Self-normalized Asymptotic Theory in Probability, Statistics and Econometrics - IMS

Self-normalized Asymptotic Theory in Probability, Statistics and Econometrics
(1 - 30 May 2014)

~ Abstracts ~

Generalized self-normalization and the subsequence principle
István Berkes, Graz University of Technology, Austria

By a classical principle of probability theory, sufficiently thin subsequences of general sequences of random variables behave like i.i.d. sequences. For example, any (dependent) sequence of r.v.'s with uniformly bounded second moments has a subsequence satisfying the central limit theorem and law of the iterated logarithm. The centering and norming factors in these results are random and can be obtained from sample means and variances of the original sequence, so we are dealing here with classical self-normalized limit theorems. In case of nonlinear limit theorems, such as extremal limit theorems, the random centering and norming factors are more complicated functions of the underlying sequence and we get self-normalized limit theorems of a new kind. In our lecture we survey this interesting theory and its applications in and outside of probability theory.

Self-normalized extreme Eigenvalues of large dimensional Cov matrices from heavy-tailed multivariate time series
Richard A. Davis, Columbia University, USA

In this paper we give an asymptotic theory for the eigenvalues of the sample covariance matrix of a multivariate time series when the number of components p goes to infinity with the sample size. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index between 0 and 4; in particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the sample covariance matrix and show point process convergence of the normalized eigenvalues as n and p go to infinity. The limiting process has an explicit form involving points of a Poisson process and eigenvalues of a non-negative definite matrix. Based on this convergence we derive limit theory for a host of other continuous functionals including the self-normalized extreme eigenvalues. In particular, we establish convergence, without any normalization, of the ratio of the largest eigenvalue to the trace of the sample covariance matrix.
(This is joint work with Thomas Mikosch and Oliver Pfaffel.)

Model selection for high-dimensional ARX models
Ching-Kang Ing, Institute of Statistical Science, Taiwan

In this talk, some recent advances in model selection for high-dimensional ARX models are introduced. Model selection for high-dimensional regression models is one of the most vibrant research topics in statistics. However, most results are obtained under the assumption that observations are independent, and hence not directly applicable to time series data. To fill this gap, we shall develop some large deviation results for self-normalized martingales, which can be used to derive error bounds for the orthogonal greedy algorithm (OGA) and to establish the optimality of the high-dimensional information criterion (HDIC, Ing and Lai, 2011) when these methods are being applied to high-dimensional ARX models.

Gaussian approximation of suprema of empirical processes
Kengo Kato, University of Tokyo, Japan

This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for func- tions. Notably, the bound in the main approximation theorem is non- asymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an effective use of Stein's method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local and se- ries empirical processes arising in nonparametric estimation via kernel and series methods, where the classes of functions change with the sample size and are non-Donsker. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples.
Joint work with V. Chernozhukov (MIT) and D. Chetverikov (UCLA).

Tutorial and applications in econometrics: Self-normalization, Gaussian approximation, and inference with many moment inequalities Part I
Kengo Kato, University of Tokyo, Japan

Inference for moment inequality models is one of central issues in the recent econometric literature. Here the focus is on the problem of inference for parameters identified by many moment inequalities; indeed there are a growing number of economic applications where the number of moment inequalities is large. To this problem, we consider inference based on inverting tests for the corresponding testing problem, and focus on the test statistic given by the maximum of (many) Studentized statistics (or t-statistics). We describe various ways of computing critical values for the test statistic, including (i) the one based on the union bound combined with a moderate deviation inequality for self-normalized sums, and (ii) the one based on the Gaussian approximation. Optimality of the tests will be also discussed.

Matrix normalization and self-normalized asymptotic theory in regression, multivariate analysis and time series
Tze Leung Lai, Stanford University, USA

After a review of matrix normalization in self-normalized asymptotic theory related to multivariate analysis of iid observations, we show how the methods can be extended to regression, first for independent regressors, and then for stochastic regressors in time series models by making use of martingale theory and a pseudo-maximization technique. Applications of the results to order selection in ARMAX models are discussed herein, and to regression in high-dimensional sparse linear models in Ching-Kang Ing's talk in this workshop.

