymptotic variance of the MLE of $\theta_g$ is, $\Lambda(\cdot)$ being the Logistic function,
\begin{align*}
&\Omega_g (\theta_g)
= \mathbb{E} ^{-1}\left[ \left. \Lambda \left(X_i \theta_g\right) \left(1- \Lambda \left(X_i \theta_g\right) \right)X_i^{\prime} X_i\right| \theta_g,i \in \mathcal{G}_g\right],
\text{ estimated by }
\widehat{\Omega}_g (\widehat{\theta}_g),
\\
&
\quad
\widehat{\Omega}_g (\tau)
=
\left[\frac{1}{n_g}
\sum_{i \in \mathcal{G}_g}
\Lambda \left(X_i \tau \right) \left(1-\Lambda \left(X_i \tau \right)\right)X_i^{\prime} X_i
\right]^{-1}.
\end{align*}
If the $X_i$'s, $i$ in $\mathcal{G}_g$ were independent of $\theta_g$, the sample mean
$G_N^{-1} \sum_{j=1}^{G_N} \widehat{\Omega}_g (\widehat{\theta}_j)$ would be a consistent estimator of $\mathbb{E} \left[\Omega_{n_{g},g} \left(\theta_g \right)\right]$ under reasonable conditions. But it would not be the case if the $X_i$'s, $i$ in $\mathcal{G}_g$ depended on $\theta_g$, as those $X_i$'s, $i$ in $\mathcal{G}_g$, are independent of $\widehat{\theta}_j$ when $j$ differs from $g$ and would bias the proposed estimator. The situation where $X_i$'s, $i$ in $\mathcal{G}_g$, depend on $\theta_g$ can arise in some econometric applications, as in \citet{fernandezval2026} who consider a second-stage regression model where $\theta_g = Z_g \beta+ \eta_g$, with possible dependence between $X_i$'s, $i$ in $\mathcal{G}_g$, and $Z_g$.
However, it is possible to match the asymptotic behavior of $W_g^{\sharp}$. If $\Omega_{n_{g},g} (\cdot)$ has a bounded derivative, the Taylor Inequality gives for all $\Sigma$,
\begin{align*}
\mathbb{E} \left[\Omega_{n_{g},g} (\theta_g)\right] = \Omega_{n_{g},g} (\theta) + O \left( \|\Sigma\|^{\frac{1}{2}}\right),
\quad
\Omega_{n_{g},g} (\theta_g)
= \Omega_{n_{g},g} (\theta) + O_{\mathbb{P}} \left( \|\Sigma\|^{\frac{1}{2}}\right),
\end{align*}
so that $\mathbb{E} \left[\Omega_{n_{g},g} (\theta_g)\right] = \Omega_{n_{g},g} (\theta_g) + O_{\mathbb{P}} \left( \|\Sigma\|^{\frac{1}{2}}\right)$, noting that the available
$\widehat{\Omega}_g \left(\widehat{\theta}_g\right)$ is an estimator of $\Omega_{n_{g},g} (\theta_g)$.
It then follows, uniformly in $\Sigma$ and when $n_g$ diverges:\footnote{Using $o_{\mathbb{P}} (a_N b_N) = o_{\mathbb{P}} (a_N^2+ b_N^2)$
since $|a_N b_N| \leq (a_N^2+b_N^2)/2$.}
\begin{align*}
W_g^{\sharp} & =n_g \left( n_g \Sigma + \mathbb{E} \left[\Omega_{n_{g},g} \left(\theta_g \right)\right]\right)^{-1}
=
n_g \left( n_g \Sigma + \Omega_{n_{g},g} \left(\theta_g \right)
+
o_{\mathbb{P}} \left(n_g^{-1/4} \left\|n_g \Sigma\right\|^{1/2}\right)
\right)^{-1}
\\
&
=
n_g \left( n_g \Sigma + \Omega_{n_{g},g} \left(\theta_g \right)
+
o_{\mathbb{P}} \left(n_g^{-1/2} + \left\|n_g \Sigma\right\|\right)
\right)^{-1}
\\
&
=
(1+ o_{\mathbb{P}} (1))
n_g \left( n_g \Sigma + \Omega_{n_{g},g} \left(\theta_g \right)
\right)^{-1},
\end{align*}
assuming the eigenvalues of $\Omega_{n_{g},g} \left(\theta_g \right)$ stay bounded away from $0$.
This suggests the more feasible population weights
\begin{align*}
W_g^{\star} (\theta_g,\Sigma) = \left(\Sigma + \frac{\Omega_{g} \left(\theta_g\right)}{n_g}\right)^{-1}.
\end{align*}
Note however that these weights depend on $\theta_g$, at the difference of $W_g^{\sharp}$, so that the mean and variance of $\sum_{g=1}^{G_N} W_g^{\star} (\theta_g,\Sigma) \delta_g$ are not explicit, unlike when using the weights $W_g^{\sharp}$.\footnote{This issue is addressed in Proposition \ref{Randweightshet}. Using $G_N^{-1} \sum_{j=1}^{G_N} \widehat{\Omega}_g \left(\widehat{\theta}_j\right)$ instead of $\widehat{\Omega}_g \left(\widehat{\theta}_g\right)$ is possible but may be subject to the same issue, because the mapping $\Omega_g (\cdot)$ may still depend on $\theta_g$. Using $\widehat{\Omega}_g \left(\widehat{\theta}_g\right)$ is simpler and can help to discriminate the subsamples with a poor estimation of $\theta_g$.
}
Recall that $\Omega_{g} \left(\theta_g\right)$ in $W^{\star} (\theta_g,\Sigma)$ can be estimated by $\widehat{\Omega}_g$. The key point is that $\Sigma$ can be estimated with an error that is negligible with respect to $1/n_g$ under or near homogeneity. Indeed, as in standard analysis of variance, the variance of the estimators $\widehat{\theta}_g$ decomposes into an inter-group variance plus an intra-group one given by the average of the
$\Omega_{n_{g},g} \left(\theta_g\right)/n_g$'s. See \citet{cochran1954}, \citet{swamy1970} or \citet{hedges1985}, who propose in a first step to correct the sample inter-group variance of the $\widehat{\theta}_g$ with an intra-group term. This suggests the variance estimator $\check{\Sigma}$:
\begin{align}
\check{\Sigma} = \widehat{\Sigma} -
\widehat{\Sigma}_w,
\text{ where }
\widehat{\Sigma} = \frac{1}{G_N}
\sum_{g=1}^{G_N}
\left(
\widehat{\theta}_g
-
\overline{\widehat{\theta}}
\right)
\left(
\widehat{\theta}_g
-
\overline{\widehat{\theta}}
\right)^{\prime}
\text{ and }
\widehat{\Sigma}_w=\frac{1}{G_N} \sum_{g=1}^{G_N} \frac{ \widehat{\Omega}_g \left(\widehat{\theta}_g\right)}{n_g},
\label{Checksig}
\end{align}
which is studied more specifically in
Proposition \ref{Varconsistency}.
Alternative estimators can involve the estimator variances $\Omega_{n_{g},g} \left(\theta_g\right)/n_g$ to weight the $\widehat{\theta}_g$'s in $\overline{\widehat{\theta}}$ and $\widehat{\Sigma}$, as in \citet{dersimonian1986}. See also \citet{veroniki2016} for a review of heterogeneity variance estimators and simulation experiments.
The associated candidate weights are therefore:
\begin{align}
\widehat{W}_g^{\star} \left(\widehat{\theta}_g\right)
&
=
\left(\check{\Sigma} + \frac{\widehat{\Omega}_{g} \left(\widehat{\theta}_g\right)}{n_g}\right)^{-1},
\label{Hatwstar}
\end{align}
with a suitable redefinition when the inverse does not exist.
These weights are termed \emph{adaptive} since they deliver asymptotically optimal variance performance under both homogeneity and heterogeneity, as shown in Theorem \ref{Opt}. The paper also covers more general sorts of weights, including the inverse-variance weights $n_g \widehat{\Omega}_g^{-1}$, which are asymptotically optimal under homogeneity.
We now introduce variance estimators for $\widehat{\theta} \left(\widehat{W}\right)$, which can be used for the construction of confidence intervals or hypothesis testing. Our estimation of the asymptotic variance of $\widehat{\theta} \left(\widehat{W}\right)$ is:
\begin{align}
&
\left(
\sum_{g=1}^{G_N}
\widehat{W}_g \left(\widehat{\theta}_g\right)
\right)^{-1}
\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right)
\left(
\sum_{g=1}^{G_N}
\widehat{W}_g^{\prime}\left(\widehat{\theta}_g\right)
\right)^{-1},
\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right)
=
\sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g\right)\left( \check{\Sigma} + \frac{\widehat{\Omega}_g\left(\widehat{\theta}_g\right)}{n_g}\right)\widehat{W}_g^{\prime}\left(\widehat{\theta}_g\right).
\label{Varest}
\end{align}
\paragraph{Homogeneity test.} Homogeneity is important for external validity in meta-analysis and for assessing the reliability of aggregated estimators. Indeed, their convergence rate is $1/\sqrt{N}$ under homogeneity, but drops down to $1/\sqrt{G_N}$ under heterogeneity, which can be much smaller. It is also related to external validity as previously mentioned.
