Joaquim J. S. Ramalho

ALTERNATIVE ESTIMATING AND TESTING EMPIRICAL STRATEGIES FOR FRACTIONAL REGRESSION MODELS

In many economic settings, the variable of interest is often a fraction or a proportion, being defined only on the unit interval. The bounded nature of such variables and, in some cases, the possibility of nontrivial probability mass accumulating at one or both boundaries raise some interesting estimation and inference issues. In this paper we: (i) provide a comprehensive survey of the main alternative models and estimation methods suitable to deal with fractional response variables; (ii) propose a full testing methodology to assess the validity of the assumptions required by each alternative estimator; and (iii) examine the finite sample properties of most of the estimators and tests discussed through an extensive Monte Carlo study. An application concerning corporate capital structure choices is also provided.

A two-part fractional regression model for the financial leverage decisions of micro, small, medium and large firms

In this paper we examine the following two hypotheses, which traditional theories of capital structure are relatively silent about: (i) the determinants of financial leverage decisions are different for micro, small, medium and large firms; and (ii) the factors that determine whether or not a firm issues debt are different from those that determine how much debt it issues. Using 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, we find strong support for both hypotheses. Confirming recent empirical evidence, we find also that, although larger firms are more likely to use debt, conditional on their having some debt, firm size is negatively related to the proportion of debt used by firms.In this paper we examine the following two hypotheses, which traditional theories of capital structure are relatively silent about: (i) the determinants of financial leverage decisions are different for micro, small, medium and large firms; and (ii) the factors that determine whether or not a firm issues debt are different from those that determine how much debt it issues. Using 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, we find strong support for both hypotheses. Confirming recent empirical evidence, we find also that, although larger firms are more likely to use debt, conditional on their having some debt, firm size is negatively related to the proportion of debt used by firms.

Regression Analysis of Multivariate Fractional Data

The present article discusses alternative regression models and estimation methods for dealing with multivariate fractional response variables. Both conditional mean models, estimable by quasi-maximum likelihood, and fully parametric models (Dirichlet and Dirichlet-multinomial), estimable by maximum likelihood, are considered. A new parameterization is proposed for the parametric models, which accommodates the most common specifications for the conditional mean (e.g., multinomial logit, nested logit, random parameters logit, dogit). The text also discusses at some length the specification analysis of fractional regression models, proposing several tests that can be performed through artificial regressions. Finally, an extensive Monte Carlo study evaluates the finite sample properties of most of the estimators and tests considered.

Generalized empirical likelihood non-nested tests

This paper examines non-nested tests for competing moment condition models using a semi-parametric generalized empirical likelihood (GEL) framework. The resultant GEL estimators are first order asymptotically equivalent to those based on generalized method of moments (GMM). Cox-type, moment encompassing and parametric encompassing non-nested tests for competing moment condition models are proposed. Simulation experiments are conducted to examine the efficacy of the proposed GEL statistics in terms of their size and power properties and to compare their properties with those of corresponding non-nested test statistics based on GMM estimation.

Asymptotic bias for GMM and GEL estimators with estimated nuisance parameters

This papers studies and compares the asymptotic bias of GMM and generalized empirical likelihood (GEL) estimators in the presence of estimated nuisance parameters. We consider cases in which the nuisance parameter is estimated from independent and identical samples. A simulation experiment is conducted for covariance structure models. Empirical likelihood offers much reduced mean and median bias, root mean squared error and mean absolute error, as compared with two-step GMM and other GEL methods. Both analytical and bootstrap bias-adjusted two-step GMM estimators are compared. Analytical bias-adjustment appears to be a serious competitor to bootstrap methods in terms of finite sample bias, root mean squared error and mean absolute error. Finite sample variance seems to be little affected.

Alternative Versions of the RESET Test for Binary Response Index Models: A Comparative Study*

Binary response index models may be affected by several forms of misspecification, which range from pure functional form problems (e.g. incorrect specification of the link function, neglected heterogeneity, heteroskedasticity) to various types of sampling issues (e.g. covariate measurement error, response misclassification, endogenous stratification, missing data). In this article we examine the ability of several versions of the RESET test to detect such misspecifications in an extensive Monte Carlo simulation study. We find that: (i) the best variants of the RESET test are clearly those based on one or two fitted powers of the response index; and (ii) the loss of power resulting from using the RESET instead of a test directed against a specific type of misspecification is very small in many cases.

A Generalized Goodness‐of‐Functional Form Test for Binary and Fractional Regression Models

This paper proposes a new conditional mean test to assess the validity of binary and fractional parametric regression models. The new test checks the joint significance of two simple functions of the fitted index and is based on a very flexible parametric generalization of the postulated model. A Monte Carlo study reveals a promising behaviour for the new test, which compares favourably with that of the well-known RESET test as well as with tests where the alternative model is nonparametric.

Small Sample Bias of Alternative Estimation Methods for Moment Condition Models: Monte Carlo Evidence for Covariance Structures

It is now widely recognized that the most commonly used efficient two-step GMM estimator may have large bias in small samples. In this paper we analyze by simulation the finite sample bias of two classes of alternative estimators. The first includes estimators which are asymptotically first-order equivalent to the GMM estimator, namely the continuous-updating, exponential tilting, and empirical likelihood estimators. Analytical and bootstrap bias-adjusted GMM estimators form the second class of alternatives. The Monte Carlo simulation study conducted in the paper for covariance structure models shows that all alternative estimators offer much reduced bias as compared to the GMM estimator, particularly the empirical likelihood and some of the bias-corrected GMM estimators.

Is neglected heterogeneity really an issue in binary and fractional regression models? A simulation exercise for logit, probit and loglog models

Theoretical and simulation analysis is performed to examine whether unobserved heterogeneity independent of the included regressors is really an issue in logit, probit and loglog models with both binary and fractional data. It is found that unobserved heterogeneity has the following effects. First, it produces an attenuation bias in the estimation of regression coefficients. Second, although it is innocuous for logit estimation of average sample partial effects, it may generate biased estimation of those effects in the probit and loglog models. Third, it has much more deleterious effects on the estimation of population partial effects. Fourth, it is only for logit models that it does not substantially affect the prediction of outcomes. Fifth, it is innocuous for the size of Wald tests for the significance of observed regressors but, in small samples, it substantially reduces their power.

Moment-Based Estimation of Nonlinear Regression Models with Boundary Outcomes and Endogeneity, with Applications to Nonnegative and Fractional Responses

ABSTRACT In this article, we suggest simple moment-based estimators to deal with unobserved heterogeneity in a special class of nonlinear regression models that includes as main particular cases exponential models for nonnegative responses and logit and complementary loglog models for fractional responses. The proposed estimators: (i) treat observed and omitted covariates in a similar manner; (ii) can deal with boundary outcomes; (iii) accommodate endogenous explanatory variables without requiring knowledge on the reduced form model, although such information may be easily incorporated in the estimation process; (iv) do not require distributional assumptions on the unobservables, a conditional mean assumption being enough for consistent estimation of the structural parameters; and (v) under the additional assumption that the dependence between observables and unobservables is restricted to the conditional mean, produce consistent estimators of partial effects conditional only on observables.