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Stanislav Volgushev

Publications and source records attributed to Stanislav Volgushev.

At least 37 records · Page 2Linked to original sources
arXiv2020

On the Unbiased Asymptotic Normality of Quantile Regression with Fixed Effects

Nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data. In a large class of such models (including probit, proportional hazard and quantile regression to name just a few) it is impossible to difference out individual effects, and inference is usually justified in a `large n large T' asymptotic framework. However, there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions (such as probit, and proportional hazard) and quantile regression. In the present paper we show that this gap can be bridged and establish asymptotic unbiased normality for quantile regression panels under conditions on n,T that are very close to what is typically assumed in standard nonlinear panels. Our results considerably improve upon existing theory and show that quantile regression is applicable to the same type of panel data (in terms of n,T) as other commonly used nonlinear panel data models. Thorough numerical experiments confirm our theoretical findings.

econ.EM↗
arXiv2019

Multiple block sizes and overlapping blocks for multivariate time series extremes

Block maxima methods constitute a fundamental part of the statistical toolbox in extreme value analysis. However, most of the corresponding theory is derived under the simplifying assumption that block maxima are independent observations from a genuine extreme value distribution. In practice however, block sizes are finite and observations from different blocks are dependent. Theory respecting the latter complications is not well developed, and, in the multivariate case, has only recently been established for disjoint blocks of a single block size. We show that using overlapping blocks instead of disjoint blocks leads to a uniform improvement in the asymptotic variance of the multivariate empirical distribution function of rescaled block maxima and any smooth functionals thereof (such as the empirical copula), without any sacrifice in the asymptotic bias. We further derive functional central limit theorems for multivariate empirical distribution functions and empirical copulas that are uniform in the block size parameter, which seems to be the first result of this kind for estimators based on block maxima in general. The theory allows for various aggregation schemes over multiple block sizes, leading to substantial improvements over the single block length case and opens the door to further methodology developments. In particular, we consider bias correction procedures that can improve the convergence rates of extreme-value estimators and shed some new light on estimation of the second-order parameter when the main purpose is bias correction.

math.ST↗
arXiv2019

Smoothed quantile regression processes for binary response models

In this paper, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearisation for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedasticity or the need for high dimensional non-parametric smoothing necessary for approaches available so far. A uniform linear representation and results on asymptotic normality are provided, and the connection to rearrangements is discussed.

stat.ME↗
arXiv2019

Model assessment for time series dynamics using copula spectral densities: a graphical tool

Finding parametric models that accurately describe the dependence structure of observed data is a central task in the analysis of time series. Classical frequency domain methods provide a popular set of tools for fitting and diagnostics of time series models, but their applicability is seriously impacted by the limitations of covariances as a measure of dependence. Motivated by recent developments of frequency domain methods that are based on copulas instead of covariances, we propose a novel graphical tool that allows to access the quality of time series models for describing dependencies that go beyond linearity. We provide a thorough theoretical justification of our approach and show in simulations that it can successfully distinguish between subtle differences of time series dynamics, including non-linear dynamics which result from GARCH and EGARCH models. We also demonstrate the utility of the proposed tools through an application to modeling returns of the S&P 500 stock market index.

stat.ME↗
arXiv2018

On Second Order Conditions in the Multivariate Block Maxima and Peak over Threshold Method

Second order conditions provide a natural framework for establishing asymptotic results about estimators for tail related quantities. Such conditions are typically tailored to the estimation principle at hand, and may be vastly different for estimators based on the block maxima (BM) method or the peak-over-threshold (POT) approach. In this paper we provide details on the relationship between typical second order conditions for BM and POT methods in the multivariate case. We show that the two conditions typically imply each other, but with a possibly different second order parameter. The latter implies that, depending on the data generating process, one of the two methods can attain faster convergence rates than the other. The class of multivariate Archimax copulas is examined in detail; we find that this class contains models for which the second order parameter is smaller for the BM method and vice versa. The theory is illustrated by a small simulation study.

math.ST↗
arXiv2018

Panel Data Quantile Regression with Grouped Fixed Effects

This paper introduces estimation methods for grouped latent heterogeneity in panel data quantile regression. We assume that the observed individuals come from a heterogeneous population with a finite number of types. The number of types and group membership is not assumed to be known in advance and is estimated by means of a convex optimization problem. We provide conditions under which group membership is estimated consistently and establish asymptotic normality of the resulting estimators. Simulations show that the method works well in finite samples when T is reasonably large. To illustrate the proposed methodology we study the effects of the adoption of Right-to-Carry concealed weapon laws on violent crime rates using panel data of 51 U.S. states from 1977 - 2010.

