Estimation of a type 2 Tobit model with generalized Box-Cox transformation

ABSTRACT

This article considers estimation of a Type 2 Tobit model with Bickel-Doksum transformation of the dependent variable of the regression equation. The basic idea is that while the transformed dependent variable is assumed to be normally distributed prior to any censoring, the inverse Bickel-Doksum transformation allows the underlying dependent variable to follow a wide variety of distributions having differing degrees of skewness and kurtosis. This adds flexibility to the shape of the distribution used to model quantitative variation in the dependent variable for the observed subsample. The log-likelihood function of this Generalized Type 2 Tobit model is globally concave conditional on the parameter of the Bickel-Doksum transformation and the correlation coefficient of the errors. A bivariate grid search over the space of these parameters may be used to find the neighbourhood of the global maximum to the log-likelihood function, provided one exists. The grid search process is important because: 1) the log-likelihood function of the Type 2 Tobit model, even with fixed functional form, often exhibits distinct local and global maxima and 2) use of consistent estimates as starting values is not sufficient to insure convergence to the global Maximum Likelihood Estimator.

KEYWORDS:

• Box-Cox
• Bickel-Doksum
• selectivity
• type 2 Tobit

JEL CLASSIFICATION:

• C10
• C24
• C52

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ The classification system of Tobit models by type may be found in a survey paper on censored regression models by Amemiya (1984); Examples of studies using the Box-Cox Tobit model include Bilger and Chaze (2008), Chaze (2005), and Han and Kronmal (2004).

² The ML estimator of an unknown lower censoring threshold, first considered in the context of a Type 1 Tobit model in Zuehlke (2003), has been used in a number of recent papers including Melstrom and Jayasekera (2017), Munch and Nguyen (2014), and Nippert, Duursma, and Marshall (2004).

³ Applications of the Box-Cox Double-Hurdle model include Aristei and Pieroni (2008), Martinez-Espineira (2006), and Moffatt (2005).

⁴ Jones and Yen (2000) impose a zero censoring threshold at the second hurdle, which corresponds to a threshold of $-1/\lambda$ after application of the Box-Cox transformation. The limit of this threshold as $\lambda$ approaches zero is minus infinity, in which case there is no censoring at the second threshold. Hence, the Jones and Yen model reduces to the semi-log Type 2 Tobit for $\lambda =0$. It is worth noting that the Box-Cox Double-Hurdle model does not nest the Type 2 Tobit model for any value of $\lambda$, including zero, if the censoring threshold at the second hurdle is positive.

⁵ Note that this comment does not apply to zeros used to represent the missing values of $Y_i$ corresponding to censored observations – a common practice in data coding.

⁶ Note that $y(\lambda ,c)$ is the Bickel-Doksum transformation of the constant $c$. It is not a function of $Y_i$ and is not observation specific.

⁷ Here, marginal effect refers to the impact of the regressor on the precensored dependent variable prior to scaling and transformation. Alternatively, one could consider the marginal effect of the regressor on the dependent variable prior to scaling and transformation, but after censoring. See Greene (2012) for details.

⁸ The correlation coefficients of the errors and regressors are being denoted by $\rho$ and $r$, respectively. This is simpler notation than using $\rho$ for both and introducing subscripts to distinguish then.

⁹ The algorithm of Marsaglia and Tsang (2000) is used to generate the independent standard normal draws.

¹⁰ Supplemental simulations, not reported here but available on request, show that as the sample size increases, the envelope of confidence intervals collapses about the horizontal line at zero. This is not a surprising result in light of the consistency of the estimator.

¹¹ As with the estimation of $\lambda$, supplemental simulations show that the envelope of confidence intervals collapses to the horizontal line at zero as the sample size increases.

¹² This data set is available in the Journal of Applied Econometrics Data Archive. See footnote (a) of Table 1 for data sources and descriptions.

¹³ It is debatable whether this strategy results in more reliable estimates. Zuehlke (1996) and Zuehlke (2017) find that deliberately misspecifying the model, either through use of exclusion restrictions or addition of quadratic and/or interaction terms, will adversely affect the mean squared error of estimation.

¹⁴ This data set is available in the data supplement to Greene (2012). See footnote (a) of Table 2 for data sources and descriptions.

¹⁵ This is a generalization of the result for classical regression found in Maddala (1977).