6

In this most upvoted CV answer on that topic the "scale" parameter (aka "sigma" in Stata) thrown in a tobit regression output is explained to be "the estimated standard deviation of the residuals". (Since the question is over three years old, I decided to open a new one.) "That value can be compared to the standard deviation of [the dependent variable]. If it is much smaller, [we] may have a reasonably good model."

However, when I look at the standard deviation of the residuals the values won't match up. Actually "scale" is more than twice the sd of the residuals!

Example

library(AER)
data(Affairs)
fit <- tobit(affairs ~ age + yearsmarried + religiousness +
               occupation + rating, data=Affairs)

fit$scale  # the "scale" value
# [1] 8.24708

sd(resid(fit))  # sd of residuals
# [1] 4.140131

sd(Affairs$affairs)  # sd of dependent variable
# [1] 3.298758

Do I misinterpret that answer, or is it flawed? Could someone clear up with this confusion about the "Scale" (Stata: "Sigma") parameter of a tobit regression? How is it calculated, and what does it tell us about the model quality?

Community
  • 1
jay.sf
  • 556
  • 3
  • 19

1 Answers1

3

Tobit regression assumes that there is a latent dependent variable that can take any value. It is part of a normal linear regression equation with a normally distributed error tern. The standard deviation of that error term is the scale parameter. However, a tobit model assumes that this latent variable is only observed when it exceeds a specific lower bound (often 0), otherwise the observed variable gets the value of that lower bound. So, yes the scale parameter is analogous to the standard deviation of the residuals, but you cannot get an estimate of those residuals by subtracting the predicted values from the observed dependent variable. That is probably what you did and it would underestimate the variance, which is what you observed.

Maarten Buis
  • 19,189
  • 29
  • 59