What are the assumptions for applying a Tobit regression model?

Question

My (very basic) knowledge of the Tobit regression model isn't from a class, like I would prefer. Instead, I have picked up pieces of information here and there through several Internet searches. My best guess at the assumptions for truncated regression are that they are very similar to the ordinary least squares (OLS) assumptions. I have no idea if that is correct, though.

Hence my question: What are the assumptions I should check for when performing Tobit regression?

Note: The original form of this question referred to truncated regression, which was not the model I was using or asking about. I have corrected the question.

Answer 1

If we go for a simple answer, the excerpt from the Wooldridge book (page 533) is very appropriate:

... both heteroskedasticity and nonnormality result in the Tobit estimator $\hat{\beta}$ being inconsistent for $\beta$. This inconsistency occurs because the derived density of $y$ given $x$ hinges crucially on $y^*|x\sim\mathrm{Normal}(x\beta,\sigma^2)$. This nonrobustness of the Tobit estimator shows that data censoring can be very costly: in the absence of censoring ($y=y^*$) $\beta$ could be consistently estimated under $E(u|x)=0$ [or even $E(x'u)=0$].

The notations in this excerpt comes from Tobit model:

\begin{align} y^{*}&=x\beta+u, \quad u|x\sim N(0,\sigma^2)\\ y^{*}&=\max(y^*,0) \end{align} where $y$ and $x$ are observed.

To sum up the difference between least squares and Tobit regression is the inherent assumption of normality in the latter.

Also I always thought that the original article of Amemyia was quite nice in laying out the theoretical foundations of the Tobit regression.

Answer 2

To echo Aniko's comment: The primary assumption is the existence of truncation. This is not the same assumption as the two other possibilities that your post suggests to me: boundedness and sample selection.

If you have a fundamentally bounded dependent variable rather than a truncated one you might want to move to a generalized linear model framework with one of the (less often chosen) distributions for Y e.g. log-normal, gamma, exponential, etc. which respect that lower bound.

Alternatively you might then ask yourself whether you think that the process that generates the zero observations in your model is the same as the one that generates the strictly positive values - prices in your application, I think. If this is not the case then something from the class of sample selection models, (e.g. Heckman models) might be appropriate. In that case you'd be in the situation of specifying one model of being willing to pay any price at all, and another model of what price your subjects would pay if they wanted to pay something.

In short, you probably want to review the difference between assuming truncated, censored, bounded, and sample selected dependent variables. Which one you want will come from the details of your application. Once that first most important assumption is made you can more easily determine whether you like the specific assumptions of any model in your chosen class. Some of the sample selection models have assumptions that are rather difficult to check...

Answer 3

@Firefeather: Does your data contain (and can only really ever contain) only positive values? If so, model it using a generalized linear model with gamma error and log link. If it contains zeros then you could consider a two stage (logistic regression for probability of zero and gamma regression for the positive values). This latter scenario can also be modeled as a single regression using a zero inflated gamma. Some great explanations of this were given on a SAS list a few years ago. Start here if interested and search for follow-ups. link text

Might help point you in another direction if the truncated regression turns out implausible.

Answer 4

As others have mentioned here, the main application of tobit regression is where there is censoring of data. Tobit is widely used in conjunction with Data Envelopment Analysis (DEA) and by the economist. In DEA, efficiency score lies in between 0 and 1, which means that the dependent variable is censored at 0 from left and 1 from right. Therefore, application of linear regression (OLS) is not feasible.

Tobit is a combination of probit and truncated regression. Care must be taken while differentiating censoring and truncating:

• Censoring: When the limit observations are in the sample. The dependent variable values hit a limit either to the left or right.
• Truncation: Observation in which certain range of dependent values is not included in the study. For example, only positive values. Truncation has greater loss of information then censoring.

Tobit = Probit + Truncation Regression

Tobit model assumes normality as the probit model does.

Steps:

1. Probit model decides whether the dependent variable is 0 or 1. $$ P(y>0) = Φ(x^{'} β) \tag{Discreet decision} $$ If the dependent variable is 1 then by how much (assuming censoring at 0).

2. $E(y│y>0)= x^{'} β+ σλ\big(\frac{x^{'} β}{σ}\big) \tag{Continuous decision}$

Coefficient $β$ is same for both the decision model. $σλ\big(\frac{x^{'} β}{σ}\big)$ is the correction term to adjust the censored values (zeros).

Please also check Cragg's model where you can use different $β$ in each step.