We consider a multivariate version of the Cochran Q-test of homogeneity, see \citet{cochran1937}, \citet{pesaran2008}, \citet{ritz2008} among others for specific models. Let $\widehat{\theta}_{IV}$ be the estimator of $\theta$ optimally weighted under homogeneity,
\begin{align}
\widehat{\theta}_{IV} = \overline{\widehat{\theta}} \left(\widehat{\Omega}^{-1}\right)
=
\left[
\sum_{g=1}^{G_N}
n_g \cdot \widehat{\Omega}_{g}^{-1} \left(\widehat{\theta}_g\right)
\right]^{-1}
\sum_{g=1}^{G_N}
n_g \cdot \widehat{\Omega}_{g}^{-1} \left(\widehat{\theta}_g\right)
\widehat{\theta}_g,
\label{IV}
\end{align}
and for $D= \mathrm{Dim} (\theta)$, define
$
\widehat{\xi}_g
=
n_g
\left(
\widehat{\theta}_g - \widehat{\theta}_{IV}
\right)^{\prime}
\widehat{\Omega}_{g}^{-1} \left(\widehat{\theta}_g\right)
\left(
\widehat{\theta}_g - \widehat{\theta}_{IV}
\right)
-
D
$.
The homogeneity test-statistic is
\begin{align}
\widehat{Q}
=
\sqrt{G_N}
\frac{\overline{\widehat{\xi}}}{\widehat{s}},
\quad
\widehat{s}^2 = \frac{1}{G_N} \sum_{g=1}^{G_N} \widehat{\xi}_g^2 -\left(\overline{\widehat{\xi}}\right)^{2}.
\label{Homstat}
\end{align}
Let $q_{\alpha}$ be the $1-\alpha$ quantile of the standard normal distribution, i.e., $\mathbb{P} \left(\mathcal{N} (0,1) \geq q_{\alpha} \right) =\alpha$.
The homogeneity test rejects the null of homogeneity if $\widehat{Q} \geq q_{\alpha}$. Alternative rejection regions and critical values can be considered when the $\widehat{\theta}_g$'s are asymptotically normal. In this case, a candidate approximation for the distribution of $\overline{\widehat{\xi}}$ is a standardized Chi-Square with $(G_N-1) \times D$ degrees of freedom under the null of homogeneity. The homogeneity test considered in the simulation experiment rejects homogeneity if $\overline{\widehat{\xi}}$ is larger than the $1-\alpha$ quantile of this distribution.
\section{Main assumptions}
\label{sec:main-assumptions}
When applied to a vector, $\left\| \cdot \right\|$ is the Euclidean norm, while for a matrix, $\left\| W \right\| = \max_{u:\left\| u \right\|=1} \left\| Wu \right\|$ is the associated operator norm.
The set $\mathcal{T}$ stands for the possible values of all the $\theta_g$.
In the main text and Appendix~\ref{app:proof-section}, $C$ stands for a positive constant that may vary from line to line. The expectations in Assumptions E and W are with respect to the sample, and then with respect to the parameter deviations $\delta_g$'s, which are random under heterogeneity but collapse to $0$ under homogeneity.
In these assumptions, $\kappa$ is a positive constant used for some inequalities. The estimators $\widehat{\theta}_g$ and data generating processes, weights and variance estimators $\widehat{\Omega}_g \left(\widehat{\theta}_g\right)$, which satisfy these inequalities define a class $\mathcal{D}_{\kappa}$, over which our results hold uniformly. Note that for any $\kappa>0$, the class $\mathcal{D}_{\kappa}$ allows for heterogeneous alternatives depending on $N$ converging to homogeneity.
\subparagraph{Assumption He.}
\emph{Under heterogeneity, the i.i.d. deviation vectors $\delta_g$ in \eqref{Par} are centered, with variance $\Sigma$ and $\mathbb{E}^{\frac{1}{8}} \left[ \left\|\Sigma^{-1/2} \delta_g\right\|^8\right] < \kappa$. The matrix $\Sigma$ stays in a class of matrices satisfying $\|\Sigma\|\leq \kappa$ and $\| \Sigma^{-1} \| \| \Sigma\|\leq \kappa$, ensuring that the highest and smallest eigenvalues of $\Sigma$ remain of the same order but can go to $0$. }
\subparagraph{Assumption E.} \emph{
$
\mathbb{E}^{\frac{1}{8}}
\left[\left\|\sqrt{n_g} \left(\widehat{\theta}_g - \theta_g \right)\right\|^8\right]
$,
$
\sup_{\tau \in \mathcal{T}} \left\| B_{n_g,g} (\tau)\right\| $,
$ \sup_{\tau \in \mathcal{T}} \left\| \Omega_{n_g,g}^{-1} (\tau)\right\| $ are smaller or equal to $\kappa$ for all $1 \leq g \leq G_N$ and $N$.
It also holds
$
\max_{1 \leq g \leq G_N} \sup_{\tau \in \mathcal{T}} \left\|B_{n_{g},g} (\tau)- B_g (\tau) \right\| = o(1)
$ and
$
\max_{1 \leq g \leq G_N}\sup_{\tau \in \mathcal{T}}\left\|\Omega_{n_{g},g} (\tau)- \Omega_g (\tau)\right\| =o(1)$.
}
\subparagraph{Assumption V.}
\emph{There is a $v>0$ such that, for all $N$ and $1 \leq g \leq G_N$,
$
\sup_{\tau \in \mathcal{T}}
n_g^{v}
\left\|
\Omega_{n_g,g} \left(\tau \right)
-
\Omega_{g} \left(\tau \right)
\right\|
$
is smaller or equal to $\kappa$, as
$\sup_{\tau \in \mathcal{T}} \left\| \Omega_{g}^{(1)} \left(\tau \right)\right\|
$
,
$
\sup_{\tau \in \mathcal{T}} \left\| \Omega_{n_g,g}^{(1)} \left(\tau \right)\right\|
$,
$ \sup_{\tau \in \mathcal{T}} \left\|\Omega_{g} \left(\tau \right)\right\|$ and $ \sup_{\tau \in \mathcal{T}} \left\|\Omega_{g}^{-1} \left(\tau \right)\right\|$.
For all $N$ and $1 \leq g \leq G_N$,
$
\mathbb{E}^{\frac{1}{8}}
\left[
\left\|
\sqrt{n_g}
\left( \widehat{\Omega}_g \left(\theta_g\right) -\Omega_{g} \left(\theta_g\right) \right)\right\|^8
\right]
$
and
$
\mathbb{E}^{\frac{1}{8}}
\left[
\sup_{\tau \in \mathcal{T}}
\left\| \widehat{\Omega}_g^{(1)} (\tau)\right\|^8 \right]
$ are lesser or equal to $\kappa$.}
\subparagraph{Assumption W.}
\emph{The population weight functions $W_g (\cdot)$ are deterministic.
The population weights are well-conditioned in the sense that, for all $N$:
\begin{align*}
\max_{1 \leq g \leq G_N} \sup_{\tau \in \mathcal{T}} \left\| W_g (\tau )\right\|\leq \kappa,
\quad
\max_{1 \leq g \leq G_N} \sup_{\tau \in \mathcal{T}} \left\| W_g ^{-1} (\tau )\right\|\leq \kappa,
\quad
\sup_{\tau \in \mathcal{T}}
\frac{\left\| \left(\sum_{g=1}^{G_N} W_g (\tau )\right)^{-1} \right\|}{ \left(\sum_{g=1}^{G_N} \left\|W_g(\tau)\right\|\right)^{-1} }
\leq
\kappa.
\end{align*}
$W_g \left(\cdot \right)$ is continuously differentiable with, for all $N$:
\begin{align*}
\max_{1 \leq g \leq G_N} \sup_{\tau \in \mathcal{T}} \left\| \left( \frac{N}{G_N} \|\Sigma\| +1\right) W_g^{(1)} (\tau) \right\| \leq \kappa.
\end{align*}}
\emph{
The weight matrices $\widehat{W}_g \left(\cdot \right)$ are continuously differentiable, and
it holds, for all $N$,
\begin{align*}
&
\max_{1 \leq g \leq G_N}
\mathbb{E}^{\frac{1}{8}}
\left[
\left\|
\left( n_g \|\Sigma\| +1 \right)
\sqrt{n_g}
\left( \widehat{W}_g \left(\theta_g\right) -W_g \left(\theta_g\right) \right)\right\|^8
\right]
\leq \kappa,
\\
&
\max_{1 \leq g \leq G_N}
\mathbb{E}^{\frac{1}{8}}
\left[
\sup_{\tau \in \mathcal{T}}
\left\| \left( n_g \|\Sigma\| +1 \right) \widehat{W}_g^{(1)} (\tau)\right\|^8 \right]
\leq \kappa .
\end{align*}
}
\subparagraph{Heterogeneity.} As with the other assumptions, Assumption He requests finite eighth moments. This moment condition allows us to avoid imposing dependence assumptions, which would not hold for weights like \eqref{Hatwstar}.
\subparagraph{Subsample estimation.} Assumption E deals with the estimation procedures, which may differ across subsamples but should all be consistent. It imposes a bias term negligible with respect to the $1/\sqrt{n_g}$ rate, as standard for estimators with a centered normal limit distribution. As mentioned above, a usual value for the bias exponent $b$ is $b=1$ (see \citet{rilstone1996} for usual econometric estimators), with higher values of $b$ possibly achieved through bias correction \citep{newey2004,yang2015,kim2016,gu2023}.