econ.EM↗
arXiv2018

Distributed inference for quantile regression processes

The increased availability of massive data sets provides a unique opportunity to discover subtle patterns in their distributions, but also imposes overwhelming computational challenges. To fully utilize the information contained in big data, we propose a two-step procedure: (i) estimate conditional quantile functions at different levels in a parallel computing environment; (ii) construct a conditional quantile regression process through projection based on these estimated quantile curves. Our general quantile regression framework covers both linear models with fixed or growing dimension and series approximation models. We prove that the proposed procedure does not sacrifice any statistical inferential accuracy provided that the number of distributed computing units and quantile levels are chosen properly. In particular, a sharp upper bound for the former and a sharp lower bound for the latter are derived to capture the minimal computational cost from a statistical perspective. As an important application, the statistical inference on conditional distribution functions is considered. Moreover, we propose computationally efficient approaches to conducting inference in the distributed estimation setting described above. Those approaches directly utilize the availability of estimators from sub-samples and can be carried out at almost no additional computational cost. Simulations confirm our statistical inferential theory.

math.ST↗
arXiv2017

Quantile Processes for Semi and Nonparametric Regression

A collection of quantile curves provides a complete picture of conditional distributions. Properly centered and scaled versions of estimated curves at various quantile levels give rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. Applications of our general process convergence results include the construction of non-crossing quantile curves and the estimation of conditional distribution functions. As a result of independent interest, we obtain a series of Bahadur representations with exponential bounds for tail probabilities of all remainder terms. Bounds of this kind are potentially useful in analyzing statistical inference procedures under divide-and-conquer setup.

math.ST↗
arXiv2017

Fourier analysis of serial dependence measures

Classical spectral analysis is based on the discrete Fourier transform of the auto-covariances. In this paper we investigate the asymptotic properties of new frequency domain methods where the auto-covariances in the spectral density are replaced by alternative dependence measures which can be estimated by U-statistics. An interesting example is given by Kendall{'}s $τ$ , for which the limiting variance exhibits a surprising behavior.

math.ST↗
arXiv2016

The independence process in conditional quantile location-scale models and an application to testing for monotonicity

In this paper the nonparametric quantile regression model is considered in a location-scale context. The asymptotic properties of the empirical independence process based on covariates and estimated residuals are investigated. In particular an asymptotic expansion and weak convergence to a Gaussian process are proved. The results can, on the one hand, be applied to test for validity of the location-scale model. On the other hand, they allow to derive various specification tests in conditional quantile location-scale models. In detail a test for monotonicity of the conditional quantile curve is investigated. For the test for validity of the location-scale model as well as for the monotonicity test smooth residual bootstrap versions of Kolmogorov-Smirnov and Cramer-von Mises type test statistics are suggested. We give rigorous proofs for bootstrap versions of the weak convergence results. The performance of the tests is demonstrated in a simulation study.

math.ST↗
arXiv2016

Quantile Spectral Analysis for Locally Stationary Time Series

Classical spectral methods are subject to two fundamental limitations: they only can account for covariance-related serial dependencies, and they require second-order stationarity. Much attention has been devoted lately to quantile-based spectral methods that go beyond covariance-based serial dependence features. At the same time, covariance-based methods relaxing stationarity into much weaker {\it local stationarity} conditions have been developed for a variety of time-series models. Here, we are combining those two approaches by proposing quantile-based spectral methods for locally stationary processes. We therefore introduce a time-varying version of the copula spectra that have been recently proposed in the literature, along with a suitable local lag-window estimator. We propose a new definition of local {\it strict} stationarity that allows us to handle completely general non-linear processes without any moment assumptions, thus accommodating our quantile-based concepts and methods. We establish a central limit theorem for the new estimators, and illustrate the power of the proposed methodology by means of a simulation study. Moreover, in two empirical studies (namely of the Standard \& Poor's 500 series and a temperature dataset recorded in Hohenpeissenberg) we demonstrate that the new approach detects important variations in serial dependence structures both across time and across quantiles. Such variations remain completely undetected, and are actually undetectable, via classical covariance-based spectral methods.

math.ST↗
arXiv2016

Equivalence of dose response curves

This paper investigates the problem whether the difference between two parametric models $m_1,m_2$ describing the relation between a response variable and several covariates in two different groups is practically irrelevant, such that inference can be performed on the basis of the pooled sample. Statistical methodology is developed to test the hypotheses $H_0 : d(m_1,m_2)\geq ε$ versus $H_1 : d(m_1,m_2) < ε$ to demonstrate equivalence between the two regression curves $m_1,m_2$ for a pre-specified threshold $ε$, where $d$ denotes a distance measuring the distance between $m_1$ and $m_2$. Our approach is based on the asymptotic properties of a suitable estimator $d(\hat{m}_1; \hat{m}_2)$ of this distance. In order to improve the approximation of the nominal level for small sample sizes a bootstrap test is developed, which addresses the specific form of the interval hypotheses. In particular, data has to be generated under the null hypothesis, which implicitly defines a manifold for the parameter vector. The results are illustrated by means of a simulation study and a data example. It is demonstrated that the new methods substantially improve currently available approaches with respect to power and approximation of the nominal level.

