Essays in Estimation and Hypothesis Testing in Non-Gaussian Nonlinear Time Series Models

Abstract

This dissertation develops new methods for estimation and hypothesis testing in strictly stationary non-Gaussian nonlinear time series models. Many economic and financial time series exhibit heavy tails, nonlinear dynamics, conditional heteroskedasticity, and locally explosive behavior that cannot be adequately captured by classical Gaussian linear frameworks. In such settings, linear autocovariances are insufficient to characterize dependence, and conventional diagnostic tools may fail to detect important dynamic features. This thesis proposes a unified approach based on nonlinear autocovariances within the Generalized Covariance (GCov) estimator and test framework to address these challenges. The first essay introduces a test of the absence of linear and nonlinear serial dependence (NLSD) for univariate and multivariate non-Gaussian processes. The proposed portmanteau-type statistic is constructed from nonlinear transformations of the data and follows an asymptotic chi-square distribution under the null hypothesis of serial independence. Building on this idea, the dissertation develops the GCov specification test for semi-parametric dynamic models with independent and identically distributed non-Gaussian errors. The asymptotic distribution of the test statistic is derived under the null and under local alternatives, providing insight into its power properties and practical implementation. The second essay addresses high-dimensional settings where the number of variables or nonlinear transformations is large relative to the sample size. A Regularized Generalized Covariance (RGCov) estimator is proposed, incorporating shrinkage techniques to stabilize covariance matrix estimation. The asymptotic properties of the estimator are established, and Monte Carlo simulations demonstrate improved finite-sample performance in terms of bias, variance, and mean squared error. The third essay extends shrinkage ideas to hypothesis testing by developing a regularized test for the absence of nonlinear serial dependence. The proposed procedure improves size control in high-dimensional environments while preserving strong power properties. The theoretical contributions are complemented by simulation studies and empirical applications to financial and commodity price data, including models with mixed causal-noncausal dynamics. Overall, the dissertation advances the econometric analysis of nonlinear and non-Gaussian time series by providing robust, theoretically grounded, and practically applicable tools for estimation and diagnostic testing.