Spline Regression in the Presence of Categorical Predictors

We consider the problem of estimating a relationship nonparametrically using regression splines when there exist both continuous and categorical predictors. We combine the global properties of regression splines with the local properties of categorical kernel functions to handle the presence of categorical predictors rather than resorting to sample splitting as is typically done to accommodate their presence. The resulting estimator possesses substantially better finite‐sample performance than either its frequency‐based peer or cross‐validated local linear kernel regression or even additive regression splines (when additivity does not hold). Theoretical underpinnings are provided and Monte Carlo simulations are undertaken to assess finite‐sample behavior; and two illustrative applications are provided. An implementation in R is available; see the R package ‘crs’ for details. Copyright © 2014 John Wiley & Sons, Ltd.     

NONPARAMETRIC AND SEMIPARAMETRIC METHODS FOR ECONOMIC RESEARCH

Abstract. Developments in the vast and growing literatures on nonparametric and semiparametric statistical estimation are reviewed. The emphasis is on useful methodology rather than statistical properties for their own sake. Some empirical applications to economic data are described. The paper deals separately with nonparametric density estimation, nonparametric regression estimation, and estimation of semiparametric models.     

Spline-based quasi-likelihood estimation of mixed Poisson regression with single-index models

We consider spline-based quasi-likelihood estimation for mixed Poisson regression with single-index models. The unknown smooth function is approximated by B-splines, and a modified Fisher scoring algorithm is employed to compute the estimates. The spline estimate of the nonparametric component is shown to achieve the optimal rate of convergence, and the asymptotic normality of the regression parameter estimates is still valid even if the variance function is misspecified. The semiparametric efficiency of the model can be established if the variance function is correctly specified. The variance of the regression parameter estimates can be consistently estimated by a simple procedure based on the least-squares estimation. The proposed method is evaluated via an extensive Monte Carlo study, and the methodology is illustrated on an air pollution study.     

Estimating class‐specific parametric models under class uncertainty: local polynomial regression clustering in an hedonic analysis of wine markets

We introduce a method for estimating multiple class regression models when class membership is uncertain. The procedure—local polynomial regression clustering—first estimates a nonparametric model via local polynomial regression, and then identifies the underlying classes by aggregating sample observations into data clusters with similar estimates of the (local) functional relationships between dependent and independent variables. Finally, parametric functions specific to each class are estimated. The technique is applied to the estimation of a multiple‐class hedonic model for wine, resulting in the identification of four distinct wine classes based on differences in implicit prices of the attributes. Copyright © 2009 John Wiley & Sons, Ltd.     

N-kernel: A review

The package performs estimation and prediction in the context of time-series or cross-section nonparametric models. It is menu-driven and very easy to operate. The manual reads well. This version has some limitations, which can easily be corrected. Nevertheless it provides a useful pedagogical and research tool, even for people not familiar with nonparametric analysis.     

A test for multimodality of regression derivatives with application to nonparametric growth regressions

This paper presents a method to test for multimodality of an estimated kernel density of derivative estimates from a nonparametric regression. The test is included in a study of nonparametric growth regressions. The results show that in the estimation of unconditional β‐convergence the distribution of the partial effects is multimodal, with one mode in the negative region (primarily OECD economies) and possibly two modes in the positive region (primarily non‐OECD economies) of the estimates. The results for conditional β‐convergence show that the density is predominantly negative and there is mixed evidence that the distribution is unimodal. Copyright © 2009 John Wiley & Sons, Ltd.     

Three Sides of Smoothing: Categorical Data Smoothing, Nonparametric Regression, and Density Estimation

The past forty years have seen a great deal of research into the construction and properties of nonparametric estimates of smooth functions. This research has focused primarily on two sides of the smoothing problem: nonparametric regression and density estimation. Theoretical results for these two situations are similar, and multivariate density estimation was an early justification for the Nadaraya-Watson kernel regression estimator.
A third, less well-explored, strand of applications of smoothing is to the estimation of probabilities in categorical data. In this paper the position of categorical data smoothing as a bridge between nonparametric regression and density estimation is explored. Nonparametric regression provides a paradigm for the construction of effective categorical smoothing estimates, and use of an appropriate likelihood function yields cell probability estimates with many desirable properties. Such estimates can be used to construct regression estimates when one or more of the categorical variables are viewed as response variables. They also lead naturally to the construction of well-behaved density estimates using local or penalized likelihood estimation, which can then be used in a regression context. Several real data sets are used to illustrate these points.     

