PTDC/ECO/64693/2006: Econometric models for fractional response variables with applications to corporate financing choices 
Description This research project was initiated in November 2007 and ended at October 2010. The submitted project had as original aim only the regression analysis of fractional data, both in theoretical terms and applied to the study of capital structure choices. However, in the first stages of the investigation we became aware of the following situations: (i) Several models, methods and statistical tests that were being developed for fractional responses could also be used, with simple adaptations, in the regression analysis of binary data, since in the latter case, similarly to the former, the variable of interest (the probability of observing a binary outcome) can take values only on the unit interval. (ii) In the investigation of capital structure choices, in addition to studying the proportion of debt in the financing mix (debt plus equity) of firms (fractional variable), it is also relevant to analyze the probability of a firm issuing debt or not (binary variable). For these two reasons, although the focus of the research project was still the econometric modeling of fractional responses, some of the papers produced also considered the case of binary data. Among the various contributions of this research project to the econometric literature, the following are probably the most relevant ones: (i) Production of a comprehensive survey of the main alternative models (logit, probit, loglog, cauchit), estimation methods (nonlinear least squares, QML, ML) and specification tests (see ii below) suitable to deal with fractional response variables, whose finitesample properties were analyzed using Monte Carlo methods. (ii) Proposal of a full testing methodology to assess the validity of the assumptions required by each alternative estimator, which included the pioneer application in the fractional response framework of tests originally proposed with very distinct aims (e.g., tests for nonnested hypotheses). (iii) Development of a twopart fractional regression model to explain the financial leverage decisions of micro, small, medium and large Portuguese firms. This twopart model, which uses a binary choice model to explain the probability of a firm raising debt and a fractional regression model to explain the relative amount of debt issued, was found to be particularly suitable to describe the capital structure choices of firms of smaller size. (iv) Examination of the consequences of neglected heterogeneity over alternative estimators for binary and fractional regression models. In particular, we found that neglected heterogeneity: produces an attenuation bias in the estimation of regression coefficients; is innocuous for logit estimation of average sample partial effects but may generate biased estimation of those effects in the probit and loglog models; has much more deleterious effects on the estimation of population partial effects; is only for logit models that it does not substantially affect the prediction of outcomes; and is innocuous for the size of Wald tests for the significance of observed regressors but, in small samples, it substantially reduces their power. (v) Development of new fractional regression models to explain, in a second stage, DEA (data envelopment analysis) efficiency scores, which, by definition, have a fractional nature. These new models are based on two alternative generalizations of standard fractional regression models, both of which use an additional parameter to introduce some flexibility in the pattern of partial effects implied by conventional models. For example, the new models do not impose a priori the condition that sampling units with a given efficiency score (e.g., 0.5 in symmetric models) are the most sensitive to changes in the explanatory variables.

Research team Joaquim
J.S. Ramalho (Universidade de Évora)  principal investigator

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

Published Papers Ramalho, J.J.S., and J.V. Silva (2013), "Functional form issues in the regression analysis of corporate capital structure", Empirical Economics, , 44(2), 799831. Ramalho, E.A. and J.J.S. Ramalho (2012), "Alternative versions of the RESET test for binary response index models: a comparative study", Oxford Bulletin of Economics and Statistics, 74(1), 107130. Ramalho, E.A., J.J.S. Ramalho and J.M.R. Murteira (2011), "Alternative estimating and testing empirical strategies for fractional regression models", Journal of Economic Surveys, 25(1), 1968. Ramalho, E.A. and J.J.S. Ramalho (2010), "Is neglected heterogeneity really an issue in binary and fractional regression models? A simulation exercise for logit, probit and loglog models", Computational Statistics and Data Analysis, 54(4), 9871001. Ramalho, E.A., J.J.S. Ramalho and P.D. Henriques (2010), "Fractional regression models for second stage DEA efficiency analyses", Journal of Productivity Analysis, 34(3), 239255. Ramalho, E.A. (2010), "Covariate measurement error: bias correction under responsebased sampling", Studies in Nonlinear Dynamics and Econometrics, 14(4), article 2. Ramalho, J.J.S. and J.V. Silva (2009), "A twopart fractional regression model for the financial leverage decisions of micro, small, medium and large firms", Quantitative Finance, 9(5), 621636.

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