However, the statistical properties of DNN estimators with dependent data are largely unknown, and existing results for general nonparametric estimators are often inapplicable to DNN estimators. As a result, empirical use of DNN estimators often lacks a theoretical foundation. This paper aims to address this deficiency by first providing general results for nonparametric sieve estimators that offer a framework that is flexible enough for studying DNN estimators under dependent data. These results are then applied to both nonparametric regression and classification contexts, yielding theoretical properties for a class of DNN architectures commonly used in applications. Notably, Brown (2024) demonstrates the practical implications of these results in a partially linear regression model with dependent data by obtaining n-asymptotic normality of the estimator for the finite dimensional parameter after first-stage DNN estimation of infinite dimensional parameters. DNN estimators can be viewed as adaptive linear sieve estimators, where inputs are passed through hidden layers that'learn' basis functions from the data by optimizing over compositions of simpler functions.

I consider inference in a partially linear regression model under stationary $\beta$-mixing data after first stage deep neural network (DNN) estimation. Using the DNN results of Brown (2024), I show that the estimator for the finite dimensional parameter, constructed using DNN-estimated nuisance components, achieves $\sqrt{n}$-consistency and asymptotic normality. By avoiding sample splitting, I address one of the key challenges in applying machine learning techniques to econometric models with dependent data. In a future version of this work, I plan to extend these results to obtain general conditions for semiparametric inference after DNN estimation of nuisance components, which will allow for considerations such as more efficient estimation procedures, and instrumental variable settings.

We describe a translation from a fragment of SUMO (SUMO-K) into higher-order set theory. The translation provides a formal semantics for portions of SUMO which are beyond first-order and which have previously only had an informal interpretation. It also for the first time embeds a large common-sense ontology into a very secure interactive theorem proving system. We further extend our previous work in finding contradictions in SUMO from first order constructs to include a portion of SUMO's higher order constructs. Finally, using the translation, we can create problems that can be proven using higher-order interactive and automated theorem provers. This is tested in several systems and can be used to form a corpus of higher-order common-sense reasoning problems.