A Comparison of Methods for Poverty Estimation in Developing Countries

Small area estimation is a widely used indirect estimation technique for micro‐level geographic profiling. Three unit level small area estimation techniques—the ELL or World Bank method, empirical best prediction (EBP) and M‐quantile (MQ) — can estimate micro‐level Foster, Greer, & Thorbecke (FGT) indicators: poverty incidence, gap and severity using both unit level survey and census data. However, they use different assumptions. The effects of using model‐based unit level census data reconstructed from cross‐tabulations and having no cluster level contextual variables for models are discussed, as are effects of small area and cluster level heterogeneity. A simulation‐based comparison of ELL, EBP and MQ uses a model‐based reconstruction of 2000/2001 data from Bangladesh and compares bias and mean square error. A three‐level ELL method is applied for comparison with the standard two‐level ELL that lacks a small area level component. An important finding is that the larger number of small areas for which ELL has been able to produce sufficiently accurate estimates in comparison with EBP and MQ has been driven more by the type of census data available or utilised than by the model per se.     

Multiple imputation of incomplete zero‐inflated count data

Empirical count data are often zero‐inflated and overdispersed. Currently, there is no software package that allows adequate imputation of these data. We present multiple‐imputation routines for these kinds of count data based on a Bayesian regression approach or alternatively based on a bootstrap approach that work as add‐ons for the popular multiple imputation by chained equations (mice ) software in R (van Buuren and Groothuis‐Oudshoorn , Journal of Statistical Software, vol. 45, 2011, p. 1). We demonstrate in a Monte Carlo simulation that our procedures are superior to currently available count data procedures. It is emphasized that thorough modeling is essential to obtain plausible imputations and that model mis‐specifications can bias parameter estimates and standard errors quite noticeably. Finally, the strengths and limitations of our procedures are discussed, and fruitful avenues for future theory and software development are outlined.     

Robust Small Area Estimation under Spatial Non‐stationarity

The effective use of spatial information in a regression‐based approach to small area estimation is an important practical issue. One approach to account for geographic information is by extending the linear mixed model to allow for spatially correlated random area effects. An alternative is to include the spatial information by a non‐parametric mixed models. Another option is geographic weighted regression where the model coefficients vary spatially across the geography of interest. Although these approaches are useful for estimating small area means efficiently under strict parametric assumptions, they can be sensitive to outliers. In this paper, we propose robust extensions of the geographically weighted empirical best linear unbiased predictor. In particular, we introduce robust projective and predictive estimators under spatial non‐stationarity. Mean squared error estimation is performed by two analytic approaches that account for the spatial structure in the data. Model‐based simulations show that the methodology proposed often leads to more efficient estimators. Furthermore, the analytic mean squared error estimators introduced have appealing properties in terms of stability and bias. Finally, we demonstrate in the application that the new methodology is a good choice for producing estimates for average rent prices of apartments in urban planning areas in Berlin.     

Maximum likelihood estimation of monotone and concave production frontiers

In this paper we bring together the previously separate parametric and nonparametric approaches to production frontier estimation by developing composed error models for maximum likelihood estimation from nonparametrically specified classes of frontiers. This approach avoids the untestable restrictions of parametric functional forms and also provides a statistical foundation for nonparametric frontier estimation. We first examine the single output setting and then extend our formulation to the multiple output setting. The key step in developing the estimation problems is to identify operational constraint sets to ensure estimation from the desired class of frontiers. We also suggest algorithms for solving the resulting constrained likelihood function optimization problems.The refereeing process of this paper was handled through R. Robert Russell. Helpful comments from Bob Russell and two anonymous referees are gratefully acknowedged. We are, of course, solely responsible for any remaining errors or omissions.     

Intensity estimation for inhomogeneous Gibbs point process with covariates‐dependent chemical activity

Recent development of intensity estimation for inhomogeneous spatial point processes with covariates suggests that kerneling in the covariate space is a competitive intensity estimation method for inhomogeneous Poisson processes. It is not known whether this advantageous performance is still valid when the points interact. In the simplest common case, this happens, for example, when the objects presented as points have a spatial dimension. In this paper, kerneling in the covariate space is extended to Gibbs processes with covariates‐dependent chemical activity and inhibitive interactions, and the performance of the approach is studied through extensive simulation experiments. It is demonstrated that under mild assumptions on the dependence of the intensity on covariates, this approach can provide better results than the classical nonparametric method based on local smoothing in the spatial domain. In comparison with the parametric pseudo‐likelihood estimation, the nonparametric approach can be more accurate particularly when the dependence on covariates is weak or if there is uncertainty about the model or about the range of interactions. An important supplementary task is the dimension reduction of the covariate space. It is shown that the techniques based on the inverse regression, previously applied to Cox processes, are useful even when the interactions are present. © 2014 The Authors. Statistica Neerlandica © 2014 VVS.     