The case $b<1$ arises, for instance, in simulation-based estimation using few simulations (\citealp{gourieroux1996}), or in semiparametric estimation involving a nonparametric step (\citealp{ichimura2007}). Moments finiteness may be an issue for econometric estimators such as instrument ones, which have a ratio expression with a denominator that can go to $0$. Assumption E can be weakened by showing that the $\widehat{\theta}_g$'s coincide with a truncation satisfying Assumption E.\footnote{See the proof of Theorem \ref{Opt} for an example of such a technique applied to weights. Suppose for instance that the decomposition $\widehat{\theta}_g = \theta_g + \frac{\widehat{N}_g}{\widehat{D}_g}$ holds, with $\max_{1 \leq g \leq G_N} \mathbb{E} \left[ \left|\sqrt{n_g}\left(\widehat{D}_g -D_g\right)\right|^8\right] = O(1)$ and $\inf_{g} |D_g|>2 \epsilon>0$, $\max_{1 \leq g \leq G_N} \mathbb{E} \left[ \left|\sqrt{n_g}\widehat{N}_g\right|^8\right] = O(1)$. Then $\widetilde{\theta}_g = \theta_g + \frac{\widehat{N}_g}{\widehat{D}_g}\mathbb{I} \left[|\widehat{D}_g-D_g|< \epsilon\right]$ will satisfy Assumption E, with
\begin{align*}
\mathbb{P} \left[ \widehat{\theta}_g \neq \widetilde{\theta}_g \text{ for some $1 \leq g \leq G_N$}\right]
\leq
\mathbb{P} \left( \max_{1 \leq g \leq G_N} \left| \widehat{D}_g - D_g \right| \geq \epsilon \right)
\leq \frac{1}{\epsilon^8} \sum_{g=1}^{G_N} \frac{\mathbb{E} \left[ \left|\sqrt{n_g}\left(\widehat{D}_g -D_g\right)\right|^8\right]}{n_g^4}
=
O\left( \frac{G_N^5}{N^4}\right),
\end{align*}
which vanishes asymptotically if $G_N = o(N^{4/5})$. Therefore results for the aggregated $\widetilde{\theta}_g$ transfer to corresponding weighted means of $\widehat{\theta}_g$.
} Note also that Assumption E imposes well-conditioned asymptotic variances.
\subparagraph{Variance estimation.}
Assumption V deals in particular with weights given by the estimator variances which are related to the optimal ones under homogeneity. Various examples of variance estimators $\widehat{\Omega}_g$ are provided in \citet{newey1994} for usual estimators; see also Gouri\'eroux and Monfort (1996) and \citet{ichimura2007} for some of the extensions mentioned above. Such variance estimators typically target the asymptotic variance $\Omega_g \left(\theta_g\right)$. Consequently, the finite-sample to asymptotic difference $\Omega_{n_g,g} \left(\theta_g\right)-\Omega_g \left(\theta_g\right)$ can be interpreted as a bias term that can affect our results, see Theorem \ref{Homtest} on our modified \citet{cochran1937} homogeneity Q-test.
\subparagraph{Weights.} Assumption W is designed for bounded versions of population weights. For instance, instead of $\widehat{W}_g^{\star} = n_g \left(n_g \check{\Sigma} + \widehat{\Omega}_g\right)^{-1}$ of \eqref{Hatwstar}, Assumption W would consider the equivalent weight:
\begin{align*}
& \widehat{W}^{\ast}_g \left(\widehat{\theta}_g,\check{\Sigma}\right)
=
\frac{G_N n_g}{N}
\left(\frac{N}{G_N} \|\Sigma \| + 1\right)
\left(n_g \check{\Sigma} + \widehat{\Omega}_g \left(\widehat{\theta}_g\right)\right)^{-1},
\text{ with population counterpart}
\\
&
W^{\ast}_g \left(\theta_g,\Sigma\right)
=
\frac{G_N n_g}{N}
\left(\frac{N}{G_N} \|\Sigma \| + 1\right)
\left(n_g \Sigma + \Omega_g (\theta_g) \right)^{-1}.
\end{align*}
That these weights are not feasible in practice as they depend on the unknown $\Sigma$ is not an issue, as these weights are equivalent to the feasible ones issued from \eqref{Hatwstar}. These bounded weights are only used for the purpose of the proof. In other words, Assumption W implicitly supposes that there is an equivalent bounded version of the considered weights that satisfies the listed statements.
Restricting to well-conditioned weights is reasonable as the subsample estimator variances are also well-conditioned. The condition on the population weight derivative $W_g^{(1)} (\cdot)$ requests that the dependence of $W_g$ on $\theta_g$ decreases when $\Sigma$ increases, and is key to controlling a bias term $\mathbb{E} \left[ W_g (\theta_g) \delta_g \right]$, see \eqref{CLT_esp} in Proposition \ref{Randweightshet}.
Moment finiteness up to order eight echoes Assumption E. In practice, it could be difficult to verify this condition for weights involving estimated inverse variance. However, for theoretical purposes, the original weights can often be replaced by truncated weights that yield asymptotically an identical estimator and satisfy the required moment conditions. See, for instance, the proof of Theorem \ref{Opt} for an example of this approach.
\section{Main results}
\label{sec:main-results}
Our main results are organised as follows. Perhaps the most useful ones are Theorem \ref{Opt}, which establishes the adaptive optimality of the weights \eqref{Hatwstar} and the asymptotic normality of the related aggregated estimator, and Theorem \ref{Homtest} on the modified \citet{cochran1937} homogeneity Q-test. Theorem \ref{Opt} crucially builds on Proposition \ref{Varconsistency}, which derives the consistency rate of $\check{\Sigma}$ under heterogeneity and homogeneity, and is fast enough near homogeneity to ensure that inference with the adaptive weight estimator is valid uniformly over $\mathcal{D}_{\kappa}$.
Theorem \ref{Opt} relies on some preliminary results, which hold in particular for weights where $\Sigma$ is considered as known, such as $\left(\Sigma + \widehat{\Omega}_g \left(\widehat{\theta}_g\right)/n_g\right)^{-1}$. Weight estimation aims to produce an aggregation procedure that achieves a performance close to the one using the corresponding population weights. However, the weight estimation error can be correlated with the group estimators, thereby generating additional bias for the final estimator as described in Proposition \ref{Expansion}. A second issue is specific to heterogeneity: as the asymptotic population weights $W_g (\theta_g)$ may depend on the parameter values $\theta_g$, they are potentially correlated with these parameter values and the $\delta_g$. This may create another bias term as $\widehat{\theta}_g$ depends on $\delta_g$ by \eqref{Representation}, and also affect the computation of the variance of the aggregated estimator, as illustrated in Proposition \ref{Randweightshet}. Proposition \ref{Inference} shows that uniform validity of standard Gaussian inference can be ensured by restricting the number $G_N$ of subsamples. Proposition \ref{Randweightshet} is also a key tool to establish Theorem \ref{Opt} .
Following \citet{lu2023} and \citet{fernandezval2026}, our results hold uniformly over the class $\mathcal{D}_{\kappa}$, allowing in particular for alternatives with a heterogeneity variance $\Sigma_N$ depending upon $N$. This covers the three cases of alternatives discussed in \citet{fernandezval2026}, i.e., strong, moderate or no heterogeneity where $(N/G_N) \Sigma_N$ diverges, has a (positive definite) limit, or asymptotically vanishes, conditioning the convergence rate and asymptotic variance of aggregated estimators. Under strong heterogeneity, $\theta$ can be estimated with the sometimes slow $\sqrt{G_N}$ rate, which improves to $\sqrt{N}$ under moderate heterogeneity or homogeneity.
\subsection{The impact of weight estimation}
\label{sec:impact-weight-estimation}
The next proposition expands the aggregated estimator $\widehat{\theta} \left(\widehat{W} \right)$ around $\widehat{\theta} \left(W\right)$. As
\begin{align*}
\widehat{\theta} \left(\widehat{W} \right)
& =\theta+ \widehat{B} + \widehat{E}_{He} + \widehat{E}_{\theta} \text{ where }
\widehat{B} = \left[\sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g \right) \right]^{-1} \sum_{g=1}^{G_N} \widehat{W}_g\left(\widehat{\theta}_g \right) \frac{B_{n_g,g} (\theta_g)}{n_g^b},
\\
&
\quad
\widehat{E}_{He}=
\left[\sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g \right) \right]^{-1} \sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g \right) \delta_g,
\text{ and }
\widehat{E}_{\theta} = \left[\sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g \right) \right]^{-1} \sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g \right)
\widehat{\varepsilon}_g,
\end{align*}
this amounts to studying the errors induced by changing $\widehat{W}_g$ to $W_g$ in the bias, heterogeneity, estimation terms $\widehat{B}$, $\widehat{E}_{He}$, and $\widehat{E}_{\theta}$, respectively. Proposition \ref{Expansion} below gives a rate for the approximation of each of these items.
\begin{proposition}
Suppose that the subsamples are balanced.
