There was a similar question a few days ago which had the relevant reference:
- Belloni, A., Chernozhukov, V., and Hansen, C. (2014) "Inference on Treatment Effects after Selection among High-Dimensional Controls", Review of Economic Studies, 81(2), pp. 608-50 (link)
At least for me the paper is a pretty tough read because the proofs behind this relatively simple are fairly elaborate. When you are interested in estimating a model like
$$y_i = \alpha T_i + X_i'\beta + \epsilon_i$$
where $y_i$ is your outcome, $T_i$ is some treatment effect of interest, and $X_i$ is a vector of potential controls. The target parameter is $\alpha$. Assuming that most of the variation in your outcome is explained by the treatment and a sparse set of controls, Belloni et al. (2014) develop a double-robust selection method which provides correct point estimates and valid confidence intervals. This sparsity assumption is important though.
If $X_i$ includes a few important predictors of $y_i$ but you don't know which they are (either single variables, their higher order polynomials, or interactions with other variables), you can perform a three step selection procedure:
- regress $y_i$ on $X_i$, their squares, and interactions, and select important predictors using LASSO
- regress $T_i$ on $X_i$, their squares, and interactions, and select important predictors using LASSO
- regress $y_i$ on $T_i$ and all the variables which were selected in either of the first two steps
They provide proofs as for why this works and why you get the correct confidence intervals etc out of this method. They also show that if you only perform a LASSO selection on the above regression and then regress the outcome on the treatment and the selected variables you get wrong point estimates and false confidence intervals, like Björn already said.
The purpose for doing this is twofold: comparing your initial model, where variable selection was guided by intuition or theory, to the double-robust selection model gives you an idea as to how good your first model was. Perhaps your first model forgot some important squared or interaction terms and thus suffers from misspecified functional form or omitted variables. Secondly, the Belloni et al. (2014) method can improve inference on your target parameter because redundant regressors were penalized away in their procedure.