galton edgeworth frisch and prospects for quantile regression in econometrics the work of three leading figures in the early history of econometrics is used to motivate some recent developments in the theory and application of quantile regression we stress not only the robustness advantages of this form of semiparametric statistical method but also the opportunity to recover a more complete description of the statistical relationship between variables a recent proposal for a more x robust form of quantile regression based on maximal depth ideas is described along with an interesting historical antecedent finally the notorious computational burden of median regression and quantile regression more generally is addressed it is argued that recent developments in interior point methods for linear programming together with some new preprocessing ideas make it possible to compute quantile regressions as quickly as least squares regressions throughout the entire range of problem sizes encountered in econometrics

nonparametric quantile estimations for dynamic smooth coefficient models in this article quantile regression methods are suggested for a class of smooth coefficient time series models we use both local polynomial and local constant fitting schemes to estimate the smooth coefficients in a quantile framework we establish the asymptotic properties of both the local polynomial and local constant estimators for mixing time series also a bandwidth selector based on the nonparametric version of the akaike information criterion is suggested together with a consistent estimate of the asymptotic covariance matrix furthermore the asymptotic behaviors of the estimators at boundaries are examined a comparison of the local polynomial quantile estimator with the local constant estimator is presented a simulation study is carried out to illustrate the performance of estimates an empirical application of the model to real data further demonstrates the potential of the proposed modeling procedures

a quantile regression approach to generating prediction intervals exponential smoothing methods do not involve a formal procedure for identifying the underlying data generating process the issue is then whether prediction intervals should be estimated by a theoretical approach with the assumption that the method is optimal in some sense or by an emp irical procedure in this paper we present an alternative hybrid approach which applies quantile regression to the empirical fit errors to produce forecast error quantile models these models are functions of the lead time as suggested by the theoretical variance expressions in addition to avoiding the optimality assumption the method is nonparametric so there is no need for the common normality assumption application of the new approach to simple holt s and damped holt s exponential smoothing using simulated and real data sets gave encouraging results

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

inference on the quantile regression process tests based on the quantile regression process can be formulated like the classical kolmogorov smirnov and cramer von mises tests of goodness of fit employing the theory of bessel processes as in kiefer 1959 however it is frequently desirable to formulate hypotheses involving unknown nuisance parameters thereby jeopardizing the distribution free character of these tests we characterize this situation as the durbin problem since it was posed in durbin 1973 for parametric empirical processes in this paper we consider an approach to the durbin problem involving a martingale transformation of the parametric empirical process suggested by khmaladze 1981 and show that it can be adapted to a wide variety of inference problems involving the quantile regression process in particular we suggest new tests of the location shift and location scale shift models that underlie much of classical econometric inference the methods are illustrated with a reanalysis of data on unemployment durations from the pennsylvania reemployment bonus experiments the pennsylvania experiments conducted in 1988 89 were designed to test the efficacy of cash bonuses paid for early reemployment in shortening the duration of insured unemployment spells

identifying risk factors for severe childhood malnutrition by boosting additive quantile regression we investigated the risk factors for childhood malnutrition in india based on the 2005 2006 demographic and health survey by applying a novel estimation technique for additive quantile regression ordinary linear and generalized linear regression models relate the mean of a response variable to a linear combination of covariate effects and as a consequence focus on average properties of the response the use of such a regression model for analyzing childhood malnutrition in developing or transition countries implies that the estimated effects describe the average nutritional status however it is of even greater interest to analyze quantiles of the response distribution such as the 5 or 10 quantile which relate to the risk of extreme malnutrition our investigation is based on a semiparametric extension of quantile regression models where different types of nonlinear effects are included in the model equation leading to additive quantile regression we addressed the variable selection and model c

robust estimation for the varying coefficient partially nonlinear models abstract in this paper we propose a robust estimation procedure based on the exponential squared loss esl function for the varying coefficient partially nonlinear model under some conditions the asymptotic properties of proposed estimators are established furthermore we develop a new minorization maximization mm algorithm to calculate the estimates for both non parametric and parametric parts and introduce a data driven procedure to select the tuning parameters simulation studies illustrate that the proposed method is more robust and efficient than the classical least squares technique when there are outliers in the dataset finally we apply the proposed methodology to analyze a real dataset the results reveal that the proposed has better the predictive ability

