Nonparametric Additive Models Research Articles

Wavelet-based robust estimation and variable selection in nonparametric additive models

This article studies M-type estimators for fitting robust additive models in the presence of anomalous data. The components in the additive model are allowed to have different degrees of smoothness. We introduce a new class of wavelet-based robust M-type estimators for performing simultaneous additive component estimation and variable selection in such inhomogeneous additive models. Each additive component is approximated by a truncated series expansion of wavelet bases, making it feasible to apply the method to nonequispaced data and sample sizes that are not necessarily a power of 2. Sparsity of the additive components together with sparsity of the wavelet coefficients within each component (group), results into a bi-level group variable selection problem. In this framework, we discuss robust estimation and variable selection. A two-stage computational algorithm, consisting of a fast accelerated proximal gradient algorithm of coordinate descend type, and thresholding, is proposed. When using nonconvex redescending loss functions, and appropriate nonconvex penalty functions at the group level, we establish optimal convergence rates of the estimates. We prove variable selection consistency under a weak compatibility condition for sparse additive models. The theoretical results are complemented with some simulations and real data analysis, as well as a comparison to other existing methods.

A non-parametric analysis of the driving factors of China's carbon prices

Carbon prices have gradually become the principal policy for controlling carbon dioxide emissions, which has attracted the attention of countries worldwide. The price factor is significantly affected by the market economy and often exhibits non-linear characteristics. This article aims to investigate the possible non-linear relationship between carbon prices and their influencing factors using a data-driven non-parametric additive regression model. Estimated results show that coal prices have an inverted U-shaped non-linear impact on carbon prices. It means that coal prices pushed up carbon prices; in the long run, the changes in coal prices will help reduce carbon prices. The real economy also exerts an inverted U-shaped effect on carbon prices, which is the main reason for the rapid expansion of energy-intensive industries in the short run. The impact of natural gas prices presents a gentle inverted U-shaped pattern, owing to the periodic changes in natural gas consumption. On the contrary, renewable energy prices have a “lower first, then increase” U-shaped impact on carbon prices. Therefore, the government should formulate differentiated policies at different time stages.

The partly parametric and partly nonparametric additive risk model

Aalen’s linear hazard rate regression model is a useful and increasingly popular alternative to Cox’ multiplicative hazard rate model. It postulates that an individual has hazard rate function h(s)=z_1alpha _1(s)+cdots +z_ralpha _r(s) in terms of his covariate values z_1,ldots ,z_r. These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions alpha _j(s) are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model’s parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

Targeting millers to improve rice marketing in Uganda

PurposeImproving milling quality is expected to improve the quality of domestic rice and hence the competitiveness of Uganda's rice industry. Therefore, this study aims to assess the determinants of four aspects of milling, namely, choice of milling technology, millers' perceptions of the importance of paddy quality attributes, milling return and milling capacity.Design/methodology/approachMultinomial logit, semi-nonparametric extended ordered probit, linear regression and additive nonparametric models are applied to cross-sectional data obtained from a sample of 196 rice millers.FindingsPhysical, economic, institutional, technological and sociodemographic factors are found to be important determinants of the four aspects of milling. Physical factors include the distance of the mill from major town and availability of storage space at the milling premises, while economic factors include milling charge and backward integration of miller into paddy production. Contracting and use of a single-pass mill are important institutional and technological factors, respectively, and miller's household size, age, gender and education are the key sociodemographic variables.Originality/valueThe study's originality lies in its scope, especially in terms of its breadth. Without compromising the needed analytical rigor, it focuses on four aspects of milling that are critical to improving the marketing of Uganda's rice. In doing so, it provides a holistic understanding of this segment of the value chain and offers specific recommendations for improving the marketing of Uganda's rice.

Estimation of varying coefficient models with measurement error

We propose a semiparametric estimator for varying coefficient models when the regressors in the nonparametric components are measured with error. Varying coefficient models are an extension of other popular semiparametric models, including partially linear and nonparametric additive models, and deliver an attractive solution to the curse-of-dimensionality. We use deconvolution kernel estimation in a two-step procedure and show that the estimator is consistent and asymptotically normally distributed. We do not assume that we know the distribution of the measurement error a priori. Instead, we suppose we have access to a repeated measurement of the noisy regressor and present results using the approach of Delaigle, Hall and Meister (2008) and, for cases when the measurement error may be asymmetric, the approach of Li and Vuong (1998) based on Kotlarski’s (1967) identity. We show that the convergence rate of the estimator is significantly reduced when the distribution of the measurement error is assumed unknown and possibly asymmetric. We study the small sample behaviour of our estimator in a simulation study and apply it to a real dataset. In particular, we consider the role of cognitive ability in augmenting the effect of risk preferences on earnings.

