monotonic transformation, probit vs logit

In my firm I am developing a model using a probit model. I noticed that when benchmarking with a logit specification, the logit slightly improves the model goodness-of-fit.

Talking with a colleague he argued that, this is purely luck because the probit and logit are very similar. He also said that I can apply a monotonic transformation to my data and get better results with a probit.

What's the intuition behind this argument?

Details: I regress a binary variable taking the value 1 if an individual is in financial distress. The data cover a period of 20 years for 1000 individuals. The probit function has the following form $P(Y=1|X)=\Phi(X\beta)$ while the logit function is given by $P(Y=1|X)=\frac{1}{1+e^{-X\beta}}$. The explanatory variables are some macro variables such as GDP, unemployment etc. I computed then an average for the actual vs predicted values across all individuals and for each year. I could then compute a $R^2$ given by the correlation between actual and predicted values squared. When repeating the process with logit I noticed a slightly better increase

• You must give more details for this to be answerable. Is this multinomial logit/probit? Commented Apr 9, 2017 at 8:34
• @kjetilbhalvorsen thanks for your comment, what do you have in mind? Commented Apr 9, 2017 at 10:07
• Well, what does your data represent? Sample sizes? Number of predictor variables? binomial logit/probit or multinomial logit/probit? What is your real problem? Commented Apr 9, 2017 at 10:29

I once had a large bioassay dataset where probit fit (marginally) better than logit. In that case the explanation is clear: the probit, going faster close to $$0 / 1$$, models better that above certain toxicity level, all the organisms die (and below certain level, there is no toxicity at all). In your case it is the opposite: The covariables can never predict with certainty a default (or not), so the probit gives oversecure predictions, and for that reason the logit is better. With your kind of data I would use the logistic fit for risk calculations in future, even if the probit had happened to fit better under training! (that of course is a kind of Bayesian thinking, if the data goes against prior information, sometimes it is better to stick to the prior!)