Under Assumptions E, He, and W, $\widehat{\theta} \left(\widehat{W} \right)$ in \eqref{Weighted} satisfies, uniformly over $\mathcal{D}_{\kappa}$,
$
\widehat{\theta} \left(\widehat{W} \right)
=
\theta
+
\left(
1
+
O_{\mathbb{P}}
\left[
\left(\frac{G_N}{N}\right)^{\frac{1}{2}}
\right]
\right)
\left[
\widetilde{B}
+
\widetilde{E}_{He} + \widetilde{E}_{\theta}
\right]
$ where
\begin{align*}
&\widetilde{B}
=
\left[\sum_{g=1}^{G_N} W_g \left(\theta_g\right) \right]^{-1}
\sum_{g=1}^{G_N} W_g\left(\theta_g\right) \frac{B_{n_g,g} \left(\theta_g\right)}{n_g^{b}},
\quad
\widetilde{E}_{He} = \left[\sum_{g=1}^{G_N} W_g \left(\theta_g\right)\right]^{-1} \sum_{g=1}^{G_N} W_g \left(\theta_g\right) \delta_g ,
\\
& \widetilde{E}_{\theta} = \left[\sum_{g=1}^{G_N} W_g\left(\theta_g\right) \right]^{-1} \sum_{g=1}^{G_N} W_g \left(\theta_g\right)
\widehat{\varepsilon}_g,
\quad r_{\theta} = \frac{G_N}{N}\frac{1}{\frac{N}{G_N} \| \Sigma\| +1 },
\quad
r_{He}
=
\frac{G_N}{N}
\frac{\left(\frac{N}{G_N} \left\|\Sigma\right\|\right)^{\frac{1}{2}}}{\frac{N}{G_N} \| \Sigma\| +1 }
.
\end{align*}
\label{Expansion}
\end{proposition}
The bias term $\widetilde{B}$, which collects the bias of all estimators $\widehat{\theta}_g$, is common to all aggregation procedures. More specific items come from the interactions of the aggregated estimators and the estimated weights.
The proof of Proposition \ref{Expansion} works by expanding $\widehat{W}_g \left(\widehat{\theta}_g\right)$ around $\widehat{W}_g \left(\theta_g\right)$, which in turn approaches $W_g (\theta_g)$. The numerators in $\widehat{E}_{He}$ and $\widehat{E}_{\theta}$ differ from those of $\widetilde{E}_{He}$ and $\widetilde{E}_{\theta}$ due to these expansion terms. In the univariate case, they are given by:
\begin{align*}
& \sum_{g=1}^{G_N} \widehat{W}_g^{(1)} \left(\theta_g\right) \left(\widehat{\theta}_g - \theta_g \right) \delta_g,
\qquad
\sum_{g=1}^{G_N} \left[\widehat{W}_g \left(\theta_g\right) - W_g \left(\theta_g\right)\right]\delta_g,&
\text{ for }\widetilde{E}_{He},
\\
& \sum_{g=1}^{G_N} \widehat{W}_g^{(1)} \left(\theta_g\right) \left(\widehat{\theta}_g - \theta_g \right)^2,
\qquad
\sum_{g=1}^{G_N} \left[\widehat{W}_g \left(\theta_g\right) - W_g \left(\theta_g\right)\right] \left(\widehat{\theta}_g - \theta_g \right),
& \text{ for }\widetilde{E}_{\theta}.
\end{align*}
Bounds for these items are collected in $r_{He}$ and $r_{\theta}$, respectively.
As a consequence of the proposition, estimating weights induces additional bias terms. Under homogeneity, i.e., $\Sigma = 0$, the order of $r_{\theta}$ is $G_N/N$, which is identical to the rate of the estimator bias term $\widetilde{B}$ when $b=1$. The magnitude of these bias terms decreases with heterogeneity, i.e., when $\| \Sigma\|$ increases.
\subsection{Random population weights and heterogeneity}
\label{sec:random-population-weights}
Compared to \citet{gu2023} who consider inverse-variance weighting under homogeneity of the parameter, our framework raises new challenges as the numerator $ \sum_{g=1}^{G_N} W_g (\theta_g) \delta_g$ of the linearized term $\widetilde{E}_{He}$ is not necessarily centered, typically when $W_g (\theta_g)$ and $\delta_g$ are correlated. Hence weighted estimators may face a heterogeneity correlation bias term. As well, the variance of $ \sum_{g=1}^{G_N} W_g (\theta_g) \delta_g$ may be difficult to estimate.
Define:
\begin{align}
\begin{array}{l}
\boldsymbol{V} \left(W\right)
=
\mathrm{Var}
\left(
\sum_{g=1}^{G_N}
W_g \left(\theta_g\right)
\left(\delta_g+\widehat{\varepsilon}_g\right)
\right),
\\
\widetilde{\boldsymbol{V}} \left(W\right)
=
\sum_{g=1}^{G_N} W_g \left(\theta_g\right) \left(\ \Sigma + n_g^{-1} \cdot \Omega_{n_g,g} \left(\theta_g\right)\right) W_g^{\prime} \left(\theta_g\right),
\label{Var}
\end{array}
\end{align}
The next proposition gives the order of the heterogeneity correlation bias and shows that the tractable $\widetilde{\boldsymbol{V}} \left(W\right)$ is a suitable approximation for the variance of $\sum_{g=1}^{G_N}
W_g \left(\theta_g\right)
\left(\delta_g+\widehat{\varepsilon}_g\right)$.
\begin{proposition}
Suppose that the subsamples are balanced.
Under Assumptions E, He, and W, it holds, uniformly over $\mathcal{D}_{\kappa}$,
\begin{align}
&
\mathbb{E}
\left[
\sum_{g=1}^{G_N}
W_g \left(\theta_g\right) \left(\delta_g + \widehat{\varepsilon}_g\right)
\right]
=
O
\left(
\frac{G_N^2}{N}
\frac{\frac{N}{G_N}\| \Sigma\|}{\frac{N}{G_N} \| \Sigma\| +1}\right),
\label{CLT_esp}
\\
&
\widetilde{\boldsymbol{V}}^{-1/2 } \left(W\right) \boldsymbol{V} \left(W\right)\widetilde{\boldsymbol{V}}^{-1/2 } \left(W\right)
=
\mathrm{Id}
+O_{\mathbb{P}}
\left[
\left(\frac{G_N}{N}\right)^{\frac{1}{2}}
+
\frac{1}{\sqrt{G_N}}
\right],
\label{Vareq}
\end{align}
implying in particular that the exact order of $\boldsymbol{V} \left(W\right)$ and $\widetilde{\boldsymbol{V}} \left(W\right)$ is
$\frac{G_N^2}{N} \left(\frac{N}{G_N} \|\Sigma\|+1\right)$ if $G_N=o(N)$ diverges. In this case, it also holds:
\begin{align}
\boldsymbol{V}^{-\frac{1}{2}}\left(W\right)
\mathbb{E}
\left[
\sum_{g=1}^{G_N}
W_g \left(\theta_g\right) \left(\delta_g + \widehat{\varepsilon}_g\right)
\right]
=
O_{\mathbb{P}}
\left[
\sqrt{\frac{G_N^2}{N}}
\frac{\frac{N}{G_N}\| \Sigma\|}{\left(\frac{N}{G_N} \| \Sigma\| +1\right)^{\frac{3}{2}}}\right],
\text{ uniformly over $\mathcal{D}_{\kappa}$.}
\label{CLT_Esp}
\end{align}
\label{Randweightshet}
\end{proposition}
\paragraph{Order of the variance.} The exact order of $\frac{G_N^2}{N} \left(\frac{N}{G_N} \|\Sigma\|+1\right)$ of the variance $\boldsymbol{V} \left(W\right)$ of $\sum_{g=1}^{G_N}
W_g \left(\theta_g\right)
\left(\delta_g+\widehat{\varepsilon}_g\right)$ exhibits different behaviors under heterogeneity and homogeneity. For a given $\Sigma \neq 0$, $\boldsymbol{V} \left(W\right)$ diverges with the exact order $G_N$, while, if $\Sigma =0 $, it goes to $0$ if $G_N = o (N^{1/2})$. As $\sum_{g=1}^{G_N} \widehat{W}_g (\widehat{\theta}_g)$ is of exact order $G_N$, the order of $\left[\sum_{g=1}^{G_N} \widehat{W}_g (\widehat{\theta}_g)\right]^{-1} \sum_{g=1}^{G_N}
W_g \left(\theta_g\right)
\left(\delta_g+\widehat{\varepsilon}_g\right)$ is $ O_{\mathbb{P}} \left[ N^{-1/2} \left(\frac{N}{G_N} \| \Sigma\| +1\right)^{1/2}\right]$, which boils down to $O_{\mathbb{P}} \left(N^{-1/2}\right)$ under homogeneity and $O_{\mathbb{P}} \left(G_{N}^{-1/2}\right)$ under heterogeneity.
\paragraph{Weight correlation and estimation bias under heterogeneity.} Under heterogeneity, i.e., if $\Sigma \neq 0$, the order $r_{He}= \left(G_N/N\right)^{3/2}$ of the estimation bias from Proposition \ref{Expansion} is dominated by $G_N^2/N$, the one of the correlation bias stated in \eqref{CLT_esp}. The corresponding quantities $r_{He} / \left\| \boldsymbol{V}^{1/2} \left(W\right)\right\|$ and \eqref{CLT_Esp} must asymptotically vanish to ensure that these bias terms do not affect the Central-Limit Theorem of Proposition \ref{Inference}. While $r_{He} / \left\| \boldsymbol{V}^{1/2} \left(W\right)\right\| =o(1)$ when $N$ diverges, \eqref{CLT_Esp} vanishes asymptotically when the stronger condition $G_N= o( N^{2/3})$ holds. This suggests that the weight correlation bias plays the strongest role under heterogeneity.
