Inference in partially identified models with many moment inequalities using Lasso

This paper considers the problem of inference in a partially identified moment (in)equality model with possibly many moment inequalities. Our contribution is to propose a two-step new inference method based on the combination of two ideas. On the one hand, our test statistic and critical values are based on those proposed by Chernozhukov, Chetverikov and Kato (2014) (CCK14, hereafter). On the other hand, we propose a new first step selection procedure based on the Lasso. Our two-step inference method can be used to conduct hypothesis tests and to construct confidence sets for the true parameter value. Our inference method is shown to have very desirable properties. First, under reasonable conditions, it is uniformly valid, both in underlying parameter $\theta$ and distribution of the data. Second, by virtue of the results in CCK14, our test is asymptotically optimal in a minimax sense. Third, we compare the power of our method with that of the corresponding two-step method proposed by CCK14, both in theory and in simulations. On the theory front, we obtain a region of underlying parameters under which the power of our method dominates. Our simulations indicate that our inference method is usually as powerful than those in CCK14, and can sometimes be more powerful. In particular, our simulations reveal that the Lasso-based first step delivers more power in designs with sparse alternatives, i.e., when only few of the moment (in)equalities are violated. Fourth, we show that our Lasso-based first step can be implemented with a thresholding least squares procedure that makes it extremely simple to compute.