固定资产投资与经济增长的非参数回归预测分析

分析了我国全社会固定资产投资与经济增长之间的关系。由于两者的长期均衡关系受到政治经济因素影响存在两个突变点,而带有虚拟变量的线性回归模型不能理想地捕获两者之间的动态关系。通过运用Local Linear非参数回归理论,对新中国成立以来的国内生产总值与固定资产投资的关系,分别建立了分段与整段非参数回归预测模型以及参数线性回归预测模型,并加以比较,结果表明:整段非参数回归模型拟合较好,并且能够做出高精度的预测。     

Spline regression for hazard rate estimation when data are censored and measured with error

In this paper, we study an estimation problem where the variables of interest are subject to both right censoring and measurement error. In this context, we propose a nonparametric estimation strategy of the hazard rate, based on a regression contrast minimized in a finite‐dimensional functional space generated by splines bases. We prove a risk bound of the estimator in terms of integrated mean square error and discuss the rate of convergence when the dimension of the projection space is adequately chosen. Then we define a data‐driven criterion of model selection and prove that the resulting estimator performs an adequate compromise. The method is illustrated via simulation experiments that show that the strategy is successful.     

Local polynomial estimation of nonparametric simultaneous equations models

We define a new procedure for consistent estimation of nonparametric simultaneous equations models under the conditional mean independence restriction of Newey et al. [1999. Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565–603]. It is based upon local polynomial regression and marginal integration techniques. We establish the asymptotic distribution of our estimator under weak data dependence conditions. Simulation evidence suggests that our estimator may significantly outperform the estimators of Pinkse [2000. Nonparametric two-step regression estimation when regressors and errors are dependent. Canadian Journal of Statistics 28, 289–300] and Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578].     

Unraveling the Relationship Between Presidential Approval and the Economy: A Multidimensional Semiparametric Approach

Empirical studies analyzing the determinants of US presidential popularity have delivered quite inconclusive results concerning the role of economic variables by assuming linear relationships. We employ penalized spline smoothing in the context of semiparametric additive mixed models and allow for flexible functional forms and thus possible nonlinear effects for the economic determinants. By controlling for the well‐known politically motivated covariables, we find strong evidence for nonlinear and negative effects of unemployment, inflation and government consumption on presidential approval. Additionally, we present new results in favor of nonparametric trivariate interaction effects between the macroeconomic covariables. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonparametric estimation and testing of fixed effects panel data models

In this paper we consider the problem of estimating nonparametric panel data models with fixed effects. We introduce an iterative nonparametric kernel estimator. We also extend the estimation method to the case of a semiparametric partially linear fixed effects model. To determine whether a parametric, semiparametric or nonparametric model is appropriate, we propose test statistics to test between the three alternatives in practice. We further propose a test statistic for testing the null hypothesis of random effects against fixed effects in a nonparametric panel data regression model. Simulations are used to examine the finite sample performance of the proposed estimators and the test statistics.     

Estimation and inference under economic restrictions

Estimation of economic relationships often requires imposition of constraints such as positivity or monotonicity on each observation. Methods to impose such constraints, however, vary depending upon the estimation technique employed. We describe a general methodology to impose (observation-specific) constraints for the class of linear regression estimators using a method known as constraint weighted bootstrapping. While this method has received attention in the nonparametric regression literature, we show how it can be applied for both parametric and nonparametric estimators. A benefit of this method is that imposing numerous constraints simultaneously can be performed seamlessly. We apply this method to Norwegian dairy farm data to estimate both unconstrained and constrained parametric and nonparametric models.     

Nonparametric methods in sample surveys. Application to the estimation of cancer prevalence

We address the problem of the estimation of the population mean and the distribution function using nonparametric regression. These methods are being used in a wide range of settings and areas of research. In particular, they are a good alternative to other classical methods in the survey sampling context, since they work under the assumption that the underlying regression function is smooth. Some relevant nonparametric regression methods in survey sampling are presented. Data on breast cancer prevalence derived from 40 European countries are used to study the application of the nonparametric estimators to the estimation of cancer prevalence. Result derived from an empirical study show that nonparametric estimators have a good empirical performance in this study on cancer prevalence.     

Designs for smoothing spline ANOVA models

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2^d factorial designs, and the glp designs is given by numerical computation. Received May 2000