This thesis delves into the numerical estimation methods of high-dimensional statistical models, a topic increasingly relevant due to the technological advancements over the past two decades that have dramatically increased the availability and complexity of datasets across various fields. Traditional statistical models often struggle to efficiently utilize the wealth of information presented by these datasets, prompting a shift towards more sophisticated approaches like Deep Neural Networks. However, these models typically lack inferential capabilities, a critical aspect for statisticians. This thesis, therefore, focuses on the development and study of models and methods that are not only flexible enough to handle large and diverse datasets but also enable meaningful inference.
The first chapter introduces a novel additive factor model, extending parametric factor models, where common components are linked to latent factors through unknown smooth functions. We outline the model's identification conditions and specifies a general nonparametric estimation procedure, demonstrating convergence as the number of observations and variables increase. The model's efficacy is illustrated through extensive numerical experiments and applied to macroeconomic data.
In the second chapter, the focus shifts to comparing the common components estimation of PCA, State-Varying Factor Models, and the additive model introduced in the first chapter. This comparison is conducted through in sample time series. Additionally, the chapter explores the application of these models in portfolio management, risk management, and market performance within the S\&P 500 context, presenting classic performance and risks ratios alongside market indices.
The third chapter reviews valuation methods for illiquid investments, particularly in private equity and real estate. It critiques traditional valuation methods like the net present value (NPV) rule, highlighting their limitations in certain scenarios. The chapter advocates for a real option valuation framework as a more robust alternative, presenting a new jump-diffusion option pricing model. This model demonstrates the framework's utility in valuing illiquid assets, especially venture capital funds, and suggests that, with harmonized parameter estimation techniques, the NPV rule can serve as a lower bound in valuation, with real option criteria offering an upper bound. This understanding of valuation bounds is posited to enhance decision-making in fund allocation to alternative investments. The estimated parameters also provide valuable insights into fair compensation for Limited Partners (LP) and General Partners (GP).
Overall, this thesis contributes to the field of statistics by exploring and enhancing numerical estimation methods for high-dimensional models, addressing the challenges posed by modern, complex datasets in various economic and financial contexts.
