Generalized least squares cross‐validation in kernel density estimation

The kernel density estimation is a popular method in density estimation. The main issue is bandwidth selection, which is a well‐known topic and is still frustrating statisticians. A robust least squares cross‐validation bandwidth is proposed, which significantly improves the classical least squares cross‐validation bandwidth for its variability and undersmoothing, adapts to different kinds of densities, and outperforms the existing bandwidths in statistical literature and software.     

A number‐of‐modes reference rule for density estimation under multimodality

We consider kernel density estimation for univariate distributions. The question of interest is as follows: given that the data analyst has some background knowledge on the modality of the data (for instance, ‘data of this type are usually bimodal’), what is the adequate bandwidth to choose? We answer this question by extending Silverman's idea of ‘normal‐reference’ to that of ‘reference to a Gaussian mixture’. The concept is illustrated in the light of real data examples.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

非线性计量经济模型的非参数估计方法研究

本文提出使用核估计的方法构造平滑转移模型(STR)的非参数模拟最大似然估计(NPSML),给出了NPSML估计量的构造方法、渐近性质以及相应的核函数和窗宽的选择准则,并利用滑动窗宽算法对估计量的构造过程进行了改进。通过Monte Carlo实验证明,该方法是可靠的,并且当误差项存在序列相关时,此种估计量是稳健的。     

A simple approach to the parametric estimation of potentially nonstationary diffusions

A simple and robust approach is proposed for the parametric estimation of scalar homogeneous stochastic differential equations. We specify a parametric class of diffusions and estimate the parameters of interest by minimizing criteria based on the integrated squared difference between kernel estimates of the drift and diffusion functions and their parametric counterparts. The procedure does not require simulations or approximations to the true transition density and has the simplicity of standard nonlinear least-squares methods in discrete time. A complete asymptotic theory for the parametric estimates is developed. The limit theory relies on infill and long span asymptotics and is robust to deviations from stationarity, requiring only recurrence.     

Fast Kernel Smoothing of Point Patterns on a Large Network using Two‐dimensional Convolution

We propose a computationally efficient and statistically principled method for kernel smoothing of point pattern data on a linear network. The point locations, and the network itself, are convolved with a two‐dimensional kernel and then combined into an intensity function on the network. This can be computed rapidly using the fast Fourier transform, even on large networks and for large bandwidths, and is robust against errors in network geometry. The estimator is consistent, and its statistical efficiency is only slightly suboptimal. We discuss bias, variance, asymptotics, bandwidth selection, variance estimation, relative risk estimation and adaptive smoothing. The methods are used to analyse spatially varying frequency of traffic accidents in Western Australia and the relative risk of different types of traffic accidents in Medellín, Colombia.     

Generalized runs tests for the IID hypothesis

We provide a family of tests for the IID hypothesis based on generalized runs, powerful against unspecified alternatives, providing a useful complement to tests designed for specific alternatives, such as serial correlation, GARCH, or structural breaks. Our tests have appealing computational simplicity in that they do not require kernel density estimation, with the associated challenge of bandwidth selection. Simulations show levels close to nominal asymptotic levels. Our tests have power against both dependent and heterogeneous alternatives, as both theory and simulations demonstrate.     

An Improved Test for Earnings Management Using Kernel Density Estimation

Abstract

This paper describes improvements on methods developed by Burgstahler and Dichev (1997, Earnings management to avoid earnings decreases and losses, Journal of Accounting and Economics, 24(1), pp. 99–126) and Bollen and Pool (2009, Do hedge fund managers misreport returns? Evidence from the pooled distribution, Journal of Finance, 64(5), pp. 2257–2288) to test for earnings management by identifying discontinuities in distributions of scaled earnings or earnings forecast errors. While existing methods use preselected bandwidths for kernel density estimation and histogram construction, the proposed test procedure addresses the key problem of bandwidth selection by using a bootstrap test to endogenise the selection step. The main advantage offered by the bootstrap procedure over prior methods is that it provides a reference distribution that cannot be globally distinguished from the empirical distribution rather than assuming a correct reference distribution. This procedure limits the researcher's degrees of freedom and offers a simple procedure to find and test a local discontinuity. I apply the bootstrap density estimation to earnings, earnings changes, and earnings forecast errors in US firms over the period 1976–2010. Significance levels found in earlier studies are greatly reduced, often to insignificant values. Discontinuities cannot be detected in analysts’ forecast errors, while such findings of discontinuities in earlier research can be explained by a simple rounding mechanism. Earnings data show a large drop in loss aversion after 2003 that cannot be detected in changes of earnings.     

