Showing papers by "Oliver Linton published in 2010"

Local Regression Models

Abstract: Local regression models are regression models where the parameters are ‘localized’, that is, they are allowed to vary with some or all of the covariates in a general way. Suppose that (Y, X) are random variables and let $$E\left( {Y|X=x} \right)=m\left( x \right)$$ (1) when it exists. The regression function m(x) is of primary interest because it describes how X affects Y One may also be interested in derivatives of m or averages thereof or in derived quantities like conditional variance var(Y|X = x) = E(2|X = x) − E2 (Y\X = x). In cases of heavy-tailed distributions, the conditional expectation may not exist, in which case one may instead work with other location functionals like trimmed mean or median. The conditional expectation is particularly easy to deal with but a lot of what is done for the mean can also be done for the median or other quantities.

Uniform bahadur representation for local polynomial estimates of m-regression and its application to the additive model

Abstract: We use local polynomial fitting to estimate the nonparametric M-regression function for strongly mixing stationary processes {(Y-i, (X) under bar (i))}. We establish a strong uniform consistency rate for the Bahadur representation of estimators of the regression function and its derivatives. These results are fundamental for statistical inference and for applications that involve plugging such estimators into other functionals where some control over higher order terms is required. We apply our results to the estimation of an additive M-regression model.

Estimation for a nonstationary semi-strong garch(1,1) model with heavy-tailed errors

Abstract: This paper studies the estimation of a semi-strong GARCH(1,1) model when it does not have a stationary solution, where semi-strong means that we do not require the errors to be independent over time. We establish necessary and sufficient conditions for a semi-strong GARCH(1,1) process to have a unique stationary solution. For the nonstationary semi-strong GARCH(1,1) model, we prove that a local minimizer of the least absolute deviations (LAD) criterion converges at the rate to a normal distribution under very mild moment conditions for the errors. Furthermore, when the distributions of the errors are in the domain of attraction of a stable law with the exponent κ (1, 2), it is shown that the asymptotic distribution of the Gaussian quasi-maximum likelihood estimator (QMLE) is non-Gaussian but is some stable law with the exponent κ (0, 2). The asymptotic distribution is difficult to estimate using standard parametric methods. Therefore, we propose a percentile-t subsampling bootstrap method to do inference when the errors are independent and identically distributed, as in Hall and Yao (2003). Our result implies that the least absolute deviations estimator (LADE) is always asymptotically normal regardless of whether there exists a stationary solution or not, even when the errors are heavy-tailed. So the LADE is more appealing when the errors are heavy-tailed. Numerical results lend further support to our theoretical results.

On internally corrected and symmetrized kernel estimators for nonparametric regression

Abstract: We investigate the properties of a kernel-type multivariate regression estimator first proposed by Mack and Muller (Sankhya 51:59–72, 1989) in the context of univariate derivative estimation. Our proposed procedure, unlike theirs, assumes that bandwidths of the same order are used throughout; this gives more realistic asymptotics for the estimation of the function itself but makes the asymptotic distribution more complicated. We also propose a modification of this estimator that has a symmetric smoother matrix, which makes it admissible, unlike some other common regression estimators. We compare the performance of the estimators in a Monte Carlo experiment. Multivariate regression - Smoothing matrix - Symmetry

Semiparametric Estimation of Locally Stationary Diffusion Models

Abstract: This paper proposes a class of locally stationary diffusion processes. The model has a time varying but locally linear drift and a volatility coefficient that is allowed to vary over time and space. We propose estimators of all the unknown quantities based on long span data. Our estimation method makes use of the local stationarity. We establish asymptotic theory for the proposed estimators as the time span increases. We apply this method to the real financial data to illustrate the validity of our model. Finally, we present a simulation study to provide the finitesample performance of the proposed estimators.

Multivariate Density Estimation Using Dimension Reducing Information and Tail Flattening Transformations

Abstract: We propose a nonparametric multiplicative bias corrected transformation estimator designed for heavy tailed data. The multiplicative correction is based on prior knowledge and has a dimension reducing effect at the same time as the original dimension of the estimation problem is retained. Adding a tail-flattening transformation improves the estimation significantly – particularly in the tail – and provides significant graphical advantages by allowing the density estimation to be visualized in a simple way. The combined method is demonstrated on a fire insurance data set and where it provides excellent performance in a data-driven simulation study.

Local Linear Fitting Under Near Epoch Dependence: Uniform consistency with Convergence Rates

Abstract: Local linear fitting is a popular nonparametric method in statistical and econometric modelling. Lu and Linton (2007) established the pointwise asymptotic distribution for the local linear estimator of a nonparametric regression function under the condition of near epoch dependence. In this paper, we further investigate the uniform consistency of this estimator. The uniform strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results regarding uniform convergence rates for nonparametric kernel-based estimators are provided. The results of this paper will be of wide potential interest in time series semiparametric modelling.

Evaluating hedge fund performance: a stochastic dominance approach

Abstract: We introduce a general and flexible framework for hedge fund performance evaluation and asset allocation: stochastic dominance (SD) theory. Our approach utilizes statistical tests for stochastic dominance to compare the returns of hedge funds. We form hedge fund portfolios by using SD criteria and examine the out-of-sample performance of these hedge fund portfolios. Compared to performance of portfolios of randomly selected hedge funds and mean–variance efficient hedge funds, our results show that fund selection method based on SD criteria greatly improves the performance of hedge fund portfolio.

