Testing many moment inequalities

This paper considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by $p$, is possibly much larger than the sample size $n$. There are variety of economic applications where the problem of testing many moment inequalities appears; a notable example is a market structure model of Ciliberto and Tamer (2009) where $p=2^{m+1}$ with $m$ being the number of firms. We consider the test statistic given by the maximum of $p$ Studentized (or $t$-type) statistics, and analyze various ways to compute critical values for the test statistic. Specifically, we consider critical values based upon (i) the union bound combined with a moderate deviation inequality for self-normalized sums, (ii) the multiplier and empirical bootstraps, and (iii) two-step and three-step variants of (i) and (ii) by incorporating selection of uninformative inequalities that are far from being binding and novel selection of weakly informative inequalities that are potentially binding but do not provide first order information. We prove validity of these methods, showing that under mild conditions, they lead to tests with error in size decreasing polynomially in $n$ while allowing for $p$ being much larger than $n$; indeed $p$ can be of order $\exp (n^{c})$ for some $c > 0$. Moreover, when $p$ grows with $n$, we show that all of our tests are (minimax) optimal in the sense that they are uniformly consistent against alternatives whose "distance" from the null is larger than the threshold $(2 (\log p)/n)^{1/2}$, while {\em any} test can only have trivial power in the worst case when the distance is smaller than the threshold. Finally, we show validity of a test based on block multiplier bootstrap in the case of dependent data under some general mixing conditions.