The lasso for high-dimensional regression with a possible change-point
Sokbae, Simon Lee, Seoul National University, Korea

We consider a high-dimensional regression model with a possible change-point due to a covariate threshold and develop the Lasso estimator of regression coefficients as well as the threshold parameter. Our Lasso estimator not only selects covariates but also selects a model between linear and threshold regression models. Under a sparsity assumption, we derive non-asymptotic oracle inequalities for both the prediction risk and the l_1 estimation loss for regression coefficients. Since the Lasso estimator selects variables simultaneously, we show that oracle inequalities can be established without pretesting the existence of the threshold effect. Furthermore, we establish conditions under which the estimation error of the unknown threshold parameter can be bounded by a nearly 1/n factor even when the number of regressors can be much larger than the sample size (n). We illustrate the usefulness of our proposed estimation method via Monte Carlo simulations and an application to real data.

The paper is available at the following link: http://arxiv.org/abs/1209.4875

Multivariate variance ratio statistics: application to stock returns
Oliver Linton, University of Cambridge, UK

We propose several multivariate variance ratio tests of the Efficient Markets Hypothesis. We derive the asymptotic distribution of the statistics and scalar functions thereof under the null hypothesis that returns are a martingale difference sequence after a constant mean adjustment. We do not impose the no leverage assumption of Lo and MacKinlay (1988) but our asymptotic standard errors are relatively simple and in particular do not require the selection of a bandwidth parameter. We extend the framework to allow for a smoothly varying risk premium in calendar time, and show that the limiting distribution is the same. We show the limiting behaviour of the statistic under a multivariate fads model. We apply the methodology to US and UK daily stock return data: in particular, five size sorted CRSP portfolios with sample period 1964-2014. We find evidence of a reduction of market inefficiency in the most recent period for small cap stocks. The main findings are not substantially affected by allowing for a time varying risk premium.

Joint work with Seok Young Hong, Oliver Linton and Hui Jun Zhang.

Impacts of high dimensionality in finite samples
Jinchi Lv, University of Southern California, USA

High-dimensional data sets are commonly collected in many contemporary applications arising in various fields of scientific research. We present two views of finite samples in high dimensions: a probabilistic one and a non-probabilistic one. With the probabilistic view, we establish the concentration property and robust spark bound for large random design matrix generated from elliptical distributions, with the former related to the sure screening property and the latter related to sparse model identifiability. An interesting concentration phenomenon in high dimensions is revealed. With the non-probabilistic view, we derive general bounds on dimensionality with some distance constraint on sparse models. These results provide new insights into the impacts of high dimensionality in finite samples.

The self-normalized sample extremogram and the self-normalized ex-periodogram
Thomas Mikosch, University of Copenhagen, Denmark

This is joint work with RA Davis (Columbia) and Y Zhao (Ulm).

The extremogram is an asymptotic correlation function measuring the extremal dependence in a strictly stationary sequence. Like the sample autocorrelation function, the sample extremogram is a self-normalized non-parametric estimator of its deterministic counterpart. The extremogram has a spectral density. Its sample version is the ex-periodogram which, again, is self-normalized. We provide limit theory for both structures, including functional central limit theory for the integrated ex-periodogram which can be used to build goodness-of-fit test statististics like Grenander-Rosenblatt and Cramer-von Mises. Since the asymptotic covariance structure of these quantities is rather complex we propose an alternative stationary bootstrap procedure for constructing confidence bands.

