We present a technique using data depth functions and resampling to perform best subset variable selection for a wide range of statistical models. We do this by assigning a score, called an $e$-value, to a candidate model, and use a fast bootstrap method to approximate sample versions of these $e$-values. Under general conditions, $e$-values can separate statistical models that adequately explain properties of the data from those that do not. This results in a fast algorithm that fits only a single model and evaluates $p +1$ models, $p$ being the number of predictors under consideration, as opposed to the traditional requirement of fitting and evaluating $2^{p}$ models. We illustrate in simulation experiments that our proposed method typically performs better than an array of currently used methods for variable selection in linear models and fixed effect selection in linear mixed models. As a real data application, we use our procedure to elicit climatic drivers of Indian summer monsoon precipitation.