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Causal inference in spatial settings is met with unique challenges and opportunities. In spatial settings, a unit's outcome might be affected by the exposure at many locations and the confounders might be spatially structured. Using causal diagrams, we investigate the complications that arise when investigating causal relationships from spatial data. We illustrate that spatial confounding and interference can manifest as each other, meaning that investigating the presence of one can lead to wrongful conclusions in the presence of the other. We also show that statistical dependencies in the exposure can render standard analyses invalid, which can have crucial implications for understanding the effect of interventions on dependent units. Based on the conclusions from this investigation, we propose a parametric approach that simultaneously accounts for interference and mitigates bias from local and neighborhood unmeasured spatial confounding. We show that incorporating an exposure model is necessary from a Bayesian perspective. Therefore, the proposed approach is based on modeling the exposure and the outcome simultaneously while accounting for the presence of common spatially-structured unmeasured predictors. We illustrate our approach with a simulation study and with an analysis of the local and interference effects of sulfur dioxide emissions from power plants on cardiovascular mortality.

Instrumental variable (IV) regression is a standard strategy for learning causal relationships between confounded treatment and outcome variables from observational data by utilizing an instrumental variable, which affects the outcome only through the treatment. In classical IV regression, learning proceeds in two stages: stage 1 performs linear regression from the instrument to the treatment; and stage 2 performs linear regression from the treatment to the outcome, conditioned on the instrument. We propose a novel method, deep feature instrumental variable regression (DFIV), to address the case where relations between instruments, treatments, and outcomes may be nonlinear. In this case, deep neural nets are trained to define informative nonlinear features on the instruments and treatments. We propose an alternating training regime for these features to ensure good end-to-end performance when composing stages 1 and 2, thus obtaining highly flexible feature maps in a computationally efficient manner. DFIV outperforms recent state-of-the-art methods on challenging IV benchmarks, including settings involving high dimensional image data. DFIV also exhibits competitive performance in off-policy policy evaluation for reinforcement learning, which can be understood as an IV regression task.

With some regularity conditions maximum likelihood estimators (MLEs) always produce asymptotically optimal (in the sense of consistency, efficiency, sufficiency, and unbiasedness) estimators. But in general, the MLEs lead to non-robust statistical inference, for example, pricing models and risk measures. Actuarial claim severity is continuous, right-skewed, and frequently heavy-tailed. The data sets that such models are usually fitted to contain outliers that are difficult to identify and separate from genuine data. Moreover, due to commonly used actuarial "loss control strategies" in financial and insurance industries, the random variables we observe and wish to model are affected by truncation (due to deductibles), censoring (due to policy limits), scaling (due to coinsurance proportions) and other transformations. To alleviate the lack of robustness of MLE-based inference in risk modeling, here in this paper, we propose and develop a new method of estimation - method of truncated moments (MTuM) and generalize it for different scenarios of loss control mechanism. Various asymptotic properties of those estimates are established by using central limit theory. New connections between different estimators are found. A comparative study of newly-designed methods with the corresponding MLEs is performed. Detail investigation has been done for a single parameter Pareto loss model including a simulation study.

In biomedical studies, estimating drug effects on chronic diseases requires a long follow-up period, which is difficult to meet in randomized clinical trials (RCTs). The use of a short-term surrogate to replace the long-term outcome for assessing the drug effect relies on stringent assumptions that empirical studies often fail to satisfy. Motivated by a kidney disease study, we investigate the drug effects on long-term outcomes by combining an RCT without observation of long-term outcome and an observational study in which the long-term outcome is observed but unmeasured confounding may exist. Under a mean exchangeability assumption weaker than the previous literature, we identify the average treatment effects in the RCT and derive the associated efficient influence function and semiparametric efficiency bound. Furthermore, we propose a locally efficient doubly robust estimator and an inverse probability weighted (IPW) estimator. The former attains the semiparametric efficiency bound if all the working models are correctly specified. The latter has a simpler form and requires much fewer model specifications. The IPW estimator using estimated propensity scores is more efficient than that using true propensity scores and achieves the semiparametric efficient bound in the case of discrete covariates and surrogates with finite support. Both estimators are shown to be consistent and asymptotically normally distributed. Extensive simulations are conducted to evaluate the finite-sample performance of the proposed estimators. We apply the proposed methods to estimate the efficacy of oral hydroxychloroquine on renal failure in a real-world data analysis.

In multivariate time series analysis, the coherence measures the linear dependency between two-time series at different frequencies. However, real data applications often exhibit nonlinear dependency in the frequency domain. Conventional coherence analysis fails to capture such dependency. The quantile coherence, on the other hand, characterizes nonlinear dependency by defining the coherence at a set of quantile levels based on trigonometric quantile regression. Although quantile coherence is a more powerful tool, its estimation remains challenging due to the high level of noise. This paper introduces a new estimation technique for quantile coherence. The proposed method is semi-parametric, which uses the parametric form of the spectrum of the vector autoregressive (VAR) model as an approximation to the quantile spectral matrix, along with nonparametric smoothing across quantiles. For each fixed quantile level, we obtain the VAR parameters from the quantile periodograms, then, using the Durbin-Levinson algorithm, we calculate the preliminary estimate of quantile coherence using the VAR parameters. Finally, we smooth the preliminary estimate of quantile coherence across quantiles using a nonparametric smoother. Numerical results show that the proposed estimation method outperforms nonparametric methods. We show that quantile coherence-based bivariate time series clustering has advantages over the ordinary VAR coherence. For applications, the identified clusters of financial stocks by quantile coherence with a market benchmark are shown to have an intriguing and more accurate structure of diversified investment portfolios that may be used by investors to make better decisions.