Asymptotic properties of change-point estimators
Chi Tim Ng, Chonnam University, Korea

Joint work with Woojoo Lee, Inha University, Seoul, Korea and Youngjo Lee, Seoul National University, Seoul, Korea

In this presentation, a penalized likelihood approach for the change point problem is discussed. A new penalty function called modified unbounded penalty is introduced to pursue consistent estimation in the number of change points and their locations and sizes. The asymptotic properties of the estimation from the modified unbounded penalty are compared with commonly used penalties, including Lasso, Scad, and Bridge. New results of the asymptotic theory for the set of local solutions are presented. Evidence from simulation results is also presented.

Key words: asymptotic normality, consistency, local solution, oracle property, penalized likelihood estimation, structural break, unbounded penalty.

Introduction to self-normalized limit theory
Qi-Man Shao, The Chinese University of Hong Kong, Hong Kong

Self-normalized processes are of common occurrence in probabilistic and statistical studies. A prototypical example is Student's t-statistic, and more generally, Studentized statistics. This tutorial will be an introduction to self-normalized limit theory and its applications. It will cover recent developments in the area, including self-normalized large and moderate deviations, weak invariance principles, Stein's method and self-normalized Berry-Esseen inequality.

Unified view of portmanteau tests for general statistical models
Masanobu Taniguchi, Waseda University, Japan

This talk consists of the two parts (I) and (II).

(I) Systematic Approach for Portmanteau Tests in View of Whittle Likelihood Ratio

Box and Pierce (1970) proposed a test statistic $T_{BP}$ which is the squared sum of $m$ sample autocorrelations of the estimated residual process of autoregressive moving average model of order (p,q). $T_{BP}$ is called the classical portmanteau test. Under the null hypothesis that the autoregressive-moving average model of order (p,q) is adequate, they suggested that the distribution of $T_{BP}$ is approximated by chi-square distribution with (m-p-q) degrees of freedom, "if $m$ is moderately large". This paper shows that $T_{BP}$ is understood as a special form of Whittle likelihood ratio test $T_{PW}$ for autoregressive-moving average spectral density with $m$-dependent residual process. Then, it is shown that, for any finite $m$, $T_{PW}$ does not converge to chi-square distribution with (m-p-q) degrees of freedom in distribution, and that, if we assume Bloomfield's exponential spectral density $T_{PW}$ is asymptotically chi-square distributed for any finite $m$. From this observation we propose a natural Whittle likelihood ratio test $T_{WLR}$ which is always asymptotically chi-square distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of $T_{WLR}$. Because many versions of the portmanteau test have been proposed, and been used in variety of fields, our systematic approach for portmanteau tests and proposal of $T_{WLR}$ will give a unified view and useful applications.

(II) A Unified View of Portmanteau Test for Diagnostic Checking :

Here we construct a portmanteau test statistic $T_{P}$ as a sort of the likelihood ratio test for general statistical models based on the procedure given in (I). Then we derive sufficient conditions that the statistic is asymptotically chi-square distributed. Furthermore, it is shown that if a time series model whose spectral density has a product structure satisfies appropriate conditions, $T_{P}$ is asymptotically chi-square distributed. We also introduce a useful application of the test for variable selection.

Nonparametric cointegrating regression with endogeneity and long memory
Qiying Wang, The University of Sydney, Australia

This paper explores nonparametric estimation, inference, and specification testing in a nonlinear cointegrating regression model where the structural equation errors are serially dependent and where the regressor is endogenous and may be driven by long memory innovations. Generalizing earlier results of Wang and Phillips (2009a,b), the conventional non-parametric local level kernel estimator is shown to be consistent and asymptotically (mixed) normal in these cases, thereby opening up inference by conventional nonparametric methods to a wide class of potentially nonlinear cointegrated relations. New results on the consistency of parametric estimates in nonlinear cointegrating regressions are provided, extending earlier research on parametric nonlinear regression and providing primitive conditions for parametric model testing. A model specification test is studied and confirmed to provide a valid mechanism for testing parametric specifications that is robust to endogeneity. But under long memory innovations the test is not pivotal, its convergence rate is parameter dependent, and its limit theory involves the local time of fractional Brownian motion. Simulation results show good performance for the nonparametric kernel estimates in cases of strong endogeneity and long memory, whereas the specification test is shown to be sensitive to the presence of long memory innovations, as predicted by asymptotic theory.
Joint work with Prof Phillips.

