Is there a theoretical upper limit to the number of parameters that be estimated with maximum likelihood estimation? My understanding is no, but that if you have too many parameters it may not be possible to find one set of parameters that uniquely optimizes the log-likelihood.

Assuming the above is correct, practically speaking, how can I determine if my model has too may parameters? Are there tests or guidelines regarding how many observed data points I need relative to my number of parameters?

  • 1
    $\begingroup$ Welcome to Cross Validated! I have addressed the issue of whether or not there is a theoretical limit to the number of parameters that can be estimated by maximizing the likelihood. As far as determining if you have too many parameters, that is a separate question that warrants posting as such. $\endgroup$
    – Dave
    Apr 7 at 11:23

1 Answer 1


Theoretical? No. Let’s look at an example where the number of parameters is unbounded.

If we assume $iid$ Gaussian error terms in a linear model, OLS coincides with maximum likelihood estimation. If there are more observations than regression parameters, then we have a model matrix $X$ that has full rank. Consequently, $(X^TX)$ exists, and the usual a$\hat\beta_{ols}=(X^TX)^{-1}X^Ty$ exists and is equivalent to maximum likelihood estimation.

Since there is no theoretical limit to the number of observations, there is no theoretical limit to the number of parameters that can be estimated.

Moving beyond this example, you are correct that having many parameters can result in the MLE not being unique, yes, but that does not keep MLEs from existing.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.