nonparametric estimation of conditional var and expected shortfall abstract this paper considers a new nonparametric estimation of conditional value at risk and expected shortfall functions conditional value at risk is estimated by inverting the weighted double kernel local linear estimate of the conditional distribution function the nonparametric estimator of conditional expected shortfall is constructed by a plugging in method both the asymptotic normality and consistency of the proposed nonparametric estimators are established at both boundary and interior points for time series data we show that the weighted double kernel local linear conditional distribution estimator has the advantages of always being a distribution continuous and differentiable besides the good properties from both the double kernel local linear and weighted nadaraya watson estimators moreover an ad hoc data driven fashion bandwidth selection method is proposed based on the nonparametric version of the akaike information criterion finally an empirical study is carried out to illustrate the finite sample performance of the proposed estimators

optimal smoothing in nonparametric conditional quantile derivative function estimation abstract marginal effect in nonparametric quantile regression is of special interest as it quantitatively measures how one unit change in explanatory variable heterogeneously affects dependent variable ceteris paribus at distinct quantiles in this paper we propose a data driven bandwidth selection procedure based on the gradient of an unknown quantile regression function our method delivers the bandwidth with the oracle property in the sense that it is asymptotically equivalent to the optimal bandwidth if the true gradient were known the results of monte carlo simulations are reported and the finite sample performance of our proposed method confirms our theoretical analysis an empirical application is also provided showing that our proposed method delivers more reasonable and reliable quantile derivative estimates than traditional cross validation method

inference for covariate adjusted regression via varying coefficient models we consider covariate adjusted regression car a regression method for situations where predictors and response are observed after being distorted by a multiplicative factor the distorting factors are unknown functions of an observable covariate where one specific distorting function is associated with each predictor or response the dependence of both response and predictors on the same confounding covariate may alter the underlying regression relation between undistorted but unobserved predictors and response we consider a class of highly flexible adjustment methods for parameter estimation in the underlying regression model which is the model of interest asymptotic normality of the estimates is obtained by establishing a connection to varying coefficient models these distribution results combined with proposed consistent estimates of the asymptotic variance are used for the construction of asymptotic confidence intervals for the regression coefficients the proposed approach is illustrated with data on serum creatinine and finite sample properties of the proposed procedures are investigated through a simulation study

efficient estimation of conditional variance functions in stochastic regression author s fan jianqing yao qiwei abstract conditional heteroscedasticity has been often used in modelling and understanding the variability of statistical data under a general setup which includes the nonlinear time series model as a special case we propose an e cient and adaptive method for estimating the conditional variance the basic idea is to apply a local linear regression to the squared residuals we demonstrate that without knowing the regression function we can estimate the conditional variance asymptotically as well as if the regression were given this asymptotic result established under the assumption that the observations are made from a strictly stationary and absolutely regular process is also veri ed via simulation further the asymptotic result paves the way for adapting an automatic bandwidth selection scheme an application with nancial data illustrates the usefulness of the proposed techniques

a semiparametric quantile panel data model with an application to estimating the growth effect of fdi abstract this paper estimates the impact of foreign direct investment on economic growth by proposing a new semiparametric quantile panel data model with correlated random effects in which some of the coefficients are allowed to depend on some smooth economic variables while other coefficients remain constant a three stage estimation procedure is proposed to estimate both constant and functional coefficients and their asymptotic properties are investigated a simple and easily implemented procedure for making inferences is proposed monte carlo simulation is conducted to examine the finite sample performance of the proposed estimators finally using the cross country panel data we find a strong empirical evidence of the existence of the absorptive capacity hypothesis together with another new finding that fdi has much stronger growth effects for countries with fast economic growth than for those with slow economic growth

functional coefficient autoregressive models in this article we propose a new class of models for nonlinear time series analysis investigate properties of the proposed model and suggest a modeling procedure for building such a model the proposed modeling procedure makes use of ideas from both parametric and nonparametric statistics a consistency result is given to support the procedure for illustration we apply the proposed model and procedure to several data sets and show that the resulting models substantially improve postsample multi step ahead forecasts over other models

generalized varying coefficient models with unknown link function summary we propose a new estimation method for generalized varying coefficient models where the link function is specified up to some smoothness conditions consistency and asymptotic normality of the estimated varying coefficient functions are established simulation results and a real data application demonstrate the usefulness of the new method