Nonlinear error correction models and multiple thresholds cointegrations
Man Wang, Dong Hua University, China

Nonlinear error correction models (ECM) with multiple regimes have been widely used in Finance and statistics. These models encompass the multiple threshold vector ECM as a special case. In this paper, the asymptotic properties of the least squares estimator of a nonlinear ECM with multiple regimes are established. For threshold cointegrated modeled of a threshold vector ECM, estimation procedures based on the least squares principle are examined. Both least squares and smoothed least squares estimations are studied and their asymptotic theories are established. In particular, the super-consistency of the least squares methods of the cointegration vector and the threshold parameters are developed. Simulation results confirm the theoretical Findings. We also study the term structure of interest rates by a two thresholds cointegration as an example. Finally we investigate the least squares estimator of smooth transition cointegration and establish the limiting distribution.

Large volatility matrix estimation for high-frequency financial data
Yazhen Wang, University of Wisconsin-Madison, USA

In financial research and practices, we often encounter a large number of assets. The availability of high-frequency financial data makes it possible to estimate the large volatility matrix of these assets.
Existing volatility matrix estimators such as kernel realized volatility and pre-averaging realized volatility perform poorly when the number of assets is very large, and in fact they are inconsistent when the number of assets and sample size go to infinity. In this paper, we introduce threshold rules to regularize kernel realized volatility, pre-averaging realized volatility, and multi-scale realized volatility. We establish asymptotic theory for these threshold estimators in the framework that allows the number of assets and sample size to go to infinity. Their convergence rates are derived under sparsity on the large integrated volatility matrix. In particular we have shown that the threshold kernel realized volatility and threshold pre-averaging realized volatility can achieve the optimal rate with respect to the sample size through proper bias corrections, but the bias adjustments causes the estimators to lose positive semi-definiteness; on the other hand, in order to be positive semi-definite, the threshold kernel realized volatility and threshold pre-averaging realized volatility have slower convergence rates with respect to the sample size. A simulation study is conducted to check the finite sample performances of the proposed threshold estimators in the case of over five hundred assets.

The cross-quantilogram: measuring quantile dependence and testing directional predictability between time series
Yoon-Jae Whang, Seoul National University, Korea

This paper proposes the cross-quantilogram to measure the quantile dependence between two time series. We apply it to test the hypothesis that one time series has no directional predictability to another time series. We establish the asymptotic distribution of the cross quantilogram and the corresponding test statistic. The limiting distributions depend on nuisance parameters. To construct consistent confidence intervals we employ the stationary bootstrap procedure; we show the consistency of this bootstrap. Also, we consider the self-normalized approach, which is shown to be asymptotically pivotal under the null hypothesis of no predictability. We provide simulation studies and two empirical applications. First, we use the cross-quantilogram to detect predictability from stock variance to excess stock return. Compared to existing tools used in the literature of stock return predictability, our method provides a more complete relationship between a predictor and stock return. Second, we investigate the systemic risk of individual financial institutions, such as JP Morgan Chase, Goldman Sachs and AIG.

Joint work with Heejoon Han, Oliver Linton, and Tatsushi Oka.

Random coefficient integer-valued moving average models
Kaizhi Yu, Southwestern University of Finance and Economics, China

The qth order random coefficient integer-valued moving average model is introduced. The results indicates that: Fixed index t, this process according to Poisson distribution; Mean,Variance and autocovariance functions are obtained; Stationary of this processes are proved, Ergodicity of the mean and autocovariance functions of this process is established; The consistent Moments estimator of parmeters are obtained in some special case. We provide some simulation results to test the performances of these estimators.

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