This thesis focuses on the development of semiparametric estimation methods for a class of time series models using shape constraints. Many of the existing time series models assume the noise follows some known parametric distributions. Typical examples are the Gaussian and t distributions. Then the model parameters are estimated by maximizing the resultant likelihood function. As an example, the autoregressive moving average (ARMA) models (Brockwell and Davis, 2009) assume Gaussian noise sequence and are estimated under the causal-invertible constraint by maximizing the Gaussian likelihood. Although the same estimates can also be used in the causal-invertible non-Gaussian case, they are not asymptotically optimal (Rosenblatt, 2012). Moreover, for the noncausal/noninvertible cases, the Gaussian likelihood estimation procedure is not applicable, since any second-order based methods cannot distinguish between causal-invertible and noncausal/noninvertible models (Brockwell and Davis,2009). As a result, many estimation methods for noncausal/noninvertible ARMA models assume the noise follows a known non-Gaussian distribution, like a Laplace distribution or a t distribution. To relax this distributional assumption and allow noncausal/noninvertible models, we borrow ideas from nonparametric shape-constraint density estimation and propose a semiparametric estimation procedure for general ARMA models by projecting the underlying noise distribution onto the space of log-concave measures (Cule and Samworth, 2010; Dümbgen et al., 2011). We show the maximum likelihood estimators in this semiparametric setting are consistent. In fact, the MLE is robust to the misspecification of log-concavity in cases where the true distribution of the noise is close to its log-concave projection. We derive a lower bound for the best asymptotic variance of regular estimators at rate sqrt(n) for AR models and construct a semiparametric efficient estimator. We also consider modeling time series of counts with shape constraints. Many of the formulated models for count time series are expressed via a pair of generalized state-space equations. In this set-up, the observation equation specifies the conditional distribution of the observation Yt at time t given a state-variable Xt. For count time series, this conditional distribution is usually specified as coming from a known parametric family such as the Poisson or the Negative Binomial distribution. To relax this formal parametric framework, we introduce a concave shape constraint into the one-parameter exponential family. This essentially amounts to assuming that the reference measure is log-concave. In this fashion, we are able to extend the class of observation-driven models studied in Davis and Liu (2016). Under this formulation, there exists a stationary and ergodic solution to the state-space model. In this new modeling framework, we consider the inference problem of estimating both the parameters of the mean model and the log-concave function, corresponding to the reference measure. We then compute and maximize the likelihood function over both the parameters associated with the mean function and the reference measure subject to a concavity constraint. The estimator of the mean function and the conditional distribution are shown to be consistent and perform well compared to a full parametric model specification. The finite sample behavior of the estimators are studied via simulation and two empirical examples are provided to illustrate the methodology.

The present work deals with selected methods suitable to work with financial time series. Firstly, univariate linear models ARMA are introduced, followed by the description of volatility models ARCH and their generalization to GARCH models. There are many modifications of standard GARCH models designed with respect to the nature of financial data, some of which are presented. Another part of the work dealing with multiple time series focuses on VAR models and bivariate GARCH models. The most important part of the work are practical examples of building the theoretically described models in various types of software with built-in procedures for time series analysis. We apply five different types of commercial and non-commercial software, namely EViews, Mathematica, R, S-PLUS and XploRe. The used software products are presented and compared in terms of their capabilities and the results obtained for particular methods.

The fundamental objective of the analysis of financial time series is to unveil the random mechanism, i.e. the probability law, underlying financial data. The effort to identify the truth that governs the observations involves proposing and estimating reasonable statistical models that well explain the empirical features of data. This thesis develops some new nonparametric tools that can be exploited in this context; the efficacy and validity of their use are supported by computational advancements and surging availability of large/complex (`big') data sets. Chapter 1 investigates the conditional first moment properties of financial returns. We propose multivariate extensions of the popular Variance Ratio (VR) statistic, aiming to test linear predictability of returns and weak-form market efficiency. We construct asymptotic distribution theories for the statistics and scalar functions thereof under the null hypothesis of no predictability. The imposed assumptions are weaker than those widely adopted in the literature, and in our view more credible with regard to the underlying data generating process we expect for stock returns. It is also shown that the limit theories can be extended to the long horizon and large dimension cases, and also to allow for a time varying risk premium. Our methods are applied to CRSP weekly returns from 1962 to 2013; the joint tests of the multivariate hypothesis reject the null at the 1% level for all horizons considered. Chapter 2 is about nonparametric estimation of conditional moments. We propose a local constant type estimator that operates with an infinite number of conditioning variables; this enables a direct estimation of many objects of econometric interest that have dependence upon the infinite past. We show pointwise and uniform consistency of the estimator and establish its asymptotic nomality in various static and dynamic regressions context. The optimal rate of estimation turns out to be of logarithmic order, and the precise rate depends on the Lambert W function, the smoothness of the regression operator and the dependence of the data in a non-trivial way. The theories are applied to investigate the intertemporal risk-return relation for the aggregate stock market. We report an overall positive risk-return relation on the S&P 500 daily data from 1950-2017, and find evidence of strong time variation and counter-cyclical behaviour in risk aversion. Lastly, Chapter 3 concerns nonparametric volatility estimation with high frequency time series. While data observed at finer time scale than daily provide rich information, their distinctive empirical properties bring new challenges in their analysis. We propose a Fourier domain based estimator for multivariate ex-post volatility that is robust to two major hurdles in high frequency finance: asynchronicity in observations and the presence of microstructure noise. Asymptotic properties are derived under some mild conditions. Simulation studies show our method outperforms time domain estimators when two assets with different liquidity are traded asynchronously.

