inverse probability weighting [AIPWT, @rrz], has a doubly-robust property in that if either the mean or propensity model is correctly specified, consistent estimates of the causal effect will be obtained. We modified PENCOMP slightly using Bayesian additive regression trees (BART), a flexible model to ease the burden of model specification by the researcher, and apply this to our proposed method.

We organize our paper as follows. We set up the framework for our problem, and provide a brief review of of MSM, PENCOMP, and Bayesian additive regression trees (BART) in Section 2. We develop our proposed method in Section 3. We then explore some of the empirical properties of our proposed method compared to a naïve method and MSM using a simulation study in Section 4. Section 5 describes the HRS data and the results of our negative wealth shock analysis. Section 6 concludes with a discussion of the implication of our results as well as future work.

Review of Relevant Methods {#chap3:review}
==========================

Setup and notation
------------------

Let $V=\{V_1,V_2,\ldots,V_p\}$ be $p$ baseline covariates, $Z_t$ be the treatment allocation at time $t=1,\ldots,T$ where $Z_t=1$ indicates a subject receiving a negative wealth shock at $t$ and $Z_t=0$ indicates no negative wealth 