Incremental intervention effects in studies with dropout and many timepoints#

Modern longitudinal studies collect feature data at many timepoints, often of the same order of sample size. Such studies are typically affected by dropout and positivity violations. We tackle these problems by generalizing effects of recent incremental interventions (which shift propensity scores rather than set treatment values deterministically) to accommodate multiple outcomes and subject dropout. We give an identifying expression for incremental intervention effects when dropout is conditionally ignorable (without requiring treatment positivity) and derive the nonparametric efficiency bound for estimating such effects. Then we present efficient nonparametric estimators, showing that they converge at fast parametric rates and yield uniform inferential guarantees, even when nuisance functions are estimated flexibly at slower rates. We also study the variance ratio of incremental intervention effects relative to more conventional deterministic effects in a novel infinite time horizon setting, where the number of timepoints can grow with sample size and show that incremental intervention effects yield near-exponential gains in statistical precision in this setup. Finally, we conclude with simulations and apply our methods in a study of the effect of low-dose aspirin on pregnancy outcomes.

Efficient Nonparametric Causal Inference with Missing Exposure Information

Missing exposure information is a very common feature of many observational studies. Here we study identifiability and efficient estimation of causal effects on vector outcomes, in such cases where treatment is unconfounded but partially missing. We consider a missing at random setting where missingness in treatment can depend not only on complex covariates, but also on post-treatment outcomes. We give a new identifying expression for average treatment effects in this setting, along with the efficient influence function for this parameter in a nonparametric model, which yields a nonparametric efficiency bound. We use this latter result to construct nonparametric estimators that are less sensitive to the curse of dimensionality than usual, e. g. by having faster rates of convergence than the complex nuisance estimators they rely on. Further we show that these estimators can be root-n consistent and asymptotically normal under weak nonparametric conditions, even when constructed using flexible machine learning. Finally we apply these results to the problem of causal inference with a partially missing instrumental variable.

Handling Missing Data in Instrumental Variable Methods for Causal Inference

In instrumental variable studies, missing instrument data are very common. For example, in the Wisconsin Longitudinal Study, one can use genotype data as a Mendelian randomization–style instrument, but this information is often missing when subjects do not contribute saliva samples or when the genotyping platform output is ambiguous. Here we review missing at random assumptions one can use to identify instrumental variable causal effects, and discuss various approaches for estimation and inference. We consider likelihood-based methods, regression and weighting estimators, and doubly robust estimators. The likelihood-based methods yield the most precise inference and are optimal under the model assumptions, while the doubly robust estimators can attain the nonparametric efficiency bound while allowing flexible nonparametric estimation of nuisance functions (e.g., instrument propensity scores). The regression and weighting estimators can sometimes be easiest to describe and implement. Our main contribution is an extensive review of this wide array of estimators under varied missing-at-random assumptions, along with discussion of asymptotic properties and inferential tools. We also implement many of the estimators in an analysis of the Wisconsin Longitudinal Study, to study effects of impaired cognitive functioning on depression.

EFFICIENT ESTIMATION OF INTEGRATED VOLATILITY AND RELATED PROCESSES

We derive nonparametric efficiency bounds for regular estimators of integrated smooth transformations of instantaneous variances, in particular, integrated power variance. We find that realized variance attains the efficiency bound for integrated variance under both regular and irregular sampling schemes. For estimating higher powers such as integrated quarticity, the block-based procedures of Mykland and Zhang (2009) can get arbitrarily close to the nonparametric bounds, when observation times are equidistant. Moreover, the estimator in Jacod and Rosenbaum (2013), whose efficiency was documented for the submodel assuming constant volatility, is efficient also for nonconstant volatility paths. When the observation times are possibly random but predictable, we provide an estimator, similar to that of Kristensen (2010), which can get arbitrarily close to the nonparametric bound. Finally, parametric information about the functional form of volatility leads to a lower efficiency bound, unless the volatility process is piecewise constant.