adaptive varying coefficient linear models varying coefficient linear models arise from multivariate nonparametric regression nonlinear time series modelling and forecasting functional data analysis longitudinal data analysis and others it has been a common practice to assume that the vary coefficients are functions of a given variable which is often called an index a frequently asked question is which variable should be used as the index in this paper we explore the class of the varying coefficient linear models in which the index is unknown and is estimated as a linear combination of regression and or other variables this will enlarge the modelling capacity substantially we search for the index such that the derived varying coefficient model provides the best approximation to the underlying unknown multi dimensional regression function in the least square sense the search is implemented through the newly proposed hybrid backfitting algorithm the core of the algorithm is the alternative iteration between estimating the index through a one step scheme and estimating coefficient functions through a one dimensional local linear smoothing the generalised cross validation method for choosing bandwidth is efficiently incorporated into the algorithm the locally significant variables are selected in terms of the combined use of t statistic and akaike information criterion we further extend the algorithm for the models with two indices simulation shows that the proposed methodology has appreciable flexibility to model complex multivariate nonlinear structure and is practically feasible with average modern computers the methods are further illustrated through the canadian mink muskrat data in 1925 1994 and the pound dollar exchange rates in 1974 1983

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

sieve instrumental variable quantile regression estimation of functional coefficient models abstract in this paper we consider sieve instrumental variable quantile regression ivqr estimation of functional coefficient models where the coefficients of endogenous regressors are unknown functions of some exogenous covariates we estimate the functional coefficients by the sieve ivqr technique and establish the uniform consistency and asymptotic normality of the estimators based on the sieve estimates we propose a nonparametric specification test for the constancy of the functional coefficients and study its asymptotic we conduct simulations to evaluate the finite sample behavior of our estimator and test statistic and apply our method to study the estimation of quantile engel curves

quantile regression methods for reference growth charts estimation of reference growth curves for children s height and weight has traditionally relied on normal theory to construct families of quantile curves based on samples from the reference population age specific parametric transformation has been used to significantly broaden the applicability of these normal theory methods non parametric quantile regression methods offer a complementary strategy for estimating conditional quantile functions we compare estimated reference curves for height using the penalized likelihood approach of cole and green statistics in medicine 1992 11 1305 1319 with quantile regression curves based on data used for modern finnish reference charts an advantage of the quantile regression approach is that it is relatively easy to incorporate prior growth and other covariates into the analysis of longitudinal growth data quantile specific autoregressive models for unequally spaced measurements are introduced and their application to diagnostic screening is illustrated copyright 2005 john wiley sons ltd

on additive conditional quantiles with high dimensional covariates we investigate the estimation of the conditional quantile when many covariates are involved in particular we model the conditional quantile of a response as a nonlinear additive function of relevant covariates for this setup we propose a nonparametric smoother to estimate the unknown functions the estimator provides direct computation of the nonlinear functions because it does not require any iteration the estimator allows fast and routine data analysis on the theoretical front we also show asymptotic properties of the estimator including mean squared error and limiting distribution the theory confirms that for moderate dimension of the covariates the estimator escapes the curse of dimensionality problem both simulated and real data examples are provided to illustrate the methodology

estimating structural changes in regression quantiles this paper considers the estimation of multiple structural changes occurring at unknown dates in one or multiple conditional quantile functions the analysis covers time series models as well as models with repeated cross sections we estimate the break dates and other parameters jointly by minimizing the check function over all permissible break dates the limiting distribution of the estimator is derived and the coverage property of the resulting confidence interval is assessed via simulations a procedure to determine the number of breaks is also discussed empirical applications to the quarterly us real gdp growth rate and the underage drunk driving data suggest that the method can deliver more informative results than the analysis of the conditional mean function alone

identification of nonlinear time series first order characterization and order determination summary we study the possibility of identifying nonlinear time series using nonparametric estimates of the conditional mean and conditional variance it is shown that most nonlinear models satisfy the assumptions needed to apply nonparametric asymptotic theory sampling variations of the conditional quantities are studied by simulation and explained by asymptotic arguments for a number of first order nonlinear autoregressive processes the conditional mean and variance can be used for identification purposes but one must be aware of bias and misspecification effects we also propose a criterion for determining the order of a general nonlinear model the criterion is justified in parts by heuristics but encouraging results are obtained from a limited set of simulation experiments several

semiparametric quantile regression estimation in dynamic models with partially varying coefficients national science foundation of china 71131008 70871003 70971113 university of north carolina at charlotte chinese ministry of education minjiang scholarship from fujian province china xiamen university boston college

