  We study the asymptotic theory of misspecified models for diffusion processes
with noisy nonsynchronous observations. Unlike with correctly specified models,
the original maximum-likelihood-type estimator has an asymptotic bias under the
misspecified setting and fails to achieve an optimal rate of convergence. To
address this, we consider a new quasi-likelihood function that arrows
constructing a maximum-likelihood-type estimator that achieves the optimal rate
of convergence. Study of misspecified models enables us to apply
machine-learning techniques to the maximum-likelihood approach. With these
techniques, we can efficiently study the microstructure of a stock market by
using rich information of high-frequency data. Neural networks have
particularly good compatibility with the maximum-likelihood approach, so we
will consider an example of using a neural network for simulation studies and
empirical analysis of high-frequency data from the Tokyo Stock Exchange. We
demonstrate that the neural network outperforms polynomial models in volatility
predictions for major stocks in Tokyo Stock Exchange.
