For regression model selection via maximum likelihood estimation, we adopt a vector representation of candidate models and study the likelihood ratio confidence region for the regression parameter vector of a full model. We show that when its confidence level increases with the sample size at a certain speed, with probability tending to one, the confidence region consists of vectors representing models containing all active variables, including the true parameter vector of the full model. Using this result, we examine the asymptotic composition of models of maximum likelihood and find the subset of such models that contain all active variables. We then devise a consistent model selection criterion which has a sparse maximum likelihood estimation interpretation and certain advantages over popular information criteria.