Making Inferences About Rich Country - Poor Country Convergence: The Polarization Trapezoid and Overlap measures

Abstract: Underlying the unresolved debate over whether the gap between rich and poor country GNP per capita has narrowed is a concern for wellbeing. The issue is really about the changing shapes of distributions of wellbeing indicators. As limiting cases con- vergence between rich and poor country groups can be brought about by countries within groups becoming less alike without any diminution of growth rate differentials between them or it can be brought about by reductions in these differentials without any diminution of within group identity. In essence the debate is about the extent to which rich and poor countries are polarizing, a subject first theoretically explored by Esteban and Ray (1994). The empirical issue is about whether separate groups can be identified in the overall distribution and whether they are tending toward common or distinct equilibria. This paper proposes two simple statistics for the problem, the Overlap measure and the Trapezoidal measure, changes in which reflect a combination of increasing (decreasing) subgroup location differences and decreasing (increasing) subgroup spreads which are the characteristics of polarization (convergence). The former statistic is of use when the sub-distributions are identified, while the latter can be used whether or not the subgroups are identified. These techniques are applied to the examination of convergence in GDP per capita between rich and poor nations when growth is viewed either as a wellbeing index or a technology index (i.e. the data are, or are not, population weighted). It turns out that such a distinction matters, viewed technologically there is divergence, viewed in a wellbeing sense there is convergence. As a collection of countries Africa is diverging from the rest of the world whatever the perspective of growth.

Semiparametric Estimation of Locally Stationary Diffusion Models

Abstract: This paper proposes a class of locally stationary diffusion processes. The modelhas a time varying but locally linear drift and a volatility coefficient that is allowed tovary over time and space. We propose estimators of all the unknown quantitiesbased on long span data. Our estimation method makes use of the localstationarity. We establish asymptotic theory for the proposed estimators as thetime span increases. We apply this method to the real financial data to illustrate thevalidity of our model. Finally, we present a simulation study to provide the finitesampleperformance of the proposed estimators.

Semiparametric Estimation of Markov Decision Processes with Continuous State Space

Abstract: We propose a general two-step estimation method for the structural parameters of popular semiparametric Markovian discrete choice models that include a class of Markovian Games and allow for continuous observable state space. The estimation procedure is simple as it directly generalizes the computationally attractive methodology of Pesendorfer and Schmidt-Dengler (2008) that assumed finite observable states. This extension is non-trivial as the value functions, to be estimated nonparametrically in the first stage, are defined recursively in a non-linear functional equation. Utilizing structural assumptions, we show how to consistently estimate the infinite dimensional parameters as the solution to some type II integral equations, the solving of which is a well-posed problem. We provide sufficient set of primitives to obtain root-T consistent estimators for the finite dimensional structural parameters and the distribution theory for the value functions in a time series framework.

Semiparametric estimation of Markov decision processeswith continuous state space

Abstract: We propose a general two-step estimation method for the structural parameters ofpopular semiparametric Markovian discrete choice models that include a class ofMarkovian Games andallow for continuous observable state space. The estimation procedure is simpleas it directly generalizes the computationally attractive methodology of Pesendorferand Schmidt-Dengler (2008) that assumed finite observable states. This extensionis non-trivial as the value functions, to be estimated nonparametrically in the firststage, are defined recursively in a non-linear functional equation. Utilizingstructural assumptions, we show how to consistently estimate the infinitedimensional parameters as the solution to some type II integral equations, thesolving of which is a well-posed problem. We provide sufficient set of primitives toobtain root-T consistent estimators for the finite dimensional structural parametersand the distribution theory for the value functions in a time series framework.

A Semiparametric Panel Model for Unbalanced Data with Application to Climate Change in the United Kingdom

Abstract: This paper is concerned with developing a semiparametric panel model to explain the trend in UK temperatures and other weather outcomes over the last century. We work with the monthly averaged maximum and minimum temperatures observed at the twenty six Meteorological Office stations. The data is an unbalanced panel. We allow the trend to evolve in a nonparametric way so that we obtain a fuller picture of the evolution of common temperature in the medium timescale. Profile likelihood estimators (PLE) are proposed and their statistical properties are studied. The proposed PLE has improved asymptotic property comparing the the sequential two-step estimators. Finally, forecasting based on the proposed model is studied.

Loch Linear Fitting Under Near Epoch Dependence: Uniform Consistency with Convergence Rate

Abstract: Local linear fitting is a popular nonparametric method in nonlinear statistical and econometric modelling. Lu and Linton (2007) established the point wise asymptotic distribution (central limit theorem) for the local linear estimator of nonparametric regression function under the condition of near epoch dependence. We further investigate the uniform consistency of this estimator. The uniformly strong and weak consistencies with convergence rates for the local linear fitting are established under mild conditions. Furthermore, general results of uniform convergence rates for nonparametric kernel-based estimators are provided. Applications of our results to conditional variance function estimation and some economic time series models are also discussed. The results of this paper will be of widely potential interest in time series semiparametric modelling.

Efficient estimation of a multivariate multiplicative volatility model

Abstract: We propose a multivariate generalization of the multiplicative volatility model of Engle and Rangel (2008), which has a nonparametric long run component and a unit multivariate GARCH short run dynamic component. We suggest various kernel-based estimation procedures for the parametric and nonparametric components, and derive the asymptotic properties thereof. For the parametric part of the model, we obtain the semiparametric efficiency bound. Our method is applied to a bivariate stock index series. We find that the univariate model of Engle and Rangel (2008) appears to be violated in the data whereas our multivariate model is more consistent with the data.