Extrapolation‐based Bandwidth Selectors: A Review and Comparative Study with Discussion on Bivariate Applications

Cross‐validation is a widely used tool in selecting the smoothing parameter in a non‐parametric procedure. However, it suffers from large sampling variation and tends to overfit the data set. Many attempts have been made to reduce the variance of cross‐validation. This paper focuses on two recent proposals of extrapolation‐based cross‐validation bandwidth selectors: indirect cross‐validation and subsampling‐extrapolation technique. In univariate case, we notice that using a fixed value parameter surrogate for indirect cross‐validation works poorly when the true density is hard to estimate, while the subsampling‐extrapolation technique is more robust to non‐normality. We investigate whether a hybrid bandwidth selector could benefit from the advantages of both approaches and compare the performance of different extrapolation‐based bandwidth selectors through simulation studies, real data analyses and large sample theory. A discussion on their extension to bivariate case is also presented.     

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.     

Robust prediction of domain compositions from uncertain data using isometric logratio transformations in a penalized multivariate Fay–Herriot model

Assessing regional population compositions is an important task in many research fields. Small area estimation with generalized linear mixed models marks a powerful tool for this purpose. However, the method has limitations in practice. When the data are subject to measurement errors, small area models produce inefficient or biased results since they cannot account for data uncertainty. This is particularly problematic for composition prediction, since generalized linear mixed models often rely on approximate likelihood inference. Obtained predictions are not reliable. We propose a robust multivariate Fay–Herriot model to solve these issues. It combines compositional data analysis with robust optimization theory. The nonlinear estimation of compositions is restated as a linear problem through isometric logratio transformations. Robust model parameter estimation is performed via penalized maximum likelihood. A robust best predictor is derived. Simulations are conducted to demonstrate the effectiveness of the approach. An application to alcohol consumption in Germany is provided.     

Robust estimation for the order of finite mixture models

In this paper, we study a robust and efficient estimation procedure for the order of finite mixture models based on the minimizing a penalized density power divergence estimator. For this task, we use the locally conic parametrization approach developed by Dacunha-Castelle and Gassiate (ESAIM Probab Stat 285–317, 1997a; Ann Stat 27:1178–1209, 1999), and verify that the minimizing a penalized density power divergence estimator is consistent. Simulation results are provided for illustration.     

A bootstrap-based non-parametric forecast density

Interest in density forecasts (as opposed to solely modeling the conditional mean) arises from the possibility of dynamics in higher moments of a time series, as well as in forecasting the probability of future events in some applications. By combining the idea of Markov bootstrapping with that of kernel density estimation, this paper presents a simple non-parametric method for estimating out-of-sample multi-step density forecasts. The paper also considers a host of evaluation tests for examining the dynamic misspecification of estimated density forecasts by targeting autocorrelation, heteroskedasticity and neglected non-linearity. These tests are useful, as a rejection of the tests gives insight into ways to improve a particular forecasting model. In an extensive Monte Carlo analysis involving a range of commonly used linear and non-linear time series processes, the non-parametric method is shown to work reasonably well across the simulated models for a suitable choice of the bandwidth (smoothing parameter). Furthermore, an application of the method to the U.S. Industrial Production series provides multi-step density forecasts that show no sign of dynamic misspecification.     

A Selective Overview and Comparison of Robust Mixture Regression Estimators

Mixture regression models have been widely used in business, marketing and social sciences to model mixed regression relationships arising from a clustered and thus heterogeneous population. The unknown mixture regression parameters are usually estimated by maximum likelihood estimators using the expectation–maximisation algorithm based on the normality assumption of component error density. However, it is well known that the normality-based maximum likelihood estimation is very sensitive to outliers or heavy-tailed error distributions. This paper aims to give a selective overview of the recently proposed robust mixture regression methods and compare their performance using simulation studies.     

