What is the meaning of a signifcant intercept in a probit model?

[Mahtab Karimi]

Hello all,

I am estimating a probit model where the intercept is significant. What does it mean? I have seen that some papers do not report the intercept of a probit model, is that something common?
It is always negative, but it becomes significant when I add some variables. Does that mean anything?

Thank you,
Mahtab

[Clyde Schechter]

If you calculate the normal distribution function of the intercept, the resulting value is the probability that the outcome variable is "true" (or "yes" or whatever) conditional on all the explanatory variables of the model being zero. You can easily find out that normal(0) = 0.5 So if you are getting a "significant" intercept, that tells you that you can reject the hypothesis that the probability is 0.5 of the outcome being true when all of the predictor variables are zero. This is not a hypothesis that anyone would typically be interested in. First of all, most real-world models use data where the condition of all predictor variables being zero is either of no special interest, or is even impossible in the real world. Even if the all predictors zero condition is of interest in your context, are you really interested in the null hypothesis that this condition is associated with a 50-50 chance of the outcome being "true?"

So, probably you should ignore it unless you have a reason to be interested in that peculiar hypothesis.

[Richard Williams]

I agree with Clyde. You rarely see much interest in hypotheses about intercepts.

Exceptions: You may see more interest if, say, the X variables have been centered, i.e. the mean is subtracted from each X value. Then the intercept = the average value. Interpretations of intercepts also change if you add interactions to the model. See

https://www3.nd.edu/~rwilliam/stats2/l53.pdf