\paragraph{Consistency.}
When $G_N$ and $N/G_N$ both diverge, $r_{\theta}$, $r_{He}$, $\widetilde{B}$ and $\left[\sum_{g=1}^{G_N} \widehat{W}_g (\widehat{\theta}_g)\right]^{-1} \sum_{g=1}^{G_N}
W_g \left(\theta_g\right)
\left(\delta_g+\widehat{\varepsilon}_g\right)= O_{\mathbb{P}} \left[ N^{-1/2} \left(\frac{N}{G_N} \| \Sigma\| +1\right)^{1/2}\right]$
all vanish uniformly over $\mathcal{D}_{\kappa} $ as discussed above. Therefore consistency of $\widehat{\theta} (\widehat{W})$ holds provided that the correlation bias term in \eqref{CLT_esp} is negligible with respect to $G_N$, which holds if $N/G_N$ diverges. Hence $\widehat{\theta} (\widehat{W})$ is consistent uniformly over $\mathcal{D}_{\kappa}$, for any $\kappa>0$, provided that $G_N$ and $N/G_N$ both diverge.
\subsection{Variance estimation}
\label{sec:variance-estimation}
Proposition \ref{Varconsistency} studies the heterogeneity variance estimator $\check{\Sigma}$ from \eqref{Checksig}, and the estimator $\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right)$ in \eqref{Varest}
of the variance term $\boldsymbol{V} \left(W\right)$.
\begin{proposition}
Suppose that the subsamples are balanced and that Assumptions E, He and V hold. Let $\underline{v} = \min \left(v,\frac{1}{2}\right)$.
Then, uniformly over $\mathcal{D}_{\kappa}$,
\begin{align}
\check{\Sigma}
& =
\Sigma
+
\left\|\Sigma\right\|^{\frac{1}{2}}
O_{\mathbb{P}}
\left[
\frac{\left\|\Sigma\right\|^{\frac{1}{2}}}{\sqrt{G_N}}
+
\frac{1}{\sqrt{N}}
+
\left(\frac{G_N}{N}\right)^{b}
\right]
+
O_{\mathbb{P}} \left[\left( \frac{G_N}{N} \right)^{1+\underline{v}}+
\left(\frac{G_N}{N}\right)^{2b}
+
\frac{\sqrt{G_N}}{N}
\right]
\nonumber \\
&=
\Sigma
\left[
1
+
O_{\mathbb{P}}
\left(
\frac{1}{\sqrt{G_N}}
\right)
\right]
+
\left\|\Sigma\right\|^{\frac{1}{2}}
O_{\mathbb{P}}
\left[
\frac{1}{\sqrt{N}}
+
\left(\frac{G_N}{N}\right)^{b}
\right]
+
o_{\mathbb{P}}
\left( \frac{G_N}{N} \right)
.
\label{Checksigwrtng}
\end{align}
Moreover, if $G_N=o(N)$ diverges and under Assumption W, it also holds uniformly over $\mathcal{D}_{\kappa}$:
\begin{align*}
\widetilde{\boldsymbol{V}}^{-1/2} \left(W\right)\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right)\widetilde{\boldsymbol{V}}^{-1/2} \left(W\right) = \mathrm{Id} + o_{\mathbb{P}} (1).
\end{align*}
\label{Varconsistency}
\end{proposition}
\vspace{-2\baselineskip}
\paragraph{Heterogeneity variance estimation.}
A key feature of $\check{\Sigma}$, which is due to the bias correction $\widehat{\Sigma}_w$ in \eqref{Checksig}, is rate adaptation which can be illustrated with the two polarised cases of heterogeneity and homogeneity, assuming that $b$ is large enough or $G_N$ small enough. Under heterogeneity, \eqref{Checksigwrtng} yields that $\check{\Sigma} = \Sigma + O_{\mathbb{P}} \left(G_N^{-1/2}\right)$, as would be feasible if the $\theta_g$ were directly observed. Under homogeneity so that $\Sigma=0$, this improves to $\check{\Sigma} = o_{\mathbb{P}} (G_N/N)$.
\paragraph{Estimation of $\widetilde{\boldsymbol{V}} \left(W\right)$.} The rate adaptation property described above is key for obtaining an estimator of $\widetilde{\boldsymbol{V}} \left(W\right)$ as in Proposition \ref{Varconsistency}. Indeed, $\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right)$ involves the item $\check{\Sigma} + \widehat{\Omega}_g \left(\widehat{\theta}_g\right)/n_g$, which is equal to $(1+o_{\mathbb{P}}(1)) \Omega_{n_g,g} \left(\theta_g\right)/n_g$ under homogeneity and $(1+o_{\mathbb{P}}(1))\Sigma$ under heterogeneity, uniformly in $g$. This is crucial to capture the different rate behaviors of $\widetilde{\boldsymbol{V}} \left(W\right)$. Note also that Proposition \ref{Varconsistency} and \eqref{Vareq} yield that
$
\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W}\right) = (1+o_{\mathbb{P}}(1)) \mathrm{Var} \left(\sum_{g=1}^{G_N} W_g (\theta_g) \left(\delta_g + \widehat{\varepsilon}_g\right)\right)
$,
which is needed for inference.
\subsection{Inference with estimated weights}
\label{sec:inference-estimated-weights}
We now study how estimated weights affect inference. While \eqref{Vareq} and Proposition \ref{Varconsistency} show that consistent variance estimation is feasible under the weak restrictions that both $G_N$ and $N/G_N$ diverge, Propositions \ref{Expansion} and \ref{Randweightshet} show that, together with the baseline estimators bias, the weight estimation and correlation between population weights and heterogeneity deviations matter. The next proposition gives conditions on the growth of the number of groups $G_N$ ensuring that standard Gaussian inference is feasible.
\begin{proposition}
Suppose that the subsamples are balanced, that $G_N$ diverges, and that Assumptions E, He, and W hold. Let $\underline{b}=\min(b,1)$. Then the distribution of the standardized estimator,
\begin{align*}
\widehat{\boldsymbol{V}}_{\Sigma}^{-\frac{1}{2}} \left(\widehat{W}\right)
\left[
\sum_{g=1}^{G_N} \widehat{W}_g \left(\widehat{\theta}_g\right)
\right] \left( \widehat{\theta}\left(\widehat{W}\right)-\theta\right)
\end{align*}
converges to a standard multivariate normal distribution, uniformly over
$\mathcal{D}_{\kappa}$ if $G_N = o \left( N^{1-\frac{1}{2\underline{b}}}\right)$, and uniformly over $\mathcal{D}_{\kappa} \cap \left\{ \frac{1}{\kappa} \leq \left\|\Sigma\right\| \right\}$ if $G_N = o \left( N^{1-\frac{1}{2\underline{b}+1}}\right)$, for any $\kappa>0$.
\label{Inference}
\end{proposition}
Under Assumption V, it is easily seen that the weights $\left(\frac{N}{G_N} \| \Sigma \| + 1\right) \frac{G_N n_g}{N}\left(n_g \Sigma + \widehat{\Omega}_g \left(\widehat{\theta}_g\right)\right)^{-1}$ satisfy Assumption W, so that Proposition \ref{Inference} applies. Specialising this result to homogeneity implies that the inverse-variance weighted estimator $\widehat{\theta}_{IV}
=
\left[
\sum_{g=1}^{G_N}
n_g \widehat{\Omega}_g^{-1} \left(\widehat{\theta}_g\right)
\right]^{-1}
\sum_{g=1}^{G_N}
n_g \widehat{\Omega}_g^{-1} \left(\widehat{\theta}_g\right)
\widehat{\theta}_g$
satisfies the Central-Limit Theorem:
\begin{align*}
\left[
\sum_{g=1}^{G_N}
n_g \widehat{\Omega}_g^{-1} \left(\widehat{\theta}_g\right)
\right]^{\frac{1}{2}}
\left(
\widehat{\theta}_{IV}
-
\theta
\right)
\stackrel{d}{\rightarrow}
\mathcal{N}
\left(0,\mathrm{Id}\right),
\end{align*}
if $\Sigma=0$ and $G_N = o \left( N^{1-\frac{1}{2\underline{b}}}\right)$,
a result which parallels \citet[Theorems 6 and 7]{gu2023} but holds without sample-splitting, at the price of a stronger growth condition for $G_N$ compared to \citet{gu2023}. Indeed, sample-splitting removes the weight estimation bias and these authors only assume that $G_N = o \left(N^{2/3}\right)$. Note however that the considered weights become infeasible under heterogeneity when $\Sigma$ is unknown, and that sample-splitting may be difficult to implement in meta-analysis.
Restricting to the class of DGP $\mathcal{D}_{\kappa} \cap \left\{ \frac{1}{\kappa} \leq \left\|\Sigma\right\| \right\}$ is mostly of theoretical interest, but useful to illustrate that heterogeneity allows for a better growth condition for $G_N$. The intuition is that $\widetilde{\boldsymbol{V}} (W)$ diverges faster under heterogeneity so that the bias-variance condition which drives the rate of admissible $G_N$ can be slightly relaxed.
\subsection{Optimal inference}
\label{sec:optimal-inference}
The next theorem complements Proposition \ref{Inference} by considering aggregation with the candidate optimal weights \eqref{Hatwstar}. It works using a second-order Taylor Inequality for this weighted estimator around the one using weights based on $\Sigma$ instead of $\check{\Sigma}$. Applying Proposition \ref{Inference} then gives the Central-Limit Theorem stated below, while the optimality statement of the aggregated estimator using the adaptive weights follows from standard arguments.
\begin{theorem}
Suppose that the subsamples are balanced, and that Assumptions E, He, and V hold. Assume that $G_N$ diverges with $G_N = o \left(N^{1-\frac{1}{2\underline{b}}}\right)$, $\underline{b} = \min (b,1)$.