Chan, Hon Kit. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (p. 57-59). / Abstracts in English and Chinese. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Cluster Analysis in Time Series --- p.1 / Chapter 1.2 --- Dissimilarity Measure --- p.2 / Chapter 1.2.1 --- Euclidean Distance --- p.3 / Chapter 1.2.2 --- Pearson's Correlation Coefficient --- p.3 / Chapter 1.2.3 --- Other Measure --- p.4 / Chapter 1.3 --- Summary --- p.5 / Chapter 2 --- Algorithm and Methodology --- p.8 / Chapter 2.1 --- Algorithm and Methodology --- p.8 / Chapter 2.2 --- Illustrative Examples --- p.14 / Chapter 3 --- Simulation Study --- p.20 / Chapter 3.1 --- Simulation Plan --- p.20 / Chapter 3.2 --- Measure of Performance --- p.24 / Chapter 3.3 --- Simulation Results --- p.27 / Chapter 3.4 --- Results of k-means Clustering --- p.33 / Chapter 4 --- Application on Gene Expression --- p.37 / Chapter 4.1 --- Dataset --- p.37 / Chapter 4.2 --- Parameter Settings --- p.38 / Chapter 4.3 --- Results --- p.38 / Chapter 5 --- Conclusion and Further Research --- p.55

Fung, Yu Hin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2011. / Includes bibliographical references (leaves 39-40). / Abstracts in English and Chinese. / List of Figures --- p.v / List of Tables --- p.vii / Chapter 1 --- Introduction --- p.1 / Chapter 2 --- Extremogram --- p.3 / Chapter 2.1 --- Strictly Stationary --- p.3 / Chapter 2.2 --- Regularly Varying: A time series {Xt} --- p.3 / Chapter 2.3 --- (Upper) tail dependence --- p.5 / Chapter 2.4 --- Extremogram --- p.6 / Chapter 3 --- Simulated Models --- p.9 / Chapter 3.1 --- Autoregressive (AR) Process --- p.9 / Chapter 3.1.1 --- The simulation --- p.9 / Chapter 3.1.2 --- Theoretical findings --- p.11 / Chapter 3.2 --- Moving Average (MA) Process --- p.12 / Chapter 3.2.1 --- The simulation --- p.12 / Chapter 3.3 --- GARCH and SV --- p.25 / Chapter 4 --- Applications to Market Data --- p.29 / Chapter 4.1 --- Case study: 2011 Japan Earthquake EOD data --- p.29 / Chapter 4.1.1 --- Data description --- p.29 / Chapter 4.1.2 --- Results --- p.30 / Chapter 4.2 --- Case study: TEPCO multi-timeframe analysis --- p.31 / Chapter 4.2.1 --- Data description --- p.31 / Chapter 4.2.2 --- Results --- p.32 / Chapter 5 --- Summary --- p.37 / References --- p.39

Digitized by Kansas Correctional Industries

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This thesis will take a look at the roots of modern-day information geometry and some applications into statistical modeling. In order to truly grasp this field, we will first provide a basic and relevant introduction to differential geometry. This includes the basic concepts of manifolds as well as key properties and theorems. We will then explore exponential families with applications of probability distributions. Finally, we select a few time series models and derive the underlying geometries of their manifolds.

Three macroeconomic issues are examined in separate self-contained studies. The first study tests the business cycle theory with application of an enhanced Augmented Dickey-Fuller test on the U.S. time series of real gross national product. Unlike previous studies, the null hypothesis of a unit root is rejected. The second study tests for IS-LM conditions in the U.S. during the post-Bretton Woods era by combining the Johansen's approach to cointegration with bootstrap algorithms. The estimated model produces a dynamic version of the IS-LM that permits short-term evaluations of fiscal and monetary policies. The third study seeks to explain the observed persistence in the Bolivan dollarization process. It is found that dollarization is now an irreversible process, with the Bolivian economy in transition toward equalization with U.S. prices and interest rates.

Department of Civil, Surveying and Environmental Engineering. Includes bibliographical references (leaves R-1--R-9)