Exact matching + multiple regression on high-dimensional treatment-control study? I'm working on a project with healthcare data where episodes of care in the treatment and control groups must be matched to estimate average treatment effect (ATE).
I have several hundred covariates including age, sex, and various utilization and health risk factors (most of which are sparse binary variables).
While my go-to model has been propensity score matching, recent research by Gary King, Richard Nielsen, and others have recommended using exact matching (or coarsened exact matching).
Question: Is it appropriate to use exact matching on a subset of the covariates, followed by multiple regression on the remaining variables in the matched data in order to estimate the ATE? The subset of covariates could be selected by choosing the largest effect sizes from a simple regression on the outcome variable.
 A: The King & Nielsen (2016) paper is misleading. It has not been peer-reviewed, and it makes a claim about the results of a testable assumption that you can assess in your own data set (i.e., whether propensity score matching produces balance). In addition, there has been some work to debunk the paper, and several instances in which coarsened exact matching does not perform well. The paper has since been peer-reviewed and accepted. The claims it makes are still empirically verifiable, though, so there is no reason to categorically avoid using propensity score matching. The paper makes testable predictions, and you can assess whether those predictions are realized in your data. The prediction is that propensity score matching will not yield balance; you can simply assess balance in your data after matching.
Update (1/1/20): A paper examining the validity of King & Nielsen (2019) in pharmacoepidemiology is Ripollone et al. (2018). They find that the propensity score paradox does occur, but far beyond recommended and common practices for propensity score matching. In applications, propensity score matching is effective at achieving bias and lowering bias. The same group (Ripollone et al. 2019), who maybe have a bone to pick with King, also evaluate the performance of CEM and find that it yields extremely high error in the effect estimates compared to propensity score matching.
The question of how to deal with high-dimensional covariates in causal inference is really hot right now, and there are several modern techniques that have been developed that you should consider before propensity score matching and regression. Matching and regression are some of the earliest causal inference techniques and there has been so much advancement upon these methods that really no one should be using their basic forms. Here are some recommendations for causal inference tools for high-dimensional data:
Targeted Minimum Loss-Based Estimation (TMLE) - TMLE is a doubly-robust effect estimator that relies on machine learning and regression to remove confounding without making functional form assumptions about the treatment or outcome model. There is a version called "Collaborative" TMLE (CTMLE), which specifically addresses the problem of high-dimensional covariates. TMLE has been shown to do very well in simulations and in a recent causal inference competition (Dorie et a., 2019). It's very easy to implement and there is an easy-to-use R package (TMLE) to do it. It is becoming the gold standard in causal inference. See Schuler & Rose (2017) for an introduction.
Bayesian Additive Regression Trees (BART) - BART is a machine learning method that uses Bayesian components both to yield good performance and inference. It works like a flexible outcome regression model, but you can include the propensity score (potentially also estimated using BART) to increase its robustness and performance. Because it only prioritizes covariates that are predictive of the outcome, it automatically selects the relevant variables from a potentially long list, and therefore is effective in high dimensions. It has also been shown to have great performance and to have done will in the causal inference competition, and there is also an easy-to-use R package (bartCause) to implement it. See Hill (2011) for an introduction. 
Group Lasso with Doubly Robust Estimation (GLIDER) - GLIDER is a double-robust propensity score weighting + regression estimator that is especially useful in high dimensions. It uses lasso to select the right covariates that predict both the outcome and the propensity score. It uses an adaptive lasso, which means the coefficients are asymptotically unbiased. It is straightforward to include many transformations of variables to account for potential nonlinearities; if they aren't useful in the model, they are lasso'ed out. See Koch, Vock, & Wolfson (2018) for an introduction.
Hopefully that should get you started. Matching and regression do not appear appropriate to me in this case, and there are several better-performing methods that would suit your goals. You should consult with a biostatistician rather than attempt to implement out-of-date methods. 

Dorie, V., Hill, J., Shalit, U., Scott, M., & Cervone, D. (2019). Automated versus Do-It-Yourself Methods for Causal Inference: Lessons Learned from a Data Analysis Competition. Statistical Science, 34(1), 43–68. https://doi.org/10.1214/18-STS667
Hill, J. L. (2011). Bayesian Nonparametric Modeling for Causal Inference. Journal of Computational and Graphical Statistics, 20(1), 217–240. https://doi.org/10.1198/jcgs.2010.08162
King, G., & Nielsen, R. (2016). Why propensity scores should not be used for matching. Retrieved from http://www.polmeth.wustl.edu/files/polmeth/psnot4.pdf
King, G., & Nielsen, R. (2019). Why Propensity Scores Should Not Be Used for Matching. Political Analysis, 1–20. https://doi.org/10.1017/pan.2019.11
Koch, B., Vock, D. M., & Wolfson, J. (2018). Covariate selection with group lasso and doubly robust estimation of causal effects. Biometrics, 74(1), 8–17. https://doi.org/10.1111/biom.12736
Ripollone, J. E., Huybrechts, K. F., Rothman, K. J., Ferguson, R. E., & Franklin, J. M. (2018). Implications of the Propensity Score Matching Paradox in Pharmacoepidemiology. American Journal of Epidemiology, 187(9), 1951–1961. https://doi.org/10.1093/aje/kwy078
Ripollone, J. E., Huybrechts, K. F., Rothman, K. J., Ferguson, R. E., & Franklin, J. M. (2019). Evaluating the Utility of Coarsened Exact Matching for Pharmacoepidemiology using Real and Simulated Claims Data. American Journal of Epidemiology, kwz268. https://doi.org/10.1093/aje/kwz268
Schuler, M. S., & Rose, S. (2017). Targeted Maximum Likelihood Estimation for Causal Inference in Observational Studies. American Journal of Epidemiology, 185(1), 65–73. https://doi.org/10.1093/aje/kww165
A: It's important to answer the question "why are we using matching in this study?" There are lots of good reasons to use matching, but wanting to estimate the ATE isn't one of them.  Most matching methods create a cohort that is excellent for estimating the effect of exposure but whose covariate distribution is no longer identical to the source population. That is, the matching methods drop or down-weight regions of poor common support in the covariate space.
If you really want to estimate the ATE in this setting, consider just fitting a regression model on the source population (possibly using a relaxed lasso to help with the numerous sparse binary variables) and then use the model to estimate your ATE. You can use the model to calculate the predicted the outcomes for everyone in the source population (once under treatment and once under control). You now have predicted (Y0, Y1) pairs for everyone, and calculating the ATE is straightforward. The estimate will only be as good as the model. For example, you may have to assume additive effects for the binary variables that are being estimated largely from one of the exposures alone. You may be doing extensive extrapolation into regions where there is little common support. That depends on your particular dataset.
One question to ask is "how important is it to estimate the ATE for this study?" Are you really interested in the average effect over the source population or are you more interested in a persuasive study design that yields a robust estimate of the treatment effect over a well defined study population? If it's the latter, the matching methods can be helpful.