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

interval forecasting an analysis based upon arch quantile estimators abstract in this paper we explore techniques for obtaining interval forecasts based on estimated time series models for processes which may exhibit autoregressive conditional heteroskedasticity arch to deal with the available variety of possible interval forecasts we propose a method for combining these forecasts based on quantile regression techniques our approach is practical rather than theoretical with attention focused directly on obtaining interval forecasts for two u s time series a measure of unemployment and a treasury bill rate we evaluate the performance of our procedures using a variety of diagnostics we find interval estimates which perform reasonably well judged by both in sample and out of sample criteria our experience suggests that a certain amount of care is required in order to obtain useful forecasts

local linear quantile regression abstract in this article we study nonparametric regression quantile estimation by kernel weighted local linear fitting two such estimators are considered one is based on localizing the characterization of a regression quantile as the minimizer of e pp y a x x where p is the appropriate check function the other follows by inverting a local linear conditional distribution estimator and involves two smoothing parameters rather than one our aim is to present fully operational versions of both approaches and to show that each works quite well although either might be used in practice we have a particular preference for the second our automatic smoothing parameter selection method is novel the main regression quantile smoothing parameters are chosen by rule of thumb adaptations of state of the art methods for smoothing parameter selection for regression mean estimation the techniques are illustrated by application to two datasets and compared in simulations

direct use of regression quantiles to construct confidence sets in linear models direct use of the empirical quantile function provides a standard distribution free approach to constructing confidence intervals and confidence bands for population quantiles we apply this method to construct confidence intervals and confidence bands for regression quantiles and to construct prediction intervals based on sample regression quantiles comparison of the direct method with the studentization and the bootstrap methods are discussed simulation results show that the direct method has the advantage of robustness against departure from the normality assumption of the error terms

nonparametric estimation of an additive quantile regression model this article is concerned with estimating the additive components of a nonparametric additive quantile regression model we develop an estimator that is asymptotically normally distributed with a rate of convergence in probability of n r 2r 1 when the additive components are r times continuously differentiable for some r 2 this result holds regardless of the dimension of the covariates and thus the new estimator has no curse of dimensionality in addition the estimator has an oracle property and is easily extended to a generalized additive quantile regression model with a link function the numerical performance and usefulness of the estimator are illustrated by monte carlo experiments and an empirical example

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

on the sum of the hermitian scalar curvatures of a compact hermitian manifold soit m une variete hermitienne compacte alors a u v 0 p m 0 m 0 b u v 0 p m 0 1 m 0 ou u et v sont les deux courbures scalaires de geometrie hermitienne et p m et q m sont les miemes plurigenres et le mieme plurigenre dual de m

quantile regression with varying coefficients quantile regression provides a framework for modeling statistical quantities of interest other than the conditional mean the regression methodology is well developed for linear models but less so for nonparametric models we consider conditional quantiles with varying coefficients and propose a methodology for their estimation and assessment using polynomial splines the proposed estimators are easy to compute via standard quantile regression algorithms and a stepwise knot selection algorithm the proposed rao score type test that assesses the model against a linear model is also easy to implement we provide asymptotic results on the convergence of the estimators and the null distribution of the test statistic empirical results are also provided including an application of the methodology to forced expiratory volume fev data

on average derivative quantile regression for fixed 0 1 the quantile regression function gives the th quantile x in the conditional distribution of a response variable y given the value x x of a vector of covariates it can be used to measure the effect of covariates not only in the center of a population but also in the upper and lower tails a functional that summarizes key features of the quantile specific relationship between x and y is the vector of weighted expected values of the vector of partial derivatives of the quantile function x in a nonparametric setting can be regarded as a vector of quantile specific nonparametric regression coefficients in survival analysis models e g cox s proportional hazard model proportional odds rate model accelerated failure time model and in monotone transformation models used in regression analysis gives the direction of the parameter vector in the parametric part of the model can also be used to estimate the direction of the parameter vector in semiparametric single index models popular in econometrics we show that under suitable regularity conditions the estimate of obtained by using the locally polynomial quantile estimate of chaudhuri 1991a is n 1 2 consistent and asymptotically normal with asymptotic variance equal to the variance of the influence function of the functional we discuss how the estimate of can be used for model diagnostics and in the construction of a link function estimate in general single index models

