Focused Information Criteria,Model Selection,and Model Averaging in a Tobit Model With a Nonzero Threshold

Claeskens and Hjort (2003 Claeskens, G. and Hjort, N. L. 2003. “The Focused Information Criterion”. Journal of the American Statistical Association, 98: 900–945. (with discussion)Taylor & Francis Online], Web of Science ®] , Google Scholar]) have developed a focused information criterion (FIC) for model selection that selects different models based on different focused functions with those functions tailored to the parameters singled out for interest. Hjort and Claeskens (2003 Hjort, N. L. and Claeskens, G. 2003. “Frequentist Model Average Estimators”. Journal of the American Statistical Association, 98: 879–899. (with discussion)Taylor & Francis Online], Web of Science ®] , Google Scholar]) also have presented model averaging as an alternative to model selection, and suggested a local misspecification framework for studying the limiting distributions and asymptotic risk properties of post-model selection and model average estimators in parametric models. Despite the burgeoning literature on Tobit models, little work has been done on model selection explicitly in the Tobit context. In this article we propose FICs for variable selection allowing for such measures as mean absolute deviation, mean squared error, and expected expected linear exponential errors in a type I Tobit model with an unknown threshold. We also develop a model average Tobit estimator using values of a smoothed version of the FIC as weights. We study the finite-sample performance of model selection and model average estimators resulting from various FICs via a Monte Carlo experiment, and demonstrate the possibility of using a model screening procedure before combining the models. Finally, we present an example from a well-known study on married women's working hours to illustrate the estimation methods discussed. This article has supplementary material online.