Econometric Models for Fractional Response Variables with Applications to Corporate Financing Choices

In many economic settings, the variables of interest are measured as proportions and fractions, i.e. they are defined and observed only in the interval [0,1]. Examples include pension plan participation rates, firm market share, fraction of total weekly hours spent working, proportion of debt in the financing mix of firms, fraction of land area allocated to agriculture, and proportion of exports in total sales. The bounded nature of such variables and the probability of observing values at the boundaries raise some interesting estimation and inference issues. In particular, the standard practice of using linear models to examine how a set of potential explanatory variables influence a given proportional or fractional response variable is not appropriate since it does not guarantee that the predicted values of the dependent variable are restricted to the [0,1] interval. Nevertheless, only recently researchers have begun to take seriously the functional form issues raised by fractional data, proposing models for the conditional mean of the fractional response variable that keep the predicted values in the unit interval. However, the methods proposed so far only deal with basic aspects of regression models and are only applicable in simple settings.

In this research project we make several important contributions for the econometric literature on fractional regression models, expanding the current methodology in many directions. First, we perform a Monte Carlo based comparison of several alternative regression models that take into account the fractional nature of the data and propose a testing and model selection methodology for selecting the fractional regression model most suitable for a particular empirical application. Up to now, most papers in this area have assumed a logistic specification, although the robustness of this formulation under different data generating processes has never been investigated. Second, based on an approach similar to that used for count data models, we develop a generalized method of moments estimator that accounts appropriately for the likely presence of unobserved heterogeneity in the data. To date, the fractional data literature has completely ignored this issue but most analyses involving micro data must somehow account for the likely presence of unobserved heterogeneity. Third, we show how to deal with multinomial data structures. Although all papers in this area have considered only univariate outcomes, many economic data structures are multivariate by nature (e.g. the proportion of income spent in different classes of goods). Fourth, we consider some sampling issues that may affect fractional data. Specifically, we construct a two-part fractional model to deal with the problem of excess of zeros that arises in many research settings and propose generalized method of moments estimators for endogenously stratified samples and some patterns of missing data. Finally, attention is devoted to models for which the dependent variable is obtained as an integers ratio. When both terms of this ratio are observed, binomial-based specifications may prove advantageous over other approaches, e.g. Bernoulli-based quasi-maximum likelihood. Binomial models, however, are used under specific assumptions, the breakdown of which can hinder possible efficiency advantages of these models. On the other hand, such failures offer a number of research opportunities, some of which are to be undertaken in the proposed work.

To illustrate the usefulness of the techniques developed in empirical work, we apply some of them to the analysis of the capital structure decisions made by Portuguese firms. Typically, empirical studies of capital structures examine the effects of a given set of potential explanatory variables on the proportion of debt used by firms to finance their activity. To the best of our knowledge, not a single study in this area has used models suitable to deal with the bounded nature of the variable of interest. Given the large percentage of firms that do not use debt at all, we develop and apply a two-part fractional regression model that recognizes the possibility that the mechanisms that explain the zero outcomes may be different from the mechanisms that determine the other fractional values. That is, we construct separate models to explain the decisions: (i) to issue or not to issue debt; and (ii) (for those firms that do decide to use debt) on how much debt to issue. The first decision is modelled as a binary choice model and the second as a fractional regression model that explains the amount of debt issued conditional upon the decision to issue debt.