Generalized Tobit models: diagnostics and application in econometrics

ABSTRACT The standard Tobit model is constructed under the assumption of a normal distribution and has been widely applied in econometrics. Atypical/extreme data have a harmful effect on the maximum likelihood estimates of the standard Tobit model parameters. Then, we need to count with diagnostic tools to evaluate the effect of extreme data. If they are detected, we must have available a Tobit model that is robust to this type of data. The family of elliptically contoured distributions has the Laplace, logistic, normal and Student-t cases as some of its members. This family has been largely used for providing generalizations of models based on the normal distribution, with excellent practical results. In particular, because the Student-t distribution has an additional parameter, we can adjust the kurtosis of the data, providing robust estimates against extreme data. We propose a methodology based on a generalization of the standard Tobit model with errors following elliptical distributions. Diagnostics in the Tobit model with elliptical errors are developed. We derive residuals and global/local influence methods considering several perturbation schemes. This is important because different diagnostic methods can detect different atypical data. We implement the proposed methodology in an R package. We illustrate the methodology with real-world econometrical data by using the R package, which shows its potential applications. The Tobit model based on the Student-t distribution with a small quantity of degrees of freedom displays an excellent performance reducing the influence of extreme cases in the maximum likelihood estimates in the application presented. It provides new empirical evidence on the capabilities of the Student-t distribution for accommodation of atypical data.

[1]  Víctor Leiva,et al.  Reparameterized Birnbaum-Saunders regression models with varying precision , 2016 .

[2]  J. A. Díaz-García,et al.  SENSITIVITY ANALYSIS IN LINEAR REGRESSION , 2022 .

[3]  J. A. Díaz-García,et al.  SINGULAR ELLIPTICAL DISTRIBUTION: DENSITY AND APPLICATIONS , 2002 .

[4]  Helton Saulo,et al.  A family of autoregressive conditional duration models applied to financial data , 2014, Comput. Stat. Data Anal..

[5]  Víctor Leiva,et al.  Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics , 2015 .

[6]  Shuangzhe Liu,et al.  On diagnostics in conditionally heteroskedastic time series models under elliptical distributions , 2004, Journal of Applied Probability.

[7]  Manuel Galea,et al.  Diagnostics in Birnbaum-Saunders accelerated life models with an application to fatigue data , 2014 .

[8]  Reinaldo Boris Arellano-Valle,et al.  Student-t censored regression model: properties and inference , 2012, Stat. Methods Appl..

[9]  Andre Lucas,et al.  Robustness of the student t based M-estimator , 1997 .

[10]  Gilberto A. Paula,et al.  On influence diagnostic in univariate elliptical linear regression models , 2003 .

[11]  E Lesaffre,et al.  Local influence in linear mixed models. , 1998, Biometrics.

[12]  G. Dagne.  A growth mixture Tobit model: application to AIDS studies , 2015 .

[13]  J. A. Calvin.  Regression Models for Categorical and Limited Dependent Variables , 1998 .

[14]  M. Kendall.  Theoretical Statistics , 1956, Nature.

[15]  D. Holden.  Testing for heteroskedasticity in the tobit and probit models , 2011 .

[16]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[17]  B. Efron,et al.  Assessing the accuracy of the maximum likelihood estimator: Observed versus expected Fisher information , 1978 .

[18]  S. Chatterjee.  Sensitivity analysis in linear regression , 1988 .

[19]  P. Grambsch,et al.  Martingale-based residuals for survival models , 1990 .

[20]  Víctor Leiva,et al.  On an extreme value version of the Birnbaum-Saunders distribution , 2012 .

[21]  Heleno Bolfarine,et al.  Local influence in elliptical linear regression models , 1997 .

[22]  Filidor V. Vilca,et al.  Influence analysis in skew-Birnbaum–Saunders regression models and applications , 2011 .

[23]  Robert G. Aykroyd,et al.  Birnbaum–Saunders autoregressive conditional duration models applied to high-frequency financial data , 2019 .

[24]  Myoung-jae Lee,et al.  Methods of moments and semiparametric econometrics for limited dependent variable models , 1996 .

[25]  Francisco José de A. Cysneiros,et al.  A Multivariate Log-Linear Model for Birnbaum-Saunders Distributions , 2016, IEEE Transactions on Reliability.

[26]  J. S. Long,et al.  Regression Models for Categorical and Limited Dependent Variables , 1997 .

[27]  Steven B. Caudill,et al.  More on testing the normality assumptionin the Tobit Model , 2009 .

[28]  Jeremy MG Taylor,et al.  Robust Statistical Modeling Using the t Distribution , 1989 .

[29]  Helton Saulo,et al.  Birnbaum–Saunders frailty regression models: Diagnostics and application to medical data , 2017, Biometrical journal. Biometrische Zeitschrift.

[30]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[31]  Peter M. Bentler,et al.  Pseudo maximum likelihood estimation in elliptical theory: effects of misspecification , 1994 .

[32]  S. Weisberg,et al.  Residuals and Influence in Regression , 1982 .

[33]  Shuangzhe Liu,et al.  On local influence for elliptical linear models , 2000 .

[34]  M. Ivette Gomes,et al.  Extreme value Birnbaum–Saunders regression models applied to environmental data , 2016, Stochastic Environmental Research and Risk Assessment.

[35]  Miguel Angel Uribe-Opazo,et al.  Birnbaum–Saunders spatial modelling and diagnostics applied to agricultural engineering data , 2016, Stochastic Environmental Research and Risk Assessment.

[36]  Shuangzhe Liu,et al.  Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions , 2015, Statistical Methods & Applications.

[37]  K. Fang,et al.  Generalized Multivariate Analysis , 1990 .

[38]  José A. Díaz-García,et al.  Influence Diagnostics for Elliptical Multivariate Linear Regression Models , 2003 .

[39]  Aldo M. Garay,et al.  Bayesian analysis of censored linear regression models with scale mixtures of normal distributions , 2015 .

[40]  Narayanaswamy Balakrishnan,et al.  Shape and change point analyses of the Birnbaum-Saunders-t hazard rate and associated estimation , 2012, Comput. Stat. Data Anal..

[41]  T. Mroz,et al.  The Sensitivity of an Empirical Model of Married Women's Hours of Work to Economic and Statistical Assumptions , 1987 .

[42]  N. Billor.  An application of the local influence approach to ridge regression , 1999 .

[43]  George G. Judge,et al.  Econometric foundations , 2000 .

[44]  Manuel González,et al.  Influence diagnostics in the tobit censored response model , 2010, Stat. Methods Appl..

[45]  Achim Zeileis,et al.  Applied Econometrics with R , 2008 .

[46]  Víctor Leiva,et al.  Birnbaum–Saunders statistical modelling: a new approach , 2014 .

[47]  Heping Zhang,et al.  A diagnostic procedure based on local influence , 2004 .

[48]  Gilberto A. Paula,et al.  Robust statistical modeling using the Birnbaum-Saunders- t distribution applied to insurance , 2012 .

[49]  Felipe Osorio,et al.  Assessment of local influence in elliptical linear models with longitudinal structure , 2007, Comput. Stat. Data Anal..

[50]  Darryl Holden,et al.  Testing the Normality Assumption in the Tobit Model , 2004 .

[51]  Juan F. Vivanco,et al.  Diagnostics in multivariate generalized Birnbaum-Saunders regression models , 2016 .

[52]  Gilberto A. Paula,et al.  Influence Diagnostics in log-Birnbaum-Saunders Regression Models , 2004 .