Statistical inference in sparse high-dimensional additive models

In this paper, we discuss the estimation of a nonparametric component f1 of a nonparametric additive model Y=f1(X1)+⋯+fq(Xq)+ϵ. We allow the number q of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating f1 in the oracle model Z=f1(X1)+ϵ, for which the additive components f2,…,fq are known. We construct a two-step presmoothing-and-resmoothing estimator of f1 and state finite-sample bounds for the difference between our estimator and a corresponding smoothing estimator fˆ1(oracle) in the oracle model. In an asymptotic setting, these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimator; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of f2,…,fq has no effect on estimation accuracy. Our first step is to estimate f1 with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. In the second step, we construct pseudo responses Yˆ by evaluating this undersmoothed estimator of f1 at the design points and then apply the smoothing method of the oracle estimator fˆ1(oracle) to the nonparametric regression problem with “responses” Yˆ and covariates X1. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.

Inference for partially linear additive higher-order spatial autoregressive model with spatial autoregressive error and unknown heteroskedasticity

This article extends spatial autoregressive model with spatial autoregressive disturbances (SARAR(1,1)) which is the most popular spatial econometric model to the case of an arbitrary finite number of nonparametric additive terms and spatial autoregressive models with spatial autoregressive disturbances of arbitrary finite order (SARAR(R,S)). We propose a sieve two-stage least squares (S2SLS) regression and generalized method of moments (GMM) procedure of the high-order spatial autoregressive parameters of the disturbance process. Under some sufficient conditions, we show that the proposed estimator for the finite dimensional parameter is consistent and asymptotically normally distributed. We show that each proposed estimator for the additive terms is consistent and also asymptotically normally distributed at a rate slower than Consistent estimators for the asymptotic variances of the proposed estimators are provided. In addition, using asymptotic properties to make statistical inference for the parametric and additive is also considered. Monte Carlo evidence suggests that the estimation procedure performs reasonably well in small samples and the proposed approach has some practical value. The proposed method is applied to analyzing factors which affect haze pollution.

Impact of renewable energy investment on carbon emissions in China - An empirical study using a nonparametric additive regression model

This study analyzed the comprehensive impact of renewable energy investment on carbon emissions in China. To achieve this, a nonparametric additive regression model was built. Using the STIRPAT model, we considered six influencing factors: economic growth, industrialization level, urbanization level, population aging, trade openness, and renewable energy investment. This enabled the exploration of the existence, direction, and intensity of the impact of renewable energy investment on carbon emissions. The results of the linear component of the model showed that renewable energy investment can slightly reduce carbon emissions. The results of the nonlinear component of the model showed that the impacts of renewable energy investment on carbon emissions were inconsistent at different stages of the investment. In the early stage, the renewable energy investment can increase carbon emissions. In the middle stage, the renewable energy investment begins to play a role in reducing emissions. In the later stage, renewable energy investment may be associated with increased carbon emissions again. The relationship between carbon emissions and the other five influencing factors can be represented by an inverted U-shaped curve, a U-shaped curve, or a slow rising curve. The results above provide useful references to adjust renewable energy investment and reduce carbon emissions.

How to achieve a low-carbon transition in the heavy industry? A nonlinear perspective

The heavy industry is the main source of CO2 emissions. Examining the influencing factors of the heavy industry's CO2 emissions is essential for this industry to achieve a low-carbon transition. Existing research literature often assumes that the relationship between the influencing factors and CO2 emissions is linear. In fact, most economic phenomena are characterized by fluctuations, due to the influence of business cycles. This often leads to more likely nonlinear relationships between economic variables. This paper uses nonparametric additive regression model to investigate CO2 emissions from China's heavy industry. The empirical findings show that economic growth, urbanization, and energy efficiency exert a “first promotion, then restriction” inverted “U-shaped” nonlinear effect on CO2 emissions. To complicate matters further, energy consumption structure and export dependence have an “N-shaped” nonlinear effect on CO2 emissions. Therefore, in the early stage, the government should actively optimize the industrial structure, strengthen the management of the real estate industry, and expand R&D investment. In the long run, the government should subsidize the heavy industry to expand clean energy consumption, and provide tax incentives to support the export of high-tech industrial products.

Additive regression for predictors of various natures and possibly incomplete Hilbertian responses

In this paper we consider a fully nonparametric additive regression model for responses and predictors of various natures. This includes the case of Hilbertian and incomplete (like censored or missing) responses, and continuous, nominal discrete and ordinal discrete predictors. We propose a backfitting technique that estimates this additive model, and establish the existence of the estimator and the convergence of the associated backfitting algorithm under minimal conditions. We also develop a general asymptotic theory for the estimator such as the rates of convergence and asymptotic distribution. We verify the practical performance of the proposed estimator in a simulation study. We also apply the method to various real data sets, including those for a density-valued response regressed on a mixture of continuous and nominal discrete predictors, for a compositional response regressed on a mixture of continuous and ordinal discrete predictors, and for a censored scalar response regressed on a mixture of continuous and nominal discrete predictors.