stat.ME↗
arXiv2016

Quantile spectral processes: Asymptotic analysis and inference

Quantile- and copula-related spectral concepts recently have been considered by various authors. Those spectra, in their most general form, provide a full characterization of the copulas associated with the pairs $(X_t,X_{t-k})$ in a process $(X_t)_{t\in\mathbb{Z}}$, and account for important dynamic features, such as changes in the conditional shape (skewness, kurtosis), time-irreversibility, or dependence in the extremes that their traditional counterparts cannot capture. Despite various proposals for estimation strategies, only quite incomplete asymptotic distributional results are available so far for the proposed estimators, which constitutes an important obstacle for their practical application. In this paper, we provide a detailed asymptotic analysis of a class of smoothed rank-based cross-periodograms associated with the copula spectral density kernels introduced in Dette et al. [Bernoulli 21 (2015) 781-831]. We show that, for a very general class of (possibly nonlinear) processes, properly scaled and centered smoothed versions of those cross-periodograms, indexed by couples of quantile levels, converge weakly, as stochastic processes, to Gaussian processes. A first application of those results is the construction of asymptotic confidence intervals for copula spectral density kernels. The same convergence results also provide asymptotic distributions (under serially dependent observations) for a new class of rank-based spectral methods involving the Fourier transforms of rank-based serial statistics such as the Spearman, Blomqvist or Gini autocovariance coefficients.

math.ST↗
arXiv2016

Testing for Homogeneity in Mixture Models

Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as $C(α)$ tests, as in Neyman (1959), and shown to be locally, asymptotically optimal. These $C(α)$ tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimator. Recent developments in convex optimization have dramatically improved upon earlier EM methods for computation of these estimators, and recent results on the large sample behavior of likelihood ratios involving such estimators yield a tractable form of asymptotic inference. Improvement in computation efficiency also facilitates the use of a bootstrap methods to determine critical values that are shown to work better than the asymptotic critical values in finite samples. Consistency of the bootstrap procedure is also formally established. We compare performance of the two approaches identifying circumstances in which each is preferred.

stat.ME↗
arXiv2015

A subsampled double bootstrap for massive data

The bootstrap is a popular and powerful method for assessing precision of estimators and inferential methods. However, for massive datasets which are increasingly prevalent, the bootstrap becomes prohibitively costly in computation and its feasibility is questionable even with modern parallel computing platforms. Recently Kleiner, Talwalkar, Sarkar, and Jordan (2014) proposed a method called BLB (Bag of Little Bootstraps) for massive data which is more computationally scalable with little sacrifice of statistical accuracy. Building on BLB and the idea of fast double bootstrap, we propose a new resampling method, the subsampled double bootstrap, for both independent data and time series data. We establish consistency of the subsampled double bootstrap under mild conditions for both independent and dependent cases. Methodologically, the subsampled double bootstrap is superior to BLB in terms of running time, more sample coverage and automatic implementation with less tuning parameters for a given time budget. Its advantage relative to BLB and bootstrap is also demonstrated in numerical simulations and a data illustration.

stat.ME↗
arXiv2015

Of copulas, quantiles, ranks and spectra: An $L_1$-approach to spectral analysis

In this paper, we present an alternative method for the spectral analysis of a univariate, strictly stationary time series $\{Y_t\}_{t\in \mathbb {Z}}$. We define a "new" spectrum as the Fourier transform of the differences between copulas of the pairs $(Y_t,Y_{t-k})$ and the independence copula. This object is called a copula spectral density kernel and allows to separate the marginal and serial aspects of a time series. We show that this spectrum is closely related to the concept of quantile regression. Like quantile regression, which provides much more information about conditional distributions than classical location-scale regression models, copula spectral density kernels are more informative than traditional spectral densities obtained from classical autocovariances. In particular, copula spectral density kernels, in their population versions, provide (asymptotically provide, in their sample versions) a complete description of the copulas of all pairs $(Y_t,Y_{t-k})$. Moreover, they inherit the robustness properties of classical quantile regression, and do not require any distributional assumptions such as the existence of finite moments. In order to estimate the copula spectral density kernel, we introduce rank-based Laplace periodograms which are calculated as bilinear forms of weighted $L_1$-projections of the ranks of the observed time series onto a harmonic regression model. We establish the asymptotic distribution of those periodograms, and the consistency of adequately smoothed versions. The finite-sample properties of the new methodology, and its potential for applications are briefly investigated by simulations and a short empirical example.

math.ST↗
arXiv2014

Weak convergence of the empirical copula process with respect to weighted metrics

The empirical copula process plays a central role in the asymptotic analysis of many statistical procedures which are based on copulas or ranks. Among other applications, results regarding its weak convergence can be used to develop asymptotic theory for estimators of dependence measures or copula densities, they allow to derive tests for stochastic independence or specific copula structures, or they may serve as a fundamental tool for the analysis of multivariate rank statistics. In the present paper, we establish weak convergence of the empirical copula process (for observations that are allowed to be serially dependent) with respect to weighted supremum distances. The usefulness of our results is illustrated by applications to general bivariate rank statistics and to estimation procedures for the Pickands dependence function arising in multivariate extreme-value theory.

math.ST↗