Just a general statistical question: when any statistical software returns log-likelihood of some model, does it account for all details in it?
For example, when we employ generalized least square method, the correlation structure and variance weighting affects the estimation of model coefficients, so the likelihood function will reflect it too.
But when we employ ordinary Ordinary Least Square with sandwich robust variance estimator post estimation, only the standard errors will be different, but not the coefficients, which remain unaffected. In this case will the likelihood function catch it somehow?
When we use Wald's testing, we can use sandwich and other adjustments to obtain more conservative estimates of covariance, therefore - standard errors and the inference. When we use the likelihood ratio testing, is this information about robust standard errors lost? So why is it said, that Likelihood Ratio testing is better than Wald, if Wald can use more information and Likelihood Ratio can not?
I found some internet materials about testing hypotheses using both methods and I try to understand the message, that says that Wald is worse because it is typically biased has increased type-1 error rate, and the Likelihood Ratio test should always be preferred. But the Wald testing can use both coefficients and the variance-covariance matrix and other adjustments, while the Likelihood Ratio seems to be missing part of the additional information!
