Test for Normality and Multicollinearity in Probit Models

[Joseph Malcontento]

Hello! I've been running post-estimation techniques after conducting a probit regression. I have two questions:

1. How can I test for normality? The normal "sktest" only works with an OLS regression. I was wondering if there is a command that can allow me to test for normality after running a probit model.

2. I want to test for the presence of multicollinearity in my probit model but just like in the previous question, the "vif" command only works after an OLS regression. I did some searching over the internet and some sources say that I should run an OLS regression and run VIF. I can disregard the OLS regression results but use the VIF in determining the presence of multicollinearity in my OLS model and apply its results to my probit model also. Can I do this or is there a more proper way of detecting this issue? I did this and the VIF, after an OLS regression, tell me that multicollinearity is unlikely present in my model because it's less than the benchmark value of 10. Is it acceptable to say that my probit model using the same set of variables is free from this issue too?

[Maarten Buis]

It is de facto impossible to test for normality in a probit model. The residual that should be normally distributed is the difference between the unobserved latent variable and the predicted values. Compare that with the residual in linear regression (OLS is the algorithm used for computing the estimates, while linear regression is the model) are the difference between the observed dependent variable and the predicted values. In essence, the normality assumption governs the functional form relating the expalantory variables with the probability. The common alternative (logit) is so similar that they are indistinguishable in most datasets.

Multicolinearity is a characteristic of the explanatory variables alone, so it does not matter which model was used to compute the VIF.

[Joseph Malcontento]

Thank you very much sir! Besides checking for multicollinearity, using robust standard errors in accounting for heteroscedasticity and conducting a link test to check for model misspecification, can you suggest other post-estimation techniques I can employ in my probit model?

[Maarten Buis]

Heteroscedasticity is a much more complicated problem in logit/probit models than in linear regression, see e.g.: http://maartenbuis.nl/wp/oddsratio.html . Robust standard errors are not a solution to that problem.

[Joao Santos Silva]

Dear Joseph and Maarten,

This may be one of those "cultural" differences, but there are tests for normality in the probit. As Maarten mentions, normality governs the functional form and the test for normality is just a standard RESET test using squares and cubes. The significance of the first term is a check for skewness and the significance of the second term is a check of for excess kurtosis. I can try to dig out the reference if you are interested. It turns out that this is also a test for a particular form of heteoskedasticity.

Best regards,

Joao

[James Park]

Dear Joseph and Maarten,

This may be one of those "cultural" differences, but there are tests for normality in the probit. As Maarten mentions, normality governs the functional form and the test for normality is just a standard RESET test using squares and cubes. The significance of the first term is a check for skewness and the significance of the second term is a check of for excess kurtosis. I can try to dig out the reference if you are interested. It turns out that this is also a test for a particular form of heteoskedasticity.

Best regards,

Joao

Dear Joao,

Could you explain how to test for normality in probit or ivprobit commands? (or heteroskedasticity)

[Joao Santos Silva]

Dear James Park,

Just perform a RESET test with squares and cubes.

Best wishes,

Joao

[Kazim Hashim]

Hey guys,

I am running a Probit model also and wish to conduct a VIF test to check for Multicollinearity. However I wanted to ask firstly if the Probit Model assumed no multicollinearity in the first place?

I appreciate any clarity you could provide

[Maarten Buis]

Mulitcolinearity is not a problem (I ignore perfect multicolinearity, as there the problem is usually a logic error by the researcher and not a data problem). Neither linear regression (some people mistakenly call it OLS) nor probit assume anything about multicolinearity.

With a regression model (linear, probit, logit, or otherwise) you are trying to separate effect of different variables, and that is harder when the variables move together. However, the standard errors acurately represents this additional uncertainty. So multicolinearity does not invalidate our model. You need to understand your data, and know how much (or little) information is present in your data, but finding multicolinearity does not require you to do anything.

Multicolineartiy is just a property of the explanatory variables alone. So you can just do a linear regression of any arbitry dependent variable with the explanatory variables of interest, and get the FIVs from the post-estimation of those models.

[Kazim Hashim]

Thank you Maarten! This has helped me a lot.