I am doing some work on conformal prediction methods, which allow for doing predictive regression-based inference with minimal assumptions. Mostly to help myself understand the methods in algorithmic terms, I created the following tutorial: link.

An accessible introduction is offered in this paper by Lei et al. (2017, arxiv), which accompanies the R package, `conformalInference`

(github). They demonstrate conformal inference methods in connection with high dimensional regression and covariate selection.

In the causal inference literature, Chernozhukov et al. (2017, arxiv) use conformal methods for robust inference with synthetic control and related panel methods. Coauthors and I are doing some more work in this area.

Chernozhukov et al. (2018, arxiv) also have new work extending conformal inference to time series and other dependent-data settings.