Then, both
\begin{align*}
\left[\sum_{g=1}^{G_N}W_g^{\star} \left(\theta_g,\Sigma\right)\right]^{\frac{1}{2}}
\left(
\widehat{\theta} \left(\widehat{W}_g^{\star}\right) - \theta
\right)
\text{ and }
\left[\sum_{g=1}^{G_N} \widehat{W}_g^{\star} \left(\widehat{\theta}_g\right)\right]^{\frac{1}{2}}
\left(
\widehat{\theta} \left(\widehat{W}_g^{\star}\right) - \theta
\right),
\end{align*}
converge in distribution to a standard multivariate normal uniformly over $\mathcal{D}_{\kappa}$.
Moreover, $ \widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ is asymptotically optimal among the class of weighted estimators $\widehat{\theta} (W)$ with asymptotic variance $\left(\sum_{g=1}^{G_N}W_g \right)^{-1} \sum_{g=1}^{G_N} W_g \left(\Sigma + \frac{\Omega_{n_g,g} (\theta_g)}{n_g}\right) W_g^{\prime} \left(\sum_{g=1}^{G_N}W_g^{\prime} \right)^{-1}$, which is asymptotically larger than the one of $\widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ uniformly in $\Sigma$,
\begin{align*}
\left(\sum_{g=1}^{G_N}W_g \right)^{-1} \sum_{g=1}^{G_N} W_g \left(\Sigma + \frac{\Omega_{n_g,g} (\theta_g)}{n_g}\right) W_g^{\prime} \left(\sum_{g=1}^{G_N}W_g^{\prime} \right)^{-1}
\succcurlyeq
(1+o_{\mathbb{P}} (1))\left[\sum_{g=1}^{G_N} \left(\Sigma + \frac{\Omega_{n_g,g} (\theta_g)}{n_g}\right)^{-1}\right]^{-1},
\end{align*}
where $\succcurlyeq$ is the usual ordering of symmetric matrices.
\label{Opt}
\end{theorem}
Theorem \ref{Opt} is valid uniformly over the class $\mathcal{D}_{\kappa}$ of DGP, including homogeneity and heterogeneity, as well as heterogeneous alternatives with $\Sigma = \Sigma_N$ such as in \citet{fernandezval2026} who consider strong (diverging $\frac{N}{G_N} \| \Sigma\|$), moderate (bounded $\frac{N}{G_N} \| \Sigma\|$) or no heterogeneity (asymptotically vanishing $\frac{N}{G_N} \| \Sigma\|$). Under strong heterogeneity, Theorem \ref{Opt} implies that $\widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ converges to $\theta$ with the rate $\sqrt{G_N}$, which improves to $\sqrt{N}$ under moderate or no heterogeneity. Moderate heterogeneity was already considered in \citet{zeng2015} for known $\Omega_{n_g,g} (\cdot)$. Their Theorem 2 states, for a one-dimensional parameter with $\Sigma_N = \frac{G_N}{N} \sigma^2>0$, a Central-Limit Theorem identical to the one above, showing in addition that $\widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ can improve on the Maximum Likelihood estimator.
Theorem \ref{Opt} therefore extends \citet{zeng2015} and \citet{gu2023} by covering a wider range of heterogeneity scenarios. Besides considering a more general class of estimators, it also extends \citet{lu2023} and \citet{fernandezval2026} due to its optimality focus. These authors indeed use for inference purposes the equal-weights estimator
\begin{align}
\widehat{\theta}_{EW}
=
\frac{1}{G_N} \sum_{g=1}^{G_N} \widehat{\theta}_g,
\label{EW}
\end{align}
which is asymptotically normal by Proposition \ref{Inference} with the asymptotic variance $\frac{1}{G_N^2}\sum_{g=1}^{G_N} \left(\Sigma + \frac{\Omega_{n_g,g} (\theta_g)}{n_g}\right)$. The latter variance is improved by the one of $\widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ by the asymptotic optimality statement of Theorem \ref{Opt}, meaning in particular that confidence intervals relying on $\widehat{\theta} \left(\widehat{W}_g^{\star}\right)$ are asymptotically shorter than counterparts based on $\widehat{\theta}_{EW}$. \citet{lu2023} and \citet{fernandezval2026} also rely on bootstrap procedures to achieve rate adaptation. This is computationally more involved than a straightforward application of the Central-Limit Theorem proposed here.
The growth rate condition $G_N = o \left(N^{1-1/(2\underline{b})}\right)$, which gives at best $G_N = o \left(N^{1/2}\right)$, is in line with the ones used in \citet{zeng2015}, \citet{lu2023} and \citet{fernandezval2026} who assume $G_N = o \left(N^{1/2}\right)$, corresponding to the case of subsample estimators with a bias of order $1/n_g$, so that $\underline{b}=1$.
In applications, it is often more convenient to relate the number $G_N$ of groups to the average subsample size $n=N/G_N$ instead of the total sample size. The condition $G_N = o \left(N^{1-1/(2\underline{b})}\right)$ then becomes $G_N = o \left(n^{2 \underline{b}-1}\right)$, which gives at best $G_N = o(n)$ when $b=1$.
\subsection{Testing for homogeneity \label{Qtest}}
\label{sec:testing-homogeneity}
Rate-adaptive inference procedures derived from Theorem \ref{Opt} do not need to involve a homogeneity testing step as they automatically adjust to the data DGP heterogeneity. However, as mentioned earlier, testing for homogeneity is interesting independently of parameter inference, as heterogeneity is a threat to external validity as noted in \citet{vivalt2020}. The next result shows that the homogeneity test based on the statistic \eqref{Homstat} has the correct asymptotic level and is asymptotically consistent.
\begin{theorem}
Suppose that Assumptions He, E, and V hold. In addition, assume that the subsamples are balanced, that, for $\underline{vb}=\min \left(2b-1,v,\frac{1}{2}\right)$, $G_N$ diverges with $G_N= o \left(N^{1-\frac{1}{2\underline{vb}+1}}\right)$, and that\\ $\min_{1 \leq g \leq G_N} \mathrm{Var} \left(n_g \widehat{\varepsilon}_g^{\prime }\Omega^{-1}_{n_g,g} (\theta_g) \widehat{\varepsilon}_g\right) \geq C>0$ for $N$ large enough.
Then $\widehat{Q}$ converges in distribution to a standard normal under homogeneity, while it diverges for any given $\Sigma \neq 0$. As a consequence,
the test which rejects the null of homogeneity if $\widehat{Q} \geq q_{\alpha}$ is an asymptotic $\alpha$-level and consistent test.
\label{Homtest}
\end{theorem}
Notably, Theorem \ref{Homtest} makes no assumption on the asymptotic distribution of the estimators $\widehat{\theta}_g$, which are not associated with any specific model.
In this sense, Theorem \ref{Homtest} extends the seminal paper of \citet{cochran1937} who considered a mean parameter under Gaussianity, as well as \citet{swamy1970} or \citet{pesaran2008} who considered panel data linear regression models. \citet{ritz2008} considered the case of general estimators but assumed that their variances are known. The condition on the number of groups $G_N$ used in Theorem \ref{Homtest} differs from the one used above for estimation because the variance estimator $\widehat{\Omega}_{g} \left(\widehat{\theta}_g\right)$ plays a crucial role in the construction of the statistic $\widehat{Q}$, especially under the null of homogeneity considered here.
\section{Simulation illustrations and application}
\label{sec:simulation-application}
\subsection{Simulation exercises}
\label{sec:simulation-exercises}
This section considers two specifications: a discrete-choice model in which the correlation between the covariates varies across groups and, reported in Appendix~\ref{app:iv-regression-specification}, an instrumental variable regression specification in which the second-stage variance depends on the considered subsamples. The subsample size is constant across groups and set to $n=1,000$. The number of groups, $G_N$, varies from $1$ to $600$, so that the total sample size $N$ ranges from $1,000$ to $600,000$. Each simulation experiment uses $100,000$ replications.
The simulation illustration first evaluates three weighting schemes, which are compared with a full-sample estimator. The three schemes are equal ($W_g=1/G_N$ as in \eqref{EW}), inverse-variance ($\widehat{W}_g \left(\widehat{\theta}_g\right) = n_g \cdot \widehat{\Omega}_g^{-1} \left( \widehat{\theta}_g \right)$), and adaptive \eqref{Hatwstar} weights.
The adaptive weights are computed using $\max \left(0,\check{\Sigma} \right)$ instead of $\check{\Sigma}$, which is sometimes negative.
The confidence intervals computed in the simulation experiments are based on the variance estimator $\widehat{\boldsymbol{V}}_{\Sigma} \left(\widehat{W} \right)$ in \eqref{Varest}. Chi-squared critical values are used for the homogeneity test.
The simulation illustration investigates the square-root Mean Squared Error (RMSE), the bias, and the confidence interval coverage of each estimation procedure. For the sake of brevity, the simulation experiments focus on the polar cases of homogeneity and heterogeneity. The level and power of the homogeneity test are also considered.
In each specification, the focus is on a univariate parameter $\theta_{1g}\sim \mathcal{N} \left(1,\sigma_{\delta}^2\right)$, with $\Sigma=0$ under homogeneity. The parameter of interest is $\mathbb{E} \left[\theta_{1g}\right]=1$. The RMSE is computed using the following average of the simulation estimates $\widehat{\theta}_{1}^s$, which accounts for the consistency rates $\sqrt{N}$ and
$\sqrt{G_N}$ under homogeneity and heterogeneity respectively:
\begin{align}
\left[
\frac{1}{100,000}
\sum_{s=1}^{100,000}
N
\left(
\widehat{\theta}_1^{s}- 1
\right)^2
\right]^{\frac{1}{2}}
\text{ and }
\left[
\frac{1}{100,000}
\sum_{s=1}^{100,000}
G_N
\left(
\widehat{\theta}_1^{s}- 1
\right)^2
\right]^{\frac{1}{2}}
.
