Likelihood Analysis of Causal Models

Abstract

We analyze several problems in causal inference from the perspective of maximum likelihood. Two archetypal likelihoods are primarily concerned: Gaussian likelihood for continuous data and multinomial likelihood for discrete data. In the first half of this dissertation, Gaussian likelihood is considered for testing and estimation. Motivated by the selection of causal graphs, in Chapter 2, we study testing between marginal and conditional independence in a Gaussian setting with the likelihood ratio test (LRT). We introduce a class of “envelope” distributions by taking pointwise suprema over asymptotic distribution functions of LRT. We show that these envelope distributions are well-behaved and lead to uniformly consistent model selection procedures. In Chapter 3, we consider the estimation of total causal effects under causal sufficiency and linearity. We derive a simple recursive least squares estimator as the MLE under Gaussian errors, which can consistently estimate any identified total effect, under either point or joint intervention. Further, this estimator is shown to be asymptotically efficient even beyond the Gaussian assumption, when compared to a reasonably large class of estimators. In the latter half, we study the inference of instrumental variable (IV) models with discrete data. In Chapter 4, we develop non-asymptotic tail bounds for the likelihood ratio statistic under multinomial sampling. Such bounds are established by bounding the moment generating function of the statistic uniformly over all multinomial parameters, which can be viewed as a finite-sample version of Wilks' theorem. Then, in Chapter 5, such bounds are combined with a convex parametrization of the IV model to streamline statistical inference as convex programming. This approach delivers strong guarantees and circumvents the difficulty in identification and post-selection inference. The approach is illustrated with a case study on the distributional effect of military service on annual earnings, using the Vietnam draft lottery as a monotone instrument. Finally, we study partial identification of the average treatment effect in a latent variable formulation and make connections to the Bell-CHSH inequalities in quantum mechanics.