# Tobit Model with a Variable Lower Censoring Bound?

I am working with NBA contract data and about 20 percent of my free agent sample do not receive contracts. Additionally, the distribution for those who do receive contracts is exponential. For each individual who did not receive a contract, I know what the league-mandated minimum salary would have been. It is a function of Years of Service, and thus I know the assessed contract value of the player is between 0 and the mandated minimum.

I am trying to estimate a regression equation on the full sample, and would like to use a tobit specification as my data is clearly left-censored. The problem is that the censoring bound is a variable that is a function of the Years of Service. Does anyone know of examples of a tobit model with a variable bound? Even better would be code to run such a model in Stata, but I'm not getting my hopes up.

• it is definitely a valid model. I have code this up in R before, but sorry, can't help with Stata. – qoheleth Aug 1 '14 at 3:53
• @qoheleth post your R code, I'd love to see it. – robin.datadrivers Feb 27 '15 at 13:55

Take a look at Stata's intreg with a logarithmic transformation of your earnings variable to get normality.

• +1 This in fact solves the technical problem. I had an opportunity recently to verify that after I modified the censReg package in R to accommodate variable censoring levels: its results agreed with intreg to eight significant figures. What I wonder about, though, is whether this model really is appropriate for a situation in which the censoring limit is a known function of a possible regressor in the model. – whuber Feb 21 '16 at 17:55
• @whuber What assumptions do you think are violated by this? – Dimitriy V. Masterov Feb 22 '16 at 17:33
• Independence of censoring level and value. – whuber Feb 22 '16 at 17:37
• @whuber I think this might be OK. On p. 784 of Wooldridge's Econometric Analysis book, he writes that the limits can be a function of the vector x_i because they are conditioned on. – Dimitriy V. Masterov Feb 22 '16 at 22:02
• Thank you for checking it. Something about this assertion feels not quite right. At a minimum, the model in the present application is not using all the available information about how the censoring levels are related to years of service. – whuber Feb 22 '16 at 22:50

Find the max salary for the ones that did not get a contract in your dataset. Set that salary as the cutoff. For example if the max salary for non_contratced=500k, The Tobit specification is the following:

Y=Y*             if  Y*>=500k
Y=500k           if   Y*<500k

• This loses information that the OP finds to be important. When there is little variation among censoring thresholds, this approach can be a reasonable choice when appropriate software is unavailable, but in this case it's not going to do the job. – whuber Aug 17 '15 at 19:05