Matthew Reimherr

BREAK DETECTION IN THE COVARIANCE STRUCTURE OF MULTIVARIATE TIME SERIES MODELS

In this paper, we introduce an asymptotic test procedure to assess the stability of volatilities and cross-volatilites of linear and nonlinear multivariate time series models. The test is very flexible as it can be applied, for example, to many of the multivariate GARCH models established in the literature, and also works well in the case of high dimensionality of the underlying data. Since it is nonparametric, the procedure avoids the difficulties associated with parametric model selection, model fitting and parameter estimation. We provide the theoretical foundation for the test and demonstrate its applicability via a simulation study and an analysis of financial data. Extensions to multiple changes and the case of infinite fourth moments are also discussed.

Determining the order of the functional autoregressive model

We propose a multistage testing procedure to determine the order p of a functional autoregressive process, FAR (p). At its core is the representation of the FAR(p) process as a fully functional linear model with dependent regressors. Estimating the kernel function in this linear model allows us to construct a test statistic which has, approximately, a chi–square distribution with the number of degrees of freedom determined by the number of functional principal components used to represent the data. The asymptotic justification relies on the concept of Lp–m–approximability which quantifies the temporal dependence of functional time series. The procedure enjoys very good finite sample properties, as confirmed by a simulation study and applications to functional time series derived from credit card transactions and Eurodollar futures data.

Predictability of Shapes of Intraday Price Curves

We develop a statistical framework, based on functional data analysis, for testing the hypothesis of the predictability of shapes of intraday price curves. We derive test statistics based on signs of the scores of the functional principal components. We establish its asymptotic properties under the null and alternative hypotheses, and demonstrate via simulations that it has excellent finite sample properties. A small empirical study shows that the shapes of the intraday price curves of large US corporations are not predictable.

A FUNCTIONAL DATA ANALYSIS APPROACH FOR GENETIC ASSOCIATION STUDIES

We present a new method based on Functional Data Analysis (FDA) for detecting associations between one or more scalar covariates and a longitudinal response, while correcting for other variables. Our methods exploit the temporal structure of longitudinal data in ways that are otherwise difficult with a multivariate approach. Our procedure, from an FDA perspective, is a departure from more established methods in two key aspects. First, the raw longitudinal phenotypes are assembled into functional trajectories prior to analysis. Second, we explore an association test that is not directly based on principal components. We instead focus on quantifying the reduction in L 2 variability as a means of detecting associations. Our procedure is motivated by longitudinal genome wide association studies and, in particular, the childhood asthma management program (CAMP) which explores the long term effects of daily asthma treatments. We conduct a simulation study to better understand the advantages (and/or disadvantages) of an FDA approach compared to a traditional multivariate one. We then apply our methodology to data coming from CAMP. We find a potentially new association with a SNP negatively affecting lung function. Furthermore, this SNP seems to have an interaction effect with one of the treatments.

On Quantifying Dependence: A Framework for Developing Interpretable Measures

We present a framework for selecting and developing measures of dependence when the goal is the quantification of a relationship between two variables, not simply the establishment of its existence. Much of the literature on dependence measures is focused, at least implicitly, on detection or revolves around the inclusion/exclusion of particular axioms and discussing which measures satisfy said axioms. In contrast, we start with only a few nonrestrictive guidelines focused on existence, range and interpretability, which provide a very open and flexible framework. For quantification, the most crucial is the notion of interpretability, whose foundation can be found in the work of Goodman and Kruskal [Measures of Association for Cross Classifications (1979) Springer], and whose importance can be seen in the popularity of tools such as the

Determining the order of the functional autoregressive model: ORDER OF FUNCTIONAL AUTOREGRESSION

R^2

Estimating Variance Components in Functional Linear Models With Applications to Genetic Heritability

in linear regression. While Goodman and Kruskal focused on probabilistic interpretations for their measures, we demonstrate how more general measures of information can be used to achieve the same goal. To that end, we present a strategy for building dependence measures that is designed to allow practitioners to tailor measures to their needs. We demonstrate how many well-known measures fit in with our framework and conclude the paper by presenting two real data examples. Our first example explores U.S. income and education where we demonstrate how this methodology can help guide the selection and development of a dependence measure. Our second example examines measures of dependence for functional data, and illustrates them using data on geomagnetic storms.

You've Gotta Be Lucky: Coverage and the Elusive Gene–Gene Interaction

We propose a multistage testing procedure to determine the order p of a functional autoregressive process, FAR (p). At its core is the representation of the FAR(p) process as a fully functional linear model with dependent regressors. Estimating the kernel function in this linear model allows us to construct a test statistic which has, approximately, a chi–square distribution with the number of degrees of freedom determined by the number of functional principal components used to represent the data. The asymptotic justification relies on the concept of Lp–m–approximability which quantifies the temporal dependence of functional time series. The procedure enjoys very good finite sample properties, as confirmed by a simulation study and applications to functional time series derived from credit card transactions and Eurodollar futures data.

Coherent synthesis of genomic associations with phenotypes and home environments

Quantifying heritability is the first step in understanding the contribution of genetic variation to the risk architecture of complex human diseases and traits. Heritability can be estimated for univariate phenotypes from nonfamily data using linear mixed effects models. There is, however, no fully developed methodology for defining or estimating heritability from longitudinal studies. By examining longitudinal studies, researchers have the opportunity to better understand the genetic influence on the temporal development of diseases, which can be vital for populations with rapidly changing phenotypes such as children or the elderly. To define and estimate heritability for longitudinally measured phenotypes, we present a framework based on functional data analysis, FDA. While our procedures have important genetic consequences, they also represent a substantial development for FDA. In particular, we present a very general methodology for constructing optimal, unbiased estimates of variance components in functional linear models. Such a problem is challenging as likelihoods and densities do not readily generalize to infinite-dimensional settings. Our procedure can be viewed as a functional generalization of the minimum norm quadratic unbiased estimation procedure, MINQUE, presented by C. R. Rao, and is equivalent to residual maximum likelihood, REML, in univariate settings. We apply our methodology to the Childhood Asthma Management Program, CAMP, a 4-year longitudinal study examining the long term effects of daily asthma medications on children.

Feature screening for time-varying coefficient models with ultrahigh-dimensional longitudinal data

Genome‐wide association studies (GWAS) have led to a large number of single‐SNP association findings, but there has been, so far, no investigation resulting in the discovery of a replicable gene–gene interaction. In this paper, we examine some of the possible explanations for the lack of findings, and argue that coverage of causal variation not only has a large effect on the loss in power, but that the effect is larger than in the single‐SNP analyses. We show that the product of linkage disequilibrium measures, r2, between causal and tested SNPs offers a good approximation to the loss in efficiency as defined by the ratio of sample sizes that lead to similar power. We also demonstrate that, in addition to the huge search space, the loss in power due to coverage when using commercially available platforms makes the search for gene–gene interactions daunting.