How to efficiently promote distributed energy resources in China: Using a nonparametric econometric method

Most of the existing related studies use linear methods to investigate distributed energy resources, ignoring the objectively existing nonlinear relationships. This article makes improvement and uses a nonparametric additive regression model to explore China’s distributed energy resources. Estimated results show that technological progress exerts an inverted “U-shaped” nonlinear effect on distributed energy resources, due to the phased changes in R&D expenditures. Energy consumption structure also has an inverted U-shaped impact, which main reason is the changes in natural gas and renewable energy consumption. On the contrary, international oil prices have a positive U-shaped impact on distributed energy resources, because international oil prices are gradually increasing over time. Foreign oil dependence has an N-shaped nonlinear impact, which is mainly due to the changes in oil imports at different stages. The relationship between urbanization and distributed energy resources also presents an N-shaped pattern. This is mainly due to the periodic changes in the growth rate of the number of residents using natural gas. The research results can provide empirical support for the government to formulate targeted distributed energy policies at different stages.

Marshallian vs Jacobs effects: Which is stronger? Evidence for Russia unemployment dynamics

This paper studies the influence of diversification and specialization on one of the main indicators of the Russian labour market: unemployment growth. The purpose of the work is to find out which effects dominate in the Russian regions, Marshallian or Jacobs, and whether this predominance is stable for different time periods. We tested empirically the following hypotheses: 1) the dependence of the unemployment growth on the concentration or diversification is nonlinear due to possible overlapping effects of urbanization and localization; 2) the influence of the concentration or diversification on the unemployment growth depends on the time period. To test these hypotheses, we use nonparametric additive models with spatial effects. Both hypotheses found empirical confirmation, with each effect prevailing in different time periods: Marshallian effects were prevalent in 2008-2010, and 2013-2016, while Jacobs effects were prevalent in 2010-2013.

Consistent Group Selection with Bayesian High Dimensional Modeling

In many applications with high dimensional covariates, the covariates are naturally structured into different groups which can be used to perform efficient statistical inference. We propose a Bayesian hierarchical model with a spike and slab prior specification to perform group selection in high dimensional linear regression models. While several penalization methods and more recently, some Bayesian approaches are proposed for group selection, theoretical properties of Bayesian approaches have not been studied extensively. In this paper, we provide novel theoretical results for group selection consistency under spike and slab priors which demonstrate that the proposed Bayesian approach has advantages compared to penalization approaches. Our theoretical results accommodate flexible conditions on the design matrix and can be applied to commonly used statistical models such as nonparametric additive models for which very limited theoretical results are available for the Bayesian methods. A shotgun stochastic search algorithm is adopted for the implementation of our proposed approach. We illustrate through simulation studies that the proposed method has better performance for group selection compared to a variety of existing methods.

Estimation of nonparametric additive models with high order spatial autoregressive errors

AbstractIn this article, we propose nonparametric generalized method of moments estimation for nonparametric additive models with high order spatial autoregressive dependence. The estimation procedure is derived in three steps by combining a spline‐backfitting method with generalized moment conditions that relieve correlations within the dependent variables. Consistency and asymptotic normality are demonstrated under mild conditions. Specifically, compared with estimators of nonparametric functions that ignore cross‐sectional dependence in errors, the resultant estimators that consider the error term are asymptotically more efficient and achieve the well‐known oracle properties. Simulation studies investigating the finite sample performance of the estimation procedure confirm the validity of our asymptotic theory. An application to the Boston housing data serves as a practical illustration.

Estimating Yield Curves by Kernel Smoothing Methods

We introduce a new method for the estimation of discount functions, yield curves and forward curves from government issued coupon bonds. Our approachis nonparametric and does not assume a particular functional form for thediscount function although we do show how to impose various restrictions inthe estimation. Our method is based on kernel smoothing and is defined asthe minimum of some localized population moment condition. The solution tothe sample problem is not explicit and our estimation procedure isiterative, rather like the backfitting method of estimating additivenonparametric models. We establish the asymptotic normality of our methodsusing the asymptotic representation of our estimator as an infinite serieswith declining coefficients. The rate of convergence is standard for onedimensional nonparametric regression.

