Inference for Subsets of Parameters in Partially Identied Models

We propose a con…dence set for the subsets of parameters under partially identi…ed models characterized by moment inequalities. The subvector inference is based on the speci…cation testing of Guggenberger, Hahn, and Kim (2006) who discuss the dual characterization between the speci…cation testing of the moment inequalities and the multi-dimensional one-sided tests. We exploit the idea that a speci…cation testing has natural implications for the construction of a CS for a subset component of a vector-valued parameter. To be precise, let be the full parameter vector in a model. We decompose = ( 1; 2) and consider a con…dence set for 1. The CS for 1 is constructed as a set of all e 1 2 1, which are not rejected by the speci…cation testing whether there exists a 2 in 2 that does not reject the model given the value of e 1. We modify CS by restricting the values that 2 can take to a …rst step con…dence set C2(1 2;e 1) that covers the true value of 2 with asymptotic coverage level equal to 1 2 given the value of e 1. Then, we collect the values of e 1 that survives this modi…ed speci…cation test. We show that the constructed CS of 1 in this way has at least 1 1 2 asymptotic coverage probability of the true value of 1 where the second step signi…cance level is set to 1 following Bonferroni-type arguments. We also …nd that our proposed CS for the subsets of parameters is asymptotically locally equivalent to the infeasible CS with known true parameter set of the other parameters. Keywords: moment inequalities, partial identi…cation, speci…cation test, testing subsets of parameters, dual characterization JEL classi…cation numbers: C12 Correspondence: [email protected], Address: 4-129 Hanson Hall, 1925 Fourth Street South, Minneapolis, MN55455