A Selective Review of Group Selection in High-Dimensional Models

Grouping structures arise naturally in many statistical modeling problems. Several methods have been proposed for variable selection that respect grouping structure in variables. Examples include the group LASSO and several concave group selection methods. In this article, we give a selective review of group selection concerning methodological developments, theoretical properties and computational algorithms. We pay particular attention to group selection methods involving concave penalties. We address both group selection and bi-level selection methods. We describe several applications of these methods in nonparametric additive models, semiparametric regression, seemingly unrelated regressions, genomic data analysis and genome wide association studies. We also highlight some issues that require further study.Comment: Published in at http://dx.doi.org/10.1214/12-STS392 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Appraisal Using Generalized Additive Models

Many of the results from real estate empirical studies depend upon using a correct functional form for their validity. Unfortunately, common parametric statistical tools cannot easily control for the possibility of misspecification. Recently, semiparametric estimators such as generalized additive models (GAMs) have arisen which can automatically control for additive (in price) or multiplicative (in ln(price)) nonlinear relations among the independent and dependent variables. As the paper shows, GAMs can empirically outperform naive parametric and polynomial models in ex-sample predictive behavior. Moreover, GAMs have well-developed statistical properties and can suggest useful transformations in parametric settings.

Overall Specialization and Income: Countries Diversity

� � � This paper gives evidence to a stylized fact often disregarded in international trade empir- ics: countries' diversification. In the last fifteen years, the growth of world trade coexisted with the tendency of countries to reduce the specialization of their export composition along the development path. On average, countries do not specialize, they diversify. Our semiparametric empirical analysis shows how this result is robust to the use of different statistical indexes used to measure trade specialization to the level of sectoral aggrega- tion and to the level of smoothing in the nonparametric term associated to income per capita. Using a General Additive Model (GAM) with country-specific fixed-effect, we show that, controlling for countries heterogeneity, sectoral export diversification increases with income. �Nonparametrics,International Trade,Specialization

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Semiparametric Bayesian inference in multiple equation models

This paper outlines an approach to Bayesian semiparametric regression in multiple equation models which can be used to carry out inference in seemingly unrelated regressions or simultaneous equations models with nonparametric components. The approach treats the points on each nonparametric regression line as unknown parameters and uses a prior on the degree of smoothness of each line to ensure valid posterior inference despite the fact that the number of parameters is greater than the number of observations. We develop an empirical Bayesian approach that allows us to estimate the prior smoothing hyperparameters from the data. An advantage of our semiparametric model is that it is written as a seemingly unrelated regressions model with independent normal-Wishart prior. Since this model is a common one, textbook results for posterior inference, model comparison, prediction and posterior computation are immediately available. We use this model in an application involving a two-equation structural model drawn from the labour and returns to schooling literatures

FOREIGN DIRECT INVESTMENT, HUMAN CAPITAL AND NONLINEARITIES IN ECONOMIC GROWTH

This paper examines the effect of FDI on the process of economic growth by allowing the impact to differ both across each country and also across each time period. We apply non-parametric techniques taking into account the previously documented nonlinear effects of initial income and human capital on economic growth. We use a wide range of countries, both developed and developing in order to be able to distinguish potential differential effects between the two groups. Our findings suggest that FDI inflows have a moderately nonlinear effect on growth and that the human capital nonlinear effect in the presence of FDI inflows is similar to the one found elsewhere in the relevant literature. Classification-JEL: O47cross country growth regressions, FDI, human capital, semi-parametric additive model

Growth and the pollution convergence hypothesis: A nonparametric approach

The pollution-convergence hypothesis is formalized in a neoclassical growth model with optimal emissions reduction: pollution growth rates are positively correlated with output growth (scale effect) but negatively correlated with emission levels (defensive effect). This dynamic law is empirically tested for two major and regulated air pollutants - nitrogen oxides (NOX) and sulfur oxides (SOX) - with a panel of 25 European countries spanning over years 1980-2005. Traditional parametric models are rejected by the data. However, more flexible regression techniques - semiparametric additive specifications and fully nonparametric regressions with discrete and continuous factors - confirm the existence of the predicted positive and defensive effects. By analyzing the spatial distributions of per capita emissions, we also show that cross-country pollution gaps have decreased over the period for both pollutants and within the Eastern as well as the Western European areas. A Markov modeling approach predicts further cross-country absolute convergence, in particular for SOX. The latter results hold in the presence of spatial non-convergence in per capita income levels within both regions.Air pollution, convergence, economic growth, mixed nonparametric regressions, distribution dynamics.

Growth and the pollution convergence hypothesis: a nonparametric approach

The pollution-convergence hypothesis is formalized in a neoclassical growth model with optimal emissions reduction: pollution growth rates are positively correlated with output growth (scale effect) but negatively correlated with emission levels (defensive effect). This dynamic law is empirically tested for two major and regulated air pollutants - nitrogen oxides (NOX) and sulfur oxides (SOX) - with a panel of 25 European countries spanning over years 1980-2005. Traditional parametric models are rejected by the data. However, more flexible regression techniques - semiparametric additive specifications and fully nonparametric regressions with discrete and continuous factors - confirm the existence of the predicted positive and defensive effects. By analyzing the spatial distributions of per capita emissions, we also show that cross-country pollution gaps have decreased over the period for both pollutants and within the Eastern as well as the Western European areas. A Markov modeling approach predicts further cross-country absolute convergence, in particular for SOX. The latter results hold in the presence of spatial non-convergence in per capita income levels within both regions.Air pollution, convergence, economic growth, mixed nonparametric regressions, distribution dynamics