This paper investigates the problem of selecting variables in regression-type models for an "instrumental" setting. Our study is motivated by empirically verifying the conditional convergence hypothesis used in the economical literature concerning the growth rate. To avoid unnecessary discussion about the choice and the pertinence of instrumental variables, we embed the model in a very high dimensional setting. We propose a selection procedure with no optimization step called LOLA, for Learning Out of Leaders with Adaptation. LOLA is an auto-driven algorithm with two thresholding steps. The consistency of the procedure is proved under sparsity conditions and simulations are conducted to illustrate the practical good performances of LOLA. The behavior of the algorithm is studied when instrumental variables are artificially added without a priori significant connection to the model. Using our algorithm, we provide a solution for modeling the link between the growth rate and the initial level of the gross domestic product and empirically prove the convergence hypothesis.