PTDC/EGEECO/119148/2010: Theoretical Developments in the Regression Analysis of Fractional Data and its Applications to Finance 
Description In several economic settings, the variable of interest in regression models is often a proportion, or a vector of proportions, corresponding to a set of shares for a given number of exhaustive, mutually exclusive categories. Examples include pension plan participation rates, fraction of land allocated to agriculture, percentage of weekly time devoted to each of a given set of human activities, fractions of income spent on various classes of goods, asset portfolio shares, and proportions of different types of debt within firms' financing mix. While in the first two cases there are only two categories (usually a characteristic and its opposite, or absence) and a single proportion is modelled, the remaining examples illustrate the more general situation where the joint behaviour of a multivariate fractional variable is of interest. The regression analysis of fractional data, inherently bounded within the unit simplex, raises a number of interesting research issues that challenge conventional approaches of estimation and inference. For the case of a single proportion, the main issues are discussed in the seminal paper by Papke and Wooldridge (1996), who propose robust quasimaximum likelihood estimation on the basis of a Bernoullibased likelihood and a logit conditional mean function. In a recent paper (Ramalho, Ramalho and Murteira, 2011), some of the authors of this research proposal survey the main alternative regression models and estimation methods that are available for dealing with (univariate) fractional response variables and propose a unified testing methodology to assess the validity of the assumptions required by each model and method. In this project, we continue the research initiated in 2007 with the FCTfunded project PTDC/ECO/64693/2006, which focused on the regression analysis of univariate fractional data using parametric methods. The application of these methods in that context is again considered but the main aim of the current research proposal is the analysis of multivariate fractional data using both parametric and nonparametric regression techniques. Moreover, while the previous project considered a single empirical application (the determinants of firms' capital structure decisions) to illustrate the usefulness of the estimation and inference tools developed, this research project considers a much wider range of applications in finance, which is an area where, in recent years, most team members have also accumulated some expertise (see, for instance, Ramalho and Silva, 2009, Santos Silva and Murteira, 2009, and Bastos, 2010). The parametric models proposed for modelling multivariate fractional responses differ on a number of respects, such as: (i) the adoption, or not, of full joint distributional assumptions for shares; (ii) the possibility, or not, of dealing with boundary observations; and (iii) in cases where shares result from ratios of known integers, the use, or not, of this additional information. In any case, all models have in common the use of functional forms for the conditional expectation of the response variables which enforce the conceptual requirement that, as for the observed shares, its elements also belong to the unit simplex. We also investigate at some length the specification analysis of these models, which is a sensitive issue that has not merited much attention in the literature on multivariate fractional regression. In some applications, it may be useful to dispose of other, less conventional regression tools to model fractional response variables. In this research project we consider two alternative techniques to parametric methods: decision tree models and artificial neural networks. Both these nonparametric methods are first adapted to the fractional context and then shown to be competitive techniques to standard methods in modelling and forecasting fractional responses. Decision tree models are also used jointly with parametric methods to model fractional response variables by groups of homogenous firms, where the definition of the groups is determined by the nonparametric procedure. A research area where many variables of interest have a fractional nature is that of finance. In this project, we intend to revisit several important issues in finance, ranging from the relationship between corporate board structure and firm performance to the classical capital structure and cashholding decisions. We will also explore other relevant topics, such as the forecasting of lossgivendefault in bank loans and the determinants of institutional investors' equity ownership. Such diversity of examples enables us to apply the methods developed in a rich variety of settings: single and multivariate shares; fractional responses obtained as ratios of known and/or unknown integers; observation, or not, of boundary values with nontrivial probability; joint or separate regressions by groups of firms; and cases where the main interest is either modelling relationships or forecasting.

Research team Joaquim
J.S. Ramalho (Universidade de Évora)  principal investigator
Research assistants Sofia
Oliveira (10/2012  09/2013)

References and sample code See "Fractional regression models"  Home Page.

Published papers Bastos, J.A. and J.J.S. Ramalho, "Nonparametric models of financial leverage decisions", Bulletin of Economic Research, forthcoming. Ramalho, E.A. and J.J.S. Ramalho, "Momentbased estimation of nonlinear regression models with boundary outcomes and endogeneity, with applications to nonnegative and fractional responses", Econometric Reviews, forthcoming. Ramalho, E.A., J.J.S. Ramalho and L.M.S. Coelho, "Exponential regression of fractionalresponse fixedeffects models with an application to firm capital structure", Journal of Econometric Methods, forthcoming. Silva, V.G., E.A. Ramalho, C. Vieira, "The use of cheques in the European Union: a crosscountry analysis", Open Economies Review, forthcoming. Murteira, J.M.R. (2016), "Goodness of link tests for multivariate regression models, Communications in Statistics  Theory and Methods, 45(24), 73677375. Murteira, J.M.R. and J.J.S. Ramalho (2016), "Regression analysis of multivariate fractional data", Econometric Reviews, 35(4), 515552. Bastos, J. (2014), "Ensemble predictions of recovery rates", Journal of Financial Services Research, 46(2), 177193. Ramalho, E.A., J.J.S. Ramalho and J.M.R. Murteira, (2014), "A generalized goodnessoffunctional form test for binary and fractional regression models", Manchester School, 82(4), 488507.

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