Many real-world decision-making tasks require learning causal relationships between a set of variables. Traditional causal discovery methods, however, require that all variables are observed, which is often not feasible in practical scenarios. Without additional assumptions about the unobserved variables, it is not possible to recover any causal relationships from observational data. Fortunately, in many applied settings, additional structure among the confounders can be expected. In particular, pervasive confounding is commonly encountered and has been utilized for consistent causal estimation in linear causal models. In this paper, we present a provably consistent method to estimate causal relationships in the non-linear, pervasive confounding setting. The core of our procedure relies on the ability to estimate the confounding variation through a simple spectral decomposition of the observed data matrix. We derive a DAG score function based on this insight, prove its consistency in recovering a correct ordering of the DAG, and empirically compare it to previous approaches. We demonstrate improved performance on both simulated and real datasets by explicitly accounting for both confounders and non-linear effects.

We develop a practical way of addressing the Errors-In-Variables (EIV) problem in the Generalized Method of Moments (GMM) framework. We focus on the settings in which the variability of the EIV is a fraction of that of the mismeasured variables, which is typical for empirical applications. For any initial set of moment conditions our approach provides a corrected set of moment conditions that are robust to the EIV. We show that the GMM estimator based on these moments is root-n-consistent, with the standard tests and confidence intervals providing valid inference. This is true even when the EIV are so large that naive estimators (that ignore the EIV problem) may be heavily biased with the confidence intervals having 0% coverage. Our approach involves no nonparametric estimation, which is particularly important for applications with multiple covariates, and settings with multivariate, serially correlated, or non-classical EIV.

Model specification searches and modifications are commonly employed in covariance structure analysis (CSA) or structural equation modeling (SEM) to improve the goodness-of-fit. However, these practices can be susceptible to capitalizing on chance, as a model that fits one sample may not generalize to another sample from the same population. This paper introduces the improved Lagrange Multipliers (LM) test, which provides a reliable method for conducting a thorough model specification search and effectively identifying missing parameters. By leveraging the stepwise bootstrap method in the standard LM and Wald tests, our data-driven approach enhances the accuracy of parameter identification. The results from Monte Carlo simulations and two empirical applications in political science demonstrate the effectiveness of the improved LM test, particularly when dealing with small sample sizes and models with large degrees of freedom. This approach contributes to better statistical fit and addresses the issue of capitalization on chance in model specification.

An N-of-1 trial is a multiple crossover trial conducted in a single individual to provide evidence to directly inform personalized treatment decisions. Advancements in wearable devices greatly improved the feasibility of adopting these trials to identify optimal individual treatment plans, particularly when treatments differ among individuals and responses are highly heterogeneous. Our work was motivated by the I-STOP-AFib Study, which examined the impact of different triggers on atrial fibrillation (AF) occurrence. We described a causal framework for 'N-of-1' trial using potential treatment selection paths and potential outcome paths. Two estimands of individual causal effect were defined:(a) the effect of continuous exposure, and (b) the effect of an individual observed behavior. We addressed three challenges: (a) imperfect compliance to the randomized treatment assignment; (b) binary treatments and binary outcomes which led to the 'non-collapsibility' issue of estimating odds ratios; and (c) serial inference in the longitudinal observations. We adopted the Bayesian IV approach where the study randomization was the IV as it impacted the choice of exposure of a subject but not directly the outcome. Estimations were through a system of two parametric Bayesian models to estimate the individual causal effect. Our model got around the non-collapsibility and non-consistency by modeling the confounding mechanism through latent structural models and by inferring with Bayesian posterior of functionals. Autocorrelation present in the repeated measurements was also accounted for. The simulation study showed our method largely reduced bias and greatly improved the coverage of the estimated causal effect, compared to existing methods (ITT, PP, and AT). We applied the method to I-STOP-AFib Study to estimate the individual effect of alcohol on AF occurrence.

Causal discovery and causal reasoning are classically treated as separate and consecutive tasks: one first infers the causal graph, and then uses it to estimate causal effects of interventions. However, such a two-stage approach is uneconomical, especially in terms of actively collected interventional data, since the causal query of interest may not require a fully-specified causal model. From a Bayesian perspective, it is also unnatural, since a causal query (e.g., the causal graph or some causal effect) can be viewed as a latent quantity subject to posterior inference -- other unobserved quantities that are not of direct interest (e.g., the full causal model) ought to be marginalized out in this process and contribute to our epistemic uncertainty. In this work, we propose Active Bayesian Causal Inference (ABCI), a fully-Bayesian active learning framework for integrated causal discovery and reasoning, which jointly infers a posterior over causal models and queries of interest. In our approach to ABCI, we focus on the class of causally-sufficient, nonlinear additive noise models, which we model using Gaussian processes. We sequentially design experiments that are maximally informative about our target causal query, collect the corresponding interventional data, and update our beliefs to choose the next experiment. Through simulations, we demonstrate that our approach is more data-efficient than several baselines that only focus on learning the full causal graph. This allows us to accurately learn downstream causal queries from fewer samples while providing well-calibrated uncertainty estimates for the quantities of interest.