estimation and testing of varying coefficients in quantile regression in this article we establish a novel connection between the null hypothesis h0 on the coefficients and a rank reducible form of the varying coefficient model in quantile regression we use b splines to approximate the varying coefficients in the rank reducible model and make use of the fact that the null hypothesis h0 implies a unidimensional structure of a transformed coefficient matrix for the b spline basis functions by evaluating the unidimensional structure we alleviate the difficulty of testing such hypotheses commonly considered in varying coefficient quantile models we demonstrate through numerical studies that the proposed method can be much more powerful than the rank score test which is widely used in the quantile regression literature supplementary materials for this article are available online

new efficient estimation and variable selection methods for semiparametric varying coefficient partially linear models the complexity of semiparametric models poses new challenges to statistical inference and model selection that frequently arise from real applications in this work we propose new estimation and variable selection procedures for the semiparametric varying coefficient partially linear model we first study quantile regression estimates for the nonparametric varying coefficient functions and the parametric regression coefficients to achieve nice efficiency properties we further develop a semiparametric composite quantile regression procedure we establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts and show that the estimators achieve the best convergence rate moreover we show that the proposed method is much more efficient than the least squares based method for many non normal errors and that it only loses a small amount of efficiency for normal errors in addition it is shown that the loss in efficiency is at most 11 1 for estimating varying coefficient functions and is no greater than 13 6 for estimating parametric components to achieve sparsity with high dimensional covariates we propose adaptive penalization methods for variable selection in the semiparametric varying coefficient partially linear model and prove that the methods possess the oracle property extensive monte carlo simulation studies are conducted to examine the finite sample performance of the proposed procedures finally we apply the new methods to analyze the plasma beta carotene level data

conditional growth charts growth charts are often more informative when they are customized per subject taking into account prior measurements and possibly other covariates of the subject we study a global semiparametric quantile regression model that has the ability to estimate conditional quantiles without the usual distributional assumptions the model can be estimated from longitudinal reference data with irregular measurement times and with some level of robustness against outliers and it is also flexible for including covariate information we propose a rank score test for large sample inference on covariates and develop a new model assessment tool for longitudinal growth data our research indicates that the global model has the potential to be a very useful tool in conditional growth chart analysis

functional coefficient regression models for nonlinear time series abstract the local linear regression technique is applied to estimation of functional coefficient regression models for time series data the models include threshold autoregressive models and functional coefficient autoregressive models as special cases but with the added advantages such as depicting finer structure of the underlying dynamics and better postsample forecasting performance also proposed are a new bootstrap test for the goodness of fit of models and a bandwidth selector based on newly defined cross validatory estimation for the expected forecasting errors the proposed methodology is data analytic and of sufficient flexibility to analyze complex and multivariate nonlinear structures without suffering from the curse of dimensionality the asymptotic properties of the proposed estimators are investigated under the mixing condition both simulated and real data examples are used for illustration

bivariate quantile smoothing splines it has long been recognized that the mean provides an inadequate summary whereas the set of quantiles can supply a more complete description of a sample we introduce bivariate quantile smoothing splines which belong to the space of bilinear tensor product splines as nonparametric estimators for the conditional quantile functions in a two dimensional design space the estimators can be computed by using standard linear programming techniques and can further be used as building blocks for conditional quantile estimations in higher dimensions for moderately large data sets we recommend penalized bivariate b splines as approximate solutions we use real and simulated data to illustrate the methodology proposed

estimation of conditional densities and sensitivity measures in nonlinear dynamical systems using locally polynomial regression we develop nonparametric estimators for the conditional density function and its square root and their partial derivatives two measures of sensitivity to initial conditions in nonlinear stochastic dynamic systems are proposed one of which relates fisher information with initial value sensitivity in dynamical systems we propose estimators for these and show asymptotic normality for one of them we further propose a simple method for choosing the bandwidth the methods are illustrated by simulation of two well known models in dynamical systems

quantile smoothing splines summary although nonparametric regression has traditionally focused on the estimation of conditional mean functions nonparametric estimation of conditional quantile functions is often of substantial practical interest we explore a class of quantile smoothing splines defined as solutions to