How to interpret coefficients in a Tobit regression?

Question

I've run a Tobit regression but I'm not sure of how I should interpret the coefficients. Do you have any suggestions? Thank you very much!

Answer 1

The tobit coefficient ("beta") estimates the linear increase of the latent variable for each unit increase of your predictor. As the latent variable is identical to your observed variable for all observations that are above the threshold, it also measures the linear increase of your predictor on your response for all observations above that threshold - just like an OLS coefficient. Interpreting the tobit coefficient directly is hence useful if...

• ... you are interested in the latent variable. This can be the case if you have a real data censoring issue and not an economic corner solution. For example, if you are interested in some variable or effect that exists in reality but just cannot be observed below a certain threshold. In this case the latent variable would be your (uncensored) variable of interest.
• ... you (for example for the sake of simplicity) are interested in the linear increase of your dependent variable over all uncensored cases and not in the non-linear increase of your dependent variable over the whole sample. Note, that in this case the tobit coefficient does not provide information about the effect of your predictor on the likelihood that your dependent variables becomes uncensored.

If you are instead interested in the non-linear increase of your dependent variable, which often describes its behaviour more accurately, you want to calculate marginal effects. As they are non-linear, the marginal effects are sensible to the position on which you calculate them. They can include both the effect of your predictor on the likelihood that your dependent variable becomes uncensored as well as on the change in magnitude of your dependent variable provoked by your independent variable. They are usually small on positions where many observations are censored and increase to the degree that observations are becoming uncensored. In practice, marginal effects are often calculated at some interesting position of your dataset, usually the mean.

I recommend reading the Woolldridge (Introductory Econometrics) chapters on tobit and censored regression models. Moreover, Hill, Griffiths and Lim's Principles of Econometrics has a nice visualisation of the tobit coefficients in their chapter on Limited Dependent Variables.

To understand the difference between the Tobit coefficient and the marginal effect, you should read Moffitt (1980): The Uses of Tobit Analysis closly.

Answer 2

First of all, the interpretation of beta in a Tobit model is different from the one in OLS. The beta measures a non-linear relationship between Y and x.

Here is a minimal reproducible example from

Farhad Hosseinzadeh Lotfi, Mohsen Vaez-Ghasemi, Ali Ebrahimnejad. 2020. Data Envelopment Analysis with r. 1st ed. Studies in Fuzziness and Soft Computing 386. Springer International Publishing.

library(Benchmarking)
library(AER)
data(norWood2004)
d <- norWood2004
x <- cbind(d$x,d$m)
y <- d$y
e <- dea(x,y)
E <- eff(e)
eOls <- lm(E ~ z1+z3+z6, data=d)

zz1 <- d$z1/1e6
eTob <- tobit(E ~ zz1+z3+z6, left=-Inf, right=1, data=d)

stargazer::stargazer(eOls, eTob, type = "text")
## 
## ===========================================================
##                               Dependent variable:          
##                     ---------------------------------------
##                                        E                   
##                              OLS                Tobit      
##                              (1)                 (2)       
## -----------------------------------------------------------
## z1                        -0.00000*                        
##                           (0.00000)                        
##                                                            
## zz1                                            -0.126*     
##                                                (0.067)     
##                                                            
## z3                         0.007**            0.008***     
##                            (0.003)             (0.003)     
##                                                            
## z6                         -0.016              -0.010      
##                            (0.055)             (0.060)     
##                                                            
## Constant                    0.185               0.166      
##                            (0.150)             (0.165)     
##                                                            
## -----------------------------------------------------------
## Observations                 113                 113       
## R2                          0.078                          
## Adjusted R2                 0.052                          
## Log Likelihood                                 -40.824     
## Residual Std. Error   0.283 (df = 109)                     
## F Statistic         3.068** (df = 3; 109)                  
## Wald Test                                 10.055** (df = 3)
## ===========================================================
## Note:                           *p<0.1; **p<0.05; ***p<0.01

We make a comparison between OLS and Tobit ones.

set.seed(123)
# The tobit model
eTob <- tobit(E ~ zz1+z3+z6, left=-Inf, right=1, data=d)
s <- sqrt(var(residuals(eTob)))
# The mean at the effect for all firms
az <- fitted(eTob)
mean_effect <- c(
    mean(coef(eTob)[2] * (pnorm((1-az)/s) - pnorm(-az/s))),
    mean(coef(eTob)[3] * (pnorm((1-az)/s) - pnorm(-az/s))),
    mean(coef(eTob)[4] * (pnorm((1-az)/s) - pnorm(-az/s)))
)
# the effect at the mean of az
az <- mean(fitted(eTob))
mean_coef <- coef(eTob) * (pnorm((1-az)/s) - pnorm(-az/s))
# the effect at the min value of az
az <- min(fitted(eTob))
min_coef <- coef(eTob) * (pnorm((1-az)/s) - pnorm(-az/s))
# the effect at the max value of az
az <- max(fitted(eTob))
max_coef <- coef(eTob) * (pnorm((1-az)/s) - pnorm(-az/s))
# the OLS model
lm_effect <- lm(E ~ zz1+z3+z6, data=d)$coef

matrix(
    c(
        mean_effect,
        mean_coef,
        min_coef,
        max_coef,
        lm_effect
    ),
    byrow = TRUE,nrow = 5
) %>% 
    `rownames<-`(c("Mean of effect for all firms", "Effect at mean value of z", "Effect at min", "Effect at max", "OLS")) %>% 
    `colnames<-`(c("Intercept", "z1 x 10e-6","z3", "z6")) %>% 
    as.data.frame() %>% 
    rownames_to_column() %>% 
    kable()

You will see, the difference between the beta of OLS and mean value of Tobit.