How to achieve green growth in China’s agricultural sector

Green development is the direction of China’s agricultural sector. Investigating the main influencing factors of the agricultural sector’s CO2 emissions will help plan agricultural development and achieve low-carbon growth. This paper selects 2001–2016 panel data of China’s 30 provinces, and uses the nonparametric additive regression models to investigate the agricultural sector’s CO2 emissions. The empirical results show that economic growth, financial capacity, and energy intensity all have a “pull first, then restrict” inverted U-shaped nonlinear effect on CO2 emissions. However, the influencing mechanisms of these three influencing factors are different and need to be paid attention by relevant government departments. More complicated, the impact of urbanization on CO2 emission shows a gentle “M-shaped” pattern. In addition, an inverse “N-shaped” nonlinear link exists between trade openness and CO2 emissions. Therefore, government departments should formulate differentiated emission reduction policies at different stages.

How does fossil energy abundance affect China's economic growth and CO2 emissions?

The paper uses the nonparametric additive regression model with data- driven characteristics to investigate the impact of fossil energy abundance on China's economic growth and carbon dioxide (CO2) emissions. The results show that the effect of fossil energy abundance on economic growth shows an inverted “U-shaped” pattern in the eastern region, due to the changes in coal mining, oil processing, and coking investments. On the contrary, fossil energy abundance exerts a positive “U-shaped” nonlinear effect on economic growth in the central region. This indicates that in the early stages fossil energy abundance did not play a role in promoting economic growth, and its driving effect was only prominent in the later stages. In addition, fossil energy abundance generates a positive “U-shaped” impact on CO2 emissions in the eastern and central regions, because of the changes in coal and oil consumption at different stages. However, fossil energy abundance has an inverted “U-shaped” nonlinear effect on CO2 emissions in the western region, on account of the phase difference in the production and consumption of natural gas and oil.

Albemarle reports 2Q 2019 growth and raises full year EPS guidance

The bulk commodities are basic goods, and their price fluctuations are directly related to the stability of a country's macro economy. Investigating the main driving forces of the commodity price fluctuations is the key to stabilizing the macro economy. Ignoring the objective facts that there are a large number of nonlinear relationships between economic variables, most of the existing scholars use the linear models to commodity price fluctuations. To compensate for the shortcomings of existing research, this paper uses the nonparametric additive regression model with data-driven features to investigate the main driving forces of China’s commodity price fluctuations. The results show that exchange rate has an “early promotion, late inhibition” inverse “U-shaped” nonlinear effect on commodity price fluctuations because the renminbi appreciates in the early stages, but depreciates in the later stages. The nonlinear impact of the real economy also appears an inverted “U-shaped” pattern, due to the changes in growth mode of the real economy. Similarly, international oil price produces an inverted “U-shaped” nonlinear effect, owing to the rise in international oil prices in the early stages and the decline in the later stages. However, the money supply has an “early inhibition, late promotion” positive “U-shaped” nonlinear effect on commodity prices. Therefore, government departments should formulate targeted policies over time to effectively stabilize China’s commodity prices.

Uncertainty Quantification for High-Dimensional Sparse Nonparametric Additive Models

Statistical inference in high-dimensional settings has recently attracted enormous attention within the literature. However, most published work focuses on the parametric linear regression problem. This article considers an important extension of this problem: statistical inference for high-dimensional sparse nonparametric additive models. To be more precise, this article develops a methodology for constructing a probability density function on the set of all candidate models. This methodology can also be applied to construct confidence intervals for various quantities of interest (such as noise variance) and confidence bands for the additive functions. This methodology is derived using a generalized fiducial inference framework. It is shown that results produced by the proposed methodology enjoy correct asymptotic frequentist properties. Empirical results obtained from numerical experimentation verify this theoretical claim. Lastly, the methodology is applied to a gene expression dataset and discovers new findings for which most existing methods based on parametric linear modeling failed to observe.

How to effectively stabilize China's commodity price fluctuations?

The bulk commodities are basic goods, and their price fluctuations are directly related to the stability of a country's macro economy. Investigating the main driving forces of the commodity price fluctuations is the key to stabilizing the macro economy. Ignoring the objective facts that there are a large number of nonlinear relationships between economic variables, most of the existing scholars use the linear models to commodity price fluctuations. To compensate for the shortcomings of existing research, this paper uses the nonparametric additive regression model with data-driven features to investigate the main driving forces of China’s commodity price fluctuations. The results show that exchange rate has an “early promotion, late inhibition” inverse “U-shaped” nonlinear effect on commodity price fluctuations because the renminbi appreciates in the early stages, but depreciates in the later stages. The nonlinear impact of the real economy also appears an inverted “U-shaped” pattern, due to the changes in growth mode of the real economy. Similarly, international oil price produces an inverted “U-shaped” nonlinear effect, owing to the rise in international oil prices in the early stages and the decline in the later stages. However, the money supply has an “early inhibition, late promotion” positive “U-shaped” nonlinear effect on commodity prices. Therefore, government departments should formulate targeted policies over time to effectively stabilize China’s commodity prices.