This paper studies treatment effect models in which individuals are classified into unobserved groups based on heterogeneous treatment rules. Using a finite mixture approach, we propose a marginal treatment effect (MTE) framework in which the treatment choice and outcome equations can be heterogeneous across groups. Under the availability of instrumental variables specific to each group, we show that the MTE for each group can be separately identified. Based on our identification result, we propose a two-step semiparametric procedure for estimating the group-wise MTE. We illustrate the usefulness of the proposed method with an application to economic returns to college education.
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The variational autoencoder (VAE) is a popular deep latent variable model used to analyse high-dimensional datasets by learning a low-dimensional latent representation of the data. It simultaneously learns a generative model and an inference network to perform approximate posterior inference. Recently proposed extensions to VAEs that can handle temporal and longitudinal data have applications in healthcare, behavioural modelling, and predictive maintenance. However, these extensions do not account for heterogeneous data (i.e., data comprising of continuous and discrete attributes), which is common in many real-life applications. In this work, we propose the heterogeneous longitudinal VAE (HL-VAE) that extends the existing temporal and longitudinal VAEs to heterogeneous data. HL-VAE provides efficient inference for high-dimensional datasets and includes likelihood models for continuous, count, categorical, and ordinal data while accounting for missing observations. We demonstrate our model's efficacy through simulated as well as clinical datasets, and show that our proposed model achieves competitive performance in missing value imputation and predictive accuracy.
Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.
Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.
We consider M-estimation problems, where the target value is determined using a minimizer of an expected functional of a Levy process. With discrete observations from the Levy process, we can produce a "quasi-path" by shuffling increments of the Levy process, we call it a quasi-process. Under a suitable sampling scheme, a quasi-process can converge weakly to the true process according to the properties of the stationary and independent increments. Using this resampling technique, we can estimate objective functionals similar to those estimated using the Monte Carlo simulations, and it is available as a contrast function. The M-estimator based on these quasi-processes can be consistent and asymptotically normal.
Background: Interpreting instrumental variable results often requires further assumptions in addition to the core assumptions of relevance, independence, and the exclusion restriction. Methods: We assess whether instrument-exposure additive homogeneity renders the Wald estimand equal to the average derivative effect (ADE) in the case of a binary instrument and a continuous exposure. Results: Instrument-exposure additive homogeneity is insufficient for ADE identification when the instrument is binary, the exposure is continuous and the effect of the exposure on the outcome is non-linear on the additive scale. For a binary exposure, the exposure-outcome effect is necessarily additive linear, so the homogeneity condition is sufficient. Conclusions: For binary instruments, instrument-exposure additive homogeneity identifies the ADE if the exposure is also binary. Otherwise, additional assumptions (such as additive linearity of the exposure-outcome effect) are required.
In randomized experiments, the actual treatments received by some experimental units may differ from their treatment assignments. This non-compliance issue often occurs in clinical trials, social experiments, and the applications of randomized experiments in many other fields. Under certain assumptions, the average treatment effect for the compliers is identifiable and equal to the ratio of the intention-to-treat effects of the potential outcomes to that of the potential treatment received. To improve the estimation efficiency, we propose three model-assisted estimators for the complier average treatment effect in randomized experiments with a binary outcome. We study their asymptotic properties, compare their efficiencies with that of the Wald estimator, and propose the Neyman-type conservative variance estimators to facilitate valid inferences. Moreover, we extend our methods and theory to estimate the multiplicative complier average treatment effect. Our analysis is randomization-based, allowing the working models to be misspecified. Finally, we conduct simulation studies to illustrate the advantages of the model-assisted methods and apply these analysis methods in a randomized experiment to evaluate the effect of academic services or incentives on academic performance.
In this paper we study the finite sample and asymptotic properties of various weighting estimators of the local average treatment effect (LATE), several of which are based on Abadie (2003)'s kappa theorem. Our framework presumes a binary endogenous explanatory variable ("treatment") and a binary instrumental variable, which may only be valid after conditioning on additional covariates. We argue that one of the Abadie estimators, which we show is weight normalized, is likely to dominate the others in many contexts. A notable exception is in settings with one-sided noncompliance, where certain unnormalized estimators have the advantage of being based on a denominator that is bounded away from zero. We use a simulation study and three empirical applications to illustrate our findings. In applications to causal effects of college education using the college proximity instrument (Card, 1995) and causal effects of childbearing using the sibling sex composition instrument (Angrist and Evans, 1998), the unnormalized estimates are clearly unreasonable, with "incorrect" signs, magnitudes, or both. Overall, our results suggest that (i) the relative performance of different kappa weighting estimators varies with features of the data-generating process; and that (ii) the normalized version of Tan (2006)'s estimator may be an attractive alternative in many contexts. Applied researchers with access to a binary instrumental variable should also consider covariate balancing or doubly robust estimators of the LATE.
The inverse probability (IPW) and doubly robust (DR) estimators are often used to estimate the average causal effect (ATE), but are vulnerable to outliers. The IPW/DR median can be used for outlier-resistant estimation of the ATE, but the outlier resistance of the median is limited and it is not resistant enough for heavy contamination. We propose extensions of the IPW/DR estimators with density power weighting, which can eliminate the influence of outliers almost completely. The outlier resistance of the proposed estimators is evaluated through the unbiasedness of the estimating equations. Unlike the median-based methods, our estimators are resistant to outliers even under heavy contamination. Interestingly, the naive extension of the DR estimator requires bias correction to keep the double robustness even under the most tractable form of contamination. In addition, the proposed estimators are found to be highly resistant to outliers in more difficult settings where the contamination ratio depends on the covariates. The outlier resistance of our estimators from the viewpoint of the influence function is also favorable. Our theoretical results are verified via Monte Carlo simulations and real data analysis. The proposed methods were found to have more outlier resistance than the median-based methods and estimated the potential mean with a smaller error than the median-based methods.
Models for dependent data are distinguished by their targets of inference. Marginal models are useful when interest lies in quantifying associations averaged across a population of clusters. When the functional form of a covariate-outcome association is unknown, flexible regression methods are needed to allow for potentially non-linear relationships. We propose a novel marginal additive model (MAM) for modelling cluster-correlated data with non-linear population-averaged associations. The proposed MAM is a unified framework for estimation and uncertainty quantification of a marginal mean model, combined with inference for between-cluster variability and cluster-specific prediction. We propose a fitting algorithm that enables efficient computation of standard errors and corrects for estimation of penalty terms. We demonstrate the proposed methods in simulations and in application to (i) a longitudinal study of beaver foraging behaviour, and (ii) a spatial analysis of Loaloa infection in West Africa. R code for implementing the proposed methodology is available at //github.com/awstringer1/mam.
In a sports competition, a team might lose a powerful incentive to exert full effort if its final rank does not depend on the outcome of the matches still to be played. Therefore, the organiser should reduce the probability of such a situation to the extent possible. Our paper provides a classification scheme to identify these weakly (where one team is indifferent) or strongly (where both teams are indifferent) stakeless games. A statistical model is estimated to simulate the UEFA Champions League groups and compare the candidate schedules used in the 2021/22 season according to the competitiveness of the matches played in the last round(s). The option followed in four of the eight groups is found to be optimal under a wide set of parameters. Minimising the number of strongly stakeless matches is verified to be a likely goal in the computer draw of the fixture that remains hidden from the public.
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