  I study the estimation of semiparametric monotone index models in the
scenario where the number of observation points $n$ is extremely large and
conventional approaches fail to work due to heavy computational burdens.
Motivated by the mini-batch gradient descent algorithm (MBGD) that is widely
used as a stochastic optimization tool in the machine learning field, I
proposes a novel subsample- and iteration-based estimation procedure. In
particular, starting from any initial guess of the true parameter, I
progressively update the parameter using a sequence of subsamples randomly
drawn from the data set whose sample size is much smaller than $n$. The update
is based on the gradient of some well-chosen loss function, where the
nonparametric component is replaced with its Nadaraya-Watson kernel estimator
based on subsamples. My proposed algorithm essentially generalizes MBGD
algorithm to the semiparametric setup. Compared with full-sample-based method,
the new method reduces the computational time by roughly $n$ times if the
subsample size and the kernel function are chosen properly, so can be easily
applied when the sample size $n$ is large. Moreover, I show that if I further
conduct averages across the estimators produced during iterations, the
difference between the average estimator and full-sample-based estimator will
be $1/\sqrt{n}$-trivial. Consequently, the average estimator is
$1/\sqrt{n}$-consistent and asymptotically normally distributed. In other
words, the new estimator substantially improves the computational speed, while
at the same time maintains the estimation accuracy.

Econometrics