Hello All,
Wooldridge's Introductory Econometrics (5th ed.) states that "Because maximum likelihood estimation is based on the distribution of y given x, the heteroskedasticity in Var(y|x) is automatically accounted for."
Does this hold also for bias-corrected or penalized maximum likelihood estimation under Firth logistic regression?
Please advise.
Thanks in advance!
Navya Kumar
For some reason, your question (that I cannot answer) reminds me of Rothchild’s Rule: 'For every phenomenon, however complex, someone will eventually come up with a simple and elegant theory. This theory will be wrong'.
I may be misunderstanding your question, but there is no constant variance assumption with logistic regression, so you do not need to worry about heteroskedasticity. In fact, heteroskedasticity is almost guaranteed with logistic regression since the variance of a binomial random variable is a function of the probability of the event happening and the probability of the event not happening, which will usually differ between observations.
I disagree with earlier responses. The logistic has two parameters, location and scale. When we fit a logistic regression, we typically treat the location parameter as a function of X and we assume the scale parameter equals 1. Thus the variance of the latent variable is constant for all values of X. If the scale is not really constant, regression parameters are inconsistent. It is possible to write the likelihood with the scale as a function of X — at least for a probit, I have not done this for a logit. Then the likelihood can be penalized. This penalization is the basis for the Firth regression. An earlier response mistakes the variance for the Bernoulli outcome with the variance for the latent variable.
William Rhodes Thank you for your response! So as I understand, given the scale is a function of X, allowing for the likelihood to be penalized, Firth logit is unlikely to automatically account for heteroskedasticity. I would need to explicitly test for heteroskedasticity post a firth logit regression...
My response was theoretical. I doubt you can perform a reliable test for heteroscedasticity. From the literature, I gather that such tests are in general controversial in binary outcome models. Given that you are dealing with a rare event problem (hence the penalized likelihood), you probably have little chance of detecting the problem and implementing an ameliorative. I think you must assume homoscedasticity, which is reasonable unless you have some reason for believing otherwise. Sorry.
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