\label{RMSE}
\end{align}
The reported bias from the simulation experiments is computed as a proportion of the RMSE, that is, under homogeneity and heterogeneity respectively:
\begin{align}
\frac{\frac{\sqrt{N}}{100,000}
\sum_{s=1}^{100,000}
\left(\widehat{\theta}_1^s-1\right)
}{RMSE}
\text { and }
\frac{\frac{\sqrt{G_N}}{100,000}
\sum_{s=1}^{100,000}
\left(\widehat{\theta}_1^s-1\right)
}{RMSE}
.
\label{Biassim}
\end{align}
For the discrete choice model considered here, it holds for all $i$ in group $\mathcal{G}_g$:
\begin{align}
\left\{
\begin{array}{ll}
y_i = \mathbb{I} \left( y_i^{\star} \geq 0\right),
\quad
y_i^{\star} = \theta_{1g} x_{1i} + \theta_2 x_{2i} - U_i,
& U_i \stackrel{i.i.d.}{\sim} \mathrm{ Logistic},
\\
\left[
\begin{array}{c}
x_{1i}
\\
x_{2i}
\end{array}
\right]
\sim
\mathcal{N}
\left(
\left[
\begin{array}{c}
0
\\
0
\end{array}
\right]
,
\left[
\begin{array}{cc}
1 & \rho_g
\\
\rho_g & 1
\end{array}
\right]
\right)
,
&
\\
\theta_{1g} \sim \mathcal{N} \left(1,\sigma_{\delta}^2 \right),
\quad
\rho_g
\sim
\left\{
\begin{array}{cc}
\mathcal{U}_{[.9,.95]}
&
\text{ with prob. $.55$},
\\
\mathcal{U}_{[0,.1]}
&
\text{ with prob. $.45$},
\end{array}
\right.
\end{array}
\right.,
&
\label{DCM}
\end{align}
where $\theta_g$ and $\rho_g$ are independent. $x_{i}=\left[x_{1i},x_{2i}\right]$ and $U_i$ are also independent given $\theta_g$ and $\rho_g$. The standard deviation $\sigma_{\delta}$ is set to $0.3$ under heterogeneity. The group correlation parameter $\rho_g$ lies in $[0.9,0.95]$ with probability $0.55$, in which case the variance of $x_i$ is close to non-invertible, making $\theta_g = \left[\theta_{1g},\theta_2\right]^{\prime}$ difficult to estimate. By contrast, $\theta_g$ is better identified when $0 \leq \rho_g \leq 0.1$, which occurs with probability $0.45$.
$\theta_g$ is estimated using a couple of Newton-Raphson iterations based on the first-order condition of the maximum likelihood estimator (MLE) initialized at $\theta_g$, so that the resulting $\widehat{\theta}_g$ is equivalent to the MLE up to an $o_{\mathbb{P}} \left(n^{-1/2}\right)$ error term. An estimator $\widehat{\theta}$ is similarly computed over the whole sample using the initial value $1$, so that $\widehat{\theta}$ is equivalent to the MLE up to an order $o_{\mathbb{P}} \left((G_N n)^{-1/2}\right)$. The asymptotic variance of $\widehat{\theta}_{1g}$ is obtained using the Hessian formula.
An important difference between the discrete choice model and the IV specification \eqref{IVmodel} is that this variance asymptotically depends on $\theta_g$. Hence, under heterogeneity, the inverse-variance weighted estimator will not estimate $\mathbb{E} \left[\widehat{\theta}_{1g}\right]$ but $\mathbb{E} \left[ \widehat{\Omega}_{g}
\left(\widehat{\theta}_g\right)^{-1} \widehat{\theta}_{1g}\right]/\mathbb{E} \left[ \widehat{\Omega}_{g}
\left(\widehat{\theta}_g\right)^{-1} \right]$. Averaging over simulations, the inverse-variance weighted estimator that aggregates $\widehat{\theta}_{1g}$, $g=1,\ldots,NG_{N}$, yields an approximate value of $0.9385$ for $\mathbb{E} \left[ \widehat{\Omega}_{g}
\left(\widehat{\theta}_g\right)^{-1} \widehat{\theta}_{1g}\right]/\mathbb{E} \left[ \widehat{\Omega}_{g}
\left(\widehat{\theta}_g\right)^{-1} \right]$. The simulation results below therefore use $0.9385$ to center the inverse-variance estimator under heterogeneity. Appendix~\ref{app:dcm-additional-results} reports simulation results when centering this estimator at $1$. As expected, the bias dominates, leading to a deterioration in both the RMSE and the confidence interval coverage as $G_N$ increases under heterogeneity with the centering of Appendix~\ref{app:dcm-additional-results}.
\subparagraph{RMSE.} Figure \ref{fig:RMSE2DCM} displays the RMSE results for this specification. Under homogeneity, the MLE asymptotically achieves the Cram\'{e}r--Rao bound. The inverse-variance and adaptive-weight estimators should also attain this bound asymptotically, yet their RMSEs remain slightly above that of the MLE. They nevertheless outperform the equal-weight estimator, with improvements of at least $30\%$, as shown in Figure \ref{fig:RMSEhomoDCM}. Under heterogeneity, Figure \ref{fig:RMSEhetero2DCM} shows that the RMSE of the MLE deteriorates when $G_N$ increases. The loss of the inverse-variance weighting procedure relative to equal and adaptive weighting is small, around $10\%$ in magnitude.
\begin{figure}[!htbp]
\centering
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{RMSEhomoDCM.png}
\caption{RMSE under homogeneity, $\theta_1=1$}
\label{fig:RMSEhomoDCM}
\end{subfigure}
~
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{RMSEhetero2DCM.png}
\caption{RMSE under heterogeneity, $\theta_1=.9385$ for inverse-variance weights}
\label{fig:RMSEhetero2DCM}
\end{subfigure}
\caption{RMSE for the DCM specification \eqref{DCM}. The RMSE is standardized as in \eqref{RMSE}. Homogeneity in the left panel and heterogeneity in the right panel. RMSE on $y$-axis with number of groups $G_N$ on $x$-axis.}
\label{fig:RMSE2DCM}
\end{figure}
\subparagraph{Coverage.}
Under homogeneity, the inverse-variance and adaptive-weight procedures deliver higher coverage than expected for $G_N$ below $100$ for equal weights, $300$ for inverse-variance, and $350$ for adaptive weights, respectively. However, coverage never exceeds $97\%$. For larger $G_N$, the coverage of the equal-weight procedure falls to about $91\%$ at $G_N=600$, whereas the random-weight procedures remain above $94\%$, compared with the nominal coverage of $95\%$. Under heterogeneity, the coverage of the three weighting procedures is nearly identical, bearing in mind that a different parameter value is used for the inverse-variance weights (see Figure~\ref{fig:CovheteroDCM} for coverage when $\theta$ is set to $1$ for all procedures). The empirical coverage of each weighted procedure is very close to the nominal $95\%$ level as soon as $G_N \geq 30$.
\begin{figure}[!htbp]
\centering
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{CovhomoDCM.png}
\caption{Coverage under homogeneity, $\theta_1=1$ for all weights}
\label{fig:CovhomoDCM}
\end{subfigure}
~
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=\textwidth]{Covhetero2DCM.png}
\caption{Coverage under heterogeneity, $\theta_1=.9385$ for inverse-variance weights}
\label{fig:Cov2heteroDCM}
\end{subfigure}
\caption{Coverage of a $95\%$ confidence interval for the DCM specification \eqref{DCM}. Homogeneity in the left panel and heterogeneity in the right panel. Coverage on $y$-axis with number of groups $G_N$ on $x$-axis.}
\label{fig:Cov2DCM}
\end{figure}
\subparagraph{Homogeneity Q test.} The simulation experiment considers a nominal level of $5\%$, for $G_N=5,10,50,100$. The $y$-axis gives power as a function of the slope heterogeneity standard deviation $\sigma_{\delta}$ in \eqref{IVmodel}. The empirical level, achieved for $\sigma_{\delta}=0$, is close to the nominal level. As expected, the power increases with $G_N$.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=.37]{QpowerDCM.png}
\caption{Homogeneity Q-test for the model \eqref{DCM}, nominal $5\%$ level. Power on the $y$-axis, heterogeneity standard deviation $\sigma_{\delta}$ on the $x$-axis.}
\label{fig:QDCM}
\end{figure}
\subsection{Racial disparities in mortgage denial}
\label{sec:hmda}
We illustrate the aggregation problem using mortgage applications from the U.S. Home Mortgage Disclosure Act (HMDA), a widely used source in empirical mortgage research on lending technology, socioeconomic disparities in mortgage outcomes, and geographic heterogeneity \citep{fuster2019,bhutta2021,bartlett2022,gerardi2023,dursundeneef2023,garcia2026}.
We focus on the conditional difference in mortgage denial between Black and White applicants within each Metropolitan Statistical Area (MSA), viewed as a specific mortgage market. A discrete-choice model as in \eqref{DCM} is estimated within each MSA subsample. Different aggregation methods are then compared and homogeneity is tested.
