A new class of HAC covariance matrix estimators is proposed based on the notion of a flat-top kernel as in Politis and Romano (1995)and Politis (2001). The new estimators are shown to be higher-order accurate when higher-order accuracy is possible, and a discussion on kernel choice is given. The higher-order accuracy of flat-top kernel estimators typically comes at the sacrifice of the positive semi-definite property. Nevertheless, we show how a modified flat-top estimator is positive semi-definite while maintaining its higher-order accuracy. In addition, an automatic and consistent procedure for optimal bandwidth choice for flat-top kernel HAC estimators is given. The general problem of spectral matrix estimation is also treated.

Functional data objects are usually collected sequentially over time exhibiting forms of dependence. Such data structure is known as functional time series. While there is plentiful literature addressing the topics of linear functional processes, relatively few contributions have dealt with nonparametric functional time series, which is the focus of this dissertation. After introducing some background and basics of functional time series in Chapter 1, I address the topics concerning the applications of kernel methods in the analysis of nonparametric functional time series. Specifically, Chapter 2 investigates the kernel estimation of the autoregressive operator in the nonparametric functional autoregression model. A componentwise bootstrap procedure is proposed in Chapter 3 which can be used for estimating the distribution of the kernel estimation and constructing the prediction regions. Chapter 4 tackles the problem of spectral density estimation of functional time series. A class of higher-order accurate spectral density kernel estimators is proposed based on the notion of flat-top kernel.

The well-known ARCH/GARCH models with normal errors account only partly for the degree of heavy tails empirically found in the distribution of financial returns series. Instead of resorting to an arbitrary nonnormal distribution for the ARCH/GARCH residuals we propose a different viewpoint via a novel normalizing and variance– stabilizing transformation (NoVaS) that can be seen as an alternative to parametric modeling. Some properties of this transformation are discussed, and algorithms for optimizing it are given. Special emphasis is given on the problem of volatility prediction and the issue of a proper measure for quality of prediction. A new prediction algorithm with favorable performance is given based on the NoVaS transformation. For motivation and illustration of this new general methodology, the NoVaS transformation is implemented in connection with three real data series: a foreign exchange series (Yen vs. Dollar), a stock index series (S&P500 index), and a stock price series (IBM).

The quest for the ‘best’ heavy-tailed distribution for ARCH/GARCH residuals appears to still be ongoing. In this connection, we propose a new distribution that arises in a natural way as an outcome of an implicit model. The challenging application of prediction of squared returns is also discussed; an optimal predictor is formulated, and the usefulness of the new distribution for prediction is demonstrated on three real datasets.

The problem of prediction is revisited with a view towards going beyond the typical nonparametric setting and reaching a fully model-free environment for predictive inference, i.e., point predictors and predictive intervals. A basic principle of model-free prediction is laid out based on the notion of transforming a given set-up into one that is easier to work with, namely i.i.d. or Gaussian. As an application, the problem of nonparametric regression is addressed in detail; the model-free predictors are worked out, and shown to be applicable under minimal assumptions. Interestingly, model-free prediction in regression is a totally automatic technique that does not necessitate the search for an optimal data transformation before model fitting. The resulting model-free predictive distributions and intervals are compared to their corresponding model-based analogs, and the use of cross-validation is extensively discussed. As an aside, improved prediction intervals in linear regression are also obtained.

Unit root process, as a process with stochastic trend and a generalization from random walk, is pervasive in physics, economics, and finance. In the hypothesis test for unit root, bootstrap methods have earned a great deal of attention. This dissertation proposes and investigates various bootstrap unit root tests. Chapter one applies linear process bootstrap to unit root test in order to alleviate the size distortions of unit root tests. While Chapter one focuses on classic unit root tests, which search for stochastic trend, Chapter two tackles seasonal unit root tests, which simultaneously check stochastic trend and stochastic seasonality. In addition, Chapter two takes into consideration seasonal heterogeneity, which pervades in seasonal processes. Specifically, Chapter two offers under seasonal heterogeneity a seasonal AR-sieve bootstrap remedy for a parametric seasonal unit root test and advocates a seasonal block bootstrap solution for a non-parametric test. This dissertation then establishes three bootstrap functional central limit theorems, via which this dissertation shows the consistency of all the aforementioned bootstrap methods.