On B-robust instrumental variable estimation of the linear model with panel data

The aim of this paper is to demonstrate how to acquire robust consistent estimates of the linear model when the fundamental orthogonality condition is not fulfilled. With this end in view, we develop two estimation procedures: Two stage generalized M (2SGM) and robust generalized method of moments (RGMM). Both estimators are B-robust, i.e. their associated influence function is bounded, consistent and asymptotic normally distributed. Our simulation results indicate that the relatively efficient RGMM estimator (in regressions with heteroskedastic and/or autocorrelated errors) provides accurate parameter estimates of a panel data model with all variables subject to measurement errors, even if a substantial portion of the data is contaminated with aberrant observations. The traditional estimation techniques such as 2SLS and GMM break down when outliers corrupt the data.     

Robust estimation for structural spurious regressions and a Hausman-type cointegration test

This paper analyzes an approach to correcting spurious regressions involving unit-root nonstationary variables by generalized least squares (GLS) using asymptotic theory. This analysis leads to a new robust estimator and a new test for dynamic regressions. The robust estimator is consistent for structural parameters not just when the regression error is stationary but also when it is unit-root nonstationary under certain conditions. We also develop a Hausman-type test for the null hypothesis of cointegration for dynamic ordinary least squares (OLS) estimation. We demonstrate our estimation and testing methods in three applications: (i) long-run money demand in the U.S., (ii) output convergence among industrial and developing countries, and (iii) purchasing power parity (PPP) for traded and non-traded goods.     

Quantile-based nonparametric inference for first-price auctions

We propose a quantile-based nonparametric approach to inference on the probability density function (PDF) of the private values in first-price sealed-bid auctions with independent private values. Our method of inference is based on a fully nonparametric kernel-based estimator of the quantiles and PDF of observable bids. Our estimator attains the optimal rate of Guerre et al. (2000), and is also asymptotically normal with an appropriate choice of the bandwidth.     

The effect of estimation method and sample size in multilevel structural equation modeling

Multilevel structural equation modeling (multilevel SEM) has become an established method to analyze multilevel multivariate data. The first useful estimation method was the pseudobalanced method. This method is approximate because it assumes that all groups have the same size, and ignores unbalance when it exists. In addition, full information maximum likelihood (ML) estimation is now available, which is often combined with robust chi‐squares and standard errors to accommodate unmodeled heterogeneity (MLR). In addition, diagonally weighted least squares (DWLS) methods have become available as estimation methods. This article compares the pseudobalanced estimation method, ML(R), and two DWLS methods by simulating a multilevel factor model with unbalanced data. The simulations included different sample sizes at the individual and group levels and different intraclass correlation (ICC). The within‐group part of the model posed no problems. In the between part of the model, the different ICC sizes had no effect. There is a clear interaction effect between number of groups and estimation method. ML reaches unbiasedness fastest, then the two DWLS methods, then MLR, and then the pseudobalanced method (which needs more than 200 groups). We conclude that both ML(R) and DWLS are genuine improvements on the pseudobalanced approximation. With small sample sizes, the robust methods are not recommended.     

Binary response panel data models with sample selection and self‐selection

We consider estimating binary response models on an unbalanced panel, where the outcome of the dependent variable may be missing due to nonrandom selection, or there is self‐selection into a treatment. In the present paper, we first consider estimation of sample selection models and treatment effects using a fully parametric approach, where the error distribution is assumed to be normal in both primary and selection equations. Arbitrary time dependence in errors is permitted. Estimation of both coefficients and partial effects, as well as tests for selection bias, are discussed. Furthermore, we consider a semiparametric estimator of binary response panel data models with sample selection that is robust to a variety of error distributions. The estimator employs a control function approach to account for endogenous selection and permits consistent estimation of scaled coefficients and relative effects.     

Labor market analysis through transformations and robust multivariate models

The work presents a robust approach to labor share analysis. The estimate of labor share presents various complexities related to the nature of the data sets to be analyzed. Typically, labor share is evaluated by using discriminant analysis and linear or generalized linear models, that do not take into account the presence of possible outliers. Moreover, the variables to be considered are often characterized by a high dimensional structure. The proposed approach has the objective of improving the estimation of the model using robust multivariate regression techniques and data transformation.