\paragraph{Data and MSA-specific estimates.}
We use 2020--2024 HMDA application-level data on originated and denied applications from non-Hispanic Black and non-Hispanic White applicants. We retain MSAs with at least 1,000 applications and positive counts in each of the four race-by-outcome cells defined by Black/White and denial/origination status. The resulting sample contains \(G=384\) MSAs and approximately 4.65 million applications. Table~\ref{tab:desc-msa-summary} summarizes the variables and the retained sample.
\begin{table}[!htbp]
\centering
\caption{Descriptive statistics and MSA sample support}
\label{tab:desc-msa-summary}
\scriptsize
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lrrr}
\toprule
quantity & observations & mean & sd \\
\midrule
\multicolumn{4}{l}{\textbf{Panel A. Application-level continuous variables}} \\
Log income & 4647610 & 4.5874 & 0.7331 \\
Log loan amount & 4647610 & 12.4171 & 0.7869 \\
CLTV ratio & 4647610 & 0.8120 & 0.1618 \\
DTI score & 4647610 & 1.6754 & 1.2608 \\
\addlinespace
\multicolumn{4}{l}{\textbf{Panel B. MSA-level sample support}} \\
MSA sample size & & 12103.1510 & 17597.8156 \\
MSA denial rate & & 0.0828 & 0.0279 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
For application \(i\) in MSA \(g\), we estimate the Logit specification,
\begin{align*}
\mathbb{P} (Denied_{ig}=1\mid X_{ig})
=
\Lambda\left(
\alpha_g+\theta_{Black,g}Black_{ig}+X_{ig}\theta_{X,g}
\right),
\end{align*}
$\Lambda (\cdot)$ being the Logistic transformation, and
where the dependent variable \(Denied_{ig}\) equals one for a denied application and zero for an originated application. The regressor \(Black_{ig}\) equals one for a non-Hispanic Black applicant, with non-Hispanic White applicants as the reference group. The controls \(X_{ig}\) include Female, log income, log loan amount, the combined loan-to-value (CLTV) ratio, and an ordinal DTI score constructed from the public HMDA debt-to-income categories. The variance estimators $\widehat{\Omega}_{g} \left(\widehat{\theta}_g\right)$ are the standard sandwich ones.
Table~\ref{tab:trimmed-dist} shows substantial cross-MSA variation. The mean Black coefficient is \(0.639\). Figure~\ref{fig:trimmed-blackmap} shows that most estimates are positive but geographically dispersed.
\begin{table}[!htbp]
\centering
\caption{Cross-MSA coefficient distributions}
\label{tab:trimmed-dist}
\scriptsize
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lrrrr}
\toprule
variable & G & mean & sd \\
\midrule
\multicolumn{5}{l}{\textbf{Panel A. Black coefficient}} \\
Black applicant & 384 & 0.6392 & 0.4168 \\
\addlinespace
\multicolumn{5}{l}{\textbf{Panel B. Other non-intercept coefficients}} \\
DTI score & 384 & 0.6800 & 0.1398 \\
CLTV ratio & 384 & 0.1168 & 0.8738 \\
Log income & 384 & 0.0131 & 0.2577 \\
Log loan amount & 384 & -0.4959 & 0.2451 \\
Female applicant & 384 & -0.1725 & 0.1511 \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{trimmed_ge1000_black_coefficient_combined.png}
\caption{Black coefficient by MSA}
\label{fig:trimmed-blackmap}
\vspace{.5em}
\begin{minipage}{.95\textwidth}
\scriptsize \textit{Note:} The left panel plots the distribution of MSA-specific Black logit coefficients. The right panel maps the same coefficients for MSA geographies.
\end{minipage}
\end{figure}
\vspace{-25pt}
\paragraph{Aggregation and heterogeneity.}
We aggregate the MSA-specific coefficients using adaptive (AD), equal (EQ), and inverse-variance (IV) weights as in \eqref{Hatwstar}, \eqref{EW} and \eqref{IV} respectively. While the coefficients of the Panel B variables of Table \ref{tab:trimmed-agg} are aggregated in a vectorial manner, a scalar aggregation is used for the Black applicant variable. While vectorial aggregation is supposed to produce a smaller asymptotic variance by Theorem \ref{Opt}, it may also combine more bias terms, so that the resulting estimator may have a larger bias.
\begin{table}[!htbp]
\centering
\caption{Aggregation of MSA-specific coefficients with $95\%$ confidence intervals}
\label{tab:trimmed-agg}
\scriptsize
\begin{adjustbox}{max width=\textwidth}
\begin{tabular}{lrlll}
\toprule
variable & G & AD & EQ & IV \\
\midrule
\multicolumn{5}{l}{\textbf{Panel A. Black coefficient}} \\
Black applicant & 384 & 0.6969 [0.6646, 0.7292] & 0.6392 [0.5976, 0.6808] & 0.7190 [0.6613, 0.7768] \\
\addlinespace
\multicolumn{5}{l}{\textbf{Panel B. Other non-intercept coefficients}} \\
DTI score & 384 & 0.6770 [0.6634, 0.6906] & 0.6800 [0.6660, 0.6939] & 0.6891 [0.6668, 0.7115] \\
CLTV ratio & 384 & 0.0463 [-0.0390, 0.1316] & 0.1168 [0.0295, 0.2041] & -0.2453 [-0.3866, -0.1040] \\
Log income & 384 & 0.0242 [-0.0007, 0.0492] & 0.0131 [-0.0126, 0.0388] & 0.1204 [0.0749, 0.1659] \\
Log loan amount & 384 & -0.4796 [-0.5037, -0.4554] & -0.4959 [-0.5203, -0.4714] & -0.3904 [-0.4336, -0.3471] \\
Female applicant & 384 & -0.1731 [-0.1855, -0.1606] & -0.1725 [-0.1876, -0.1574] & -0.1900 [-0.2071, -0.1730] \\
\bottomrule
\end{tabular}
\end{adjustbox}
\end{table}
For the Black coefficient, adaptive weighting gives \(0.697\), equal weighting \(0.639\), and inverse-variance weighting \(0.719\). While these values are economically similar, the random-weights estimators seem substantially higher than the equal weight ones. In particular, the AD and IV based confidence intervals do not contain the EQ point estimation. The standard deviation of the AD procedure is around $30\%$ lower than those of the EQ and IV estimators.
For the Panel B coefficients which are aggregated in a vectorial manner, the AD procedure also produces a smaller estimated standard deviation, but marginally compared to the EQ estimator. The IV one has a larger estimated standard deviation than its two competitors. The AD and EQ procedures are in agreement for the significance of all coefficients except for the CLTV ratio variable, which is viewed as non-significant with AD inference. Note also that the EQ and IV estimates of this coefficient have opposite signs, which may be due to bias.
Consider now homogeneity of the Black applicant coefficient. The left panel of Figure \ref{fig:trimmed-blackmap} and its standard deviation of 0.4168 across MSAs in Table \ref{tab:trimmed-dist} both suggest some heterogeneity. The corrected standard deviation computed from $\check{\Sigma}$ in \eqref{Checksig} yields a smaller 0.2498, suggesting that estimation uncertainty is an important source of spurious heterogeneity.
However, computation of the modified Cochran homogeneity statistic $\widehat{Q}$ in \eqref{Homstat} provides strong evidence of parameter heterogeneity, with a value of 6.1598 for the Black coefficient alone and of 14.0173 for the variables of Panel B, leading to a rejection of the null of homogeneity at any usual statistical level.
\section{Concluding remarks}
\label{sec:concluding-remarks}
Using estimated weights to aggregate subsample estimators is common in meta-analysis and is also relevant when applying distributed estimation to batches of observations from different origins, which may affect the parameter of interest and/or the variances of the estimators. The equal weighting procedure often considered in the distributed estimation literature may be inappropriate in many economic datasets where heterogeneity affects the dispersion of the estimators.
However, optimal deterministic weights based on the inverse of the sum of the heterogeneity variance and expected estimation variance can be difficult to apply. In full generality, the estimation variance may depend on the parameter of interest in an unknown way, so that its expectation may be difficult to estimate under parameter heterogeneity. This paper proposes an adaptive weighting scheme which addresses this issue in a uniform manner, at the price of using estimated weights which asymptotically depend on the subsample parameter. This creates an estimation bias and, under heterogeneity, a correlation bias due to the dependence of the weights on the subsample parameter. The latter also implies that the variance of the weighted estimator is difficult to compute.
We show that, under suitable conditions, these issues can be addressed when the number of groups $G_N$ is negligible with respect to the (average) subsample size $n= N/G_N$, where $N$ is the total sample size. Under such conditions, standard Gaussian inference applies to the adaptive weighting scheme, which is also optimal. This uses a heterogeneity-variance estimator which asymptotically vanishes with a fast rate under homogeneity, as in \citet{hedges1985}. As our framework covers a large range of estimation methods,
these results extend \citet{gu2023}, who restrict to the homogeneous case, and extend \citet{lu2023}, \citet{fernandezval2026}, who both consider equal-weight averages which may be inefficient for confidence intervals.
As heterogeneity is a threat to external validity as noted in \citet{vivalt2020}, the paper also considers a homogeneity test which extends \citet{cochran1937}, \citet{pesaran2008}, \citet{ritz2008}.
Our simulations show that our theoretical results seem applicable when $G_N$ is negligible with respect to $n$, typically less than $ n/10$, $n$ being in the range of a thousand. An application to Black--White differences in mortgage denial across U.S. metropolitan areas illustrates our methodology.
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