Essays in high dimensional nonlinear time series analysis
In this thesis, I study high dimensional nonlinear time series analysis, and its applications in financial forecasting and identifying risk in highly interconnected financial networks. The first chapter is devoted to the testing for nonlinearity in financial time series. I present a tentative classification of the various linearity tests that have been proposed in the literature. Then I investigate nonlinear features of real financial series to determine if the data justify the use of nonlinear techniques, such as those inspired by machine learning theories. In Chapter 3 5, I develop forecasting strategies with a high dimensional panel of predictors while considering nonlinear dynamics. Combining these two elements is a developing area of research. In the third chapter, I propose a nonlinear generalization of the statistical factor models. As a first step, factor estimation, I employ an auto associative neural network to estimate nonlinear factors from predictors. In the second step, forecasting equation, I apply a nonlinear function feedforward neural networkon estimated factors for prediction. I show that these features can go beyond covariance analysis and enhance forecast accuracy. I apply this approach to forecast equity returns, and show that capturing nonlinear dynamics between equities significantly improves the quality of forecasts over current univariate and multivariate factor models. In Chapter 5, I propose a high dimensional learning based on a shrinkage estimation of a backpropagation algorithm for skip layer neural networks. This thesis emphasizes that linear models can be represented as special cases of these two aforementioned models, which basically means that if there is no nonlinearity between series, the proposed models will reduce to a linear model. This thesis also includes a chapter (chapter 4, with Negar Kiyavash and Seyedjalal Etesami) which in this chapter, we propose a new approach for identifying and measuring systemic risk in financial networks by introducing a nonlinearly modified Granger causality network based on directed information graphs. The suggested method allows for nonlinearity and has predictive power over future economic activity through a time varying network of interconnections. We apply the method to the daily returns of U.S. financial Institutions including banks, brokers and insurance companiesto identifythe level of systemic risk inthe financial sector and the contribution of each financial institution.