On the Econometric Modeling of Non-Linear Relationships: the Gumbel Regression Model

Abstract

Nonlinear relationships among random variables often come out in all fields of economics. The academic debate on how to deal with nonlinearities, from a statistical point of view, has been centered in developing new estimation methods or modifying the specification of the classic linear econometric models. Here, we propose to face this issue by deriving population regression models from conditional distributions with genuine nonlinear conditional means. Such a mathematical procedure guarantees not only a consistent derivation of the conditional mean that gives rise to a nonlinear econometric model, but also a proper analysis of the causal effects among the involved economic variables (i.e., partial effects). Finally, we exemplify the workings of this approach by specifying a nonlinear and heteroskedastic econometric model based on the Gumbel distribution. Resumen Relaciones de tipo no lineal entre variables aparecen de manera frecuente en todos los campos de la economía. El debate académico sobre como modelar dichas relaciones, desde un punto de vista estadístico, ha estado centrado en el desarrollo de nuevos métodos de estimación o en la especificación de los componentes del modelo clásico de regresión lineal. En este artículo proponemos enfrentar dicho problema derivando los modelos de regresión poblacional a partir de funciones de densidad condicionales con medias no-lineales genuinas. Este procedimiento matemático garantiza no sólo una consistente derivación de la media condicional que da origen a un modelo econométrico no lineal, sino también un análisis más apropiado de los efectos causales entre las variables económicas involucradas (efectos marginales). Finalmente, se presenta un ejemplo del funcionamiento de dicho procedimiento mediante la especificación de un modelo econométrico no lineal y heteroscedastico que surge de una distribución Gumbel. JEL Classification:C1; C51. 1. Introduction The existence of nonlinear relationships among economic variables is always controversial. This issue becomes more complicated when applied researchers find nonlinearities that the economic theory does not predict or estimate linear models that do not properly fit the data. Here, we argue that, from a statistical point of view, we can avoid controversies by using conditional probability densities f(y|x) as the basis to derive the nonlinear conditional means E(y | x) = m(x, θ) that give rise to reliable econometric models y = m(x, θ)+e, rather than assuming the standard functional forms of the conditional mean suggested in the econometrics textbooks. That is, we propose starting the specification of the econometric model by assuming a proper conditional density, for the data on hand, and derive the associated regression and skedastic functions from it by taking the expected value of the explained variable y given the set of explanatory variables x. In other words, we show how to use a statistical procedure to derive proper econometric models that capture genuine nonlinear relationships. To do so, we assume suitable conditional probability distributions that give rise to nonlinear regression functions (Spanos, 1986). We exemplify this approach by deriving a population regression model that can be useful to analyze a nonlinear relationship between economic variables. Specifically, we use a Gumbel conditional distribution as the basis to derive the nonlinear regression curves that might describe such type of phenomena. The Gumbel regression model allows us to briefly illustrate two interesting facts. First, we show how a nonlinear relationship might be well represented by an exponential distribution and its associated regression model (Gumbel 1960); which has a nonlinear conditional mean and a heteroskedastic conditional variance. Second, we show that a non linear model like the Gumbel regression exhibits changing partial effects of the explanatory variables over the entire distribution of the explained variable, which is not the case in a normal-linear model. These facts might be useful to elucidate controversial economic arguments when the empirical data exhibit nonlinearities. This paper is structured as follows. The second section briefly discusses the general statistical approach to derive linear or non-linear econometric models in a stochastic setting. In the third section, we exemplify the statistical approach by discussing the specification, estimation, and validation of the Gumbel regression model. In the last section, we make some remarks on the implications of the employed approach. 2. Deriving Nonlinear Regression Models In the context of a modern approach to econometrics any linear or nonlinear model can be specified by making assumptions on two components: 1) the population regression model, and 2) the sampling model (Wooldrige, 2010). The first assumption refers to the functional form of the conditional mean that describes the stochastic relationship between y and x. The second assumption refers to the probabilistic behavior of the sample. Here, we only deal with the derivation of the population regression model that gives rise to the nonlinear relationships among a set of economic variables so, for the sake of simplicity, we assume that we have an independent and identically distributed sample (iid) in the rest of the paper.