  In this paper we propose new approaches to estimating large dimensional
monotone index models. This class of models has been popular in the applied and
theoretical econometrics literatures as it includes discrete choice,
nonparametric transformation, and duration models. A main advantage of our
approach is computational. For instance, rank estimation procedures such as
those proposed in Han (1987) and Cavanagh and Sherman (1998) that optimize a
nonsmooth, non convex objective function are difficult to use with more than a
few regressors and so limits their use in with economic data sets. For such
monotone index models with increasing dimension, we propose to use a new class
of estimators based on batched gradient descent (BGD) involving nonparametric
methods such as kernel estimation or sieve estimation, and study their
asymptotic properties. The BGD algorithm uses an iterative procedure where the
key step exploits a strictly convex objective function, resulting in
computational advantages. A contribution of our approach is that our model is
large dimensional and semiparametric and so does not require the use of
parametric distributional assumptions.
