This paper presents the pseudo maximum likelihood method, which involves estimating parameters by maximizing a likelihood function associated with a probability distribution that may not be the true one. The method is applied to exponential families of type I and II, which are shown to provide consistent and asymptotically normal estimators for parameters in the first and second moments of the true distribution. The paper discusses the properties of these estimators, their asymptotic normality, and the conditions under which they are consistent. It also extends the method to cases where second-order moments can be consistently estimated. Theoretical results are derived, including the asymptotic covariance matrix of the pseudo maximum likelihood estimator. The paper also introduces the quasi-generalized pseudo maximum likelihood (QGPML) estimator, which incorporates a nuisance parameter and is shown to be asymptotically normal and efficient under certain conditions. The paper concludes that exponential families of type I and II are essential for achieving consistent and asymptotically normal estimators, and that the pseudo maximum likelihood method is a robust and general approach applicable to various econometric models.This paper presents the pseudo maximum likelihood method, which involves estimating parameters by maximizing a likelihood function associated with a probability distribution that may not be the true one. The method is applied to exponential families of type I and II, which are shown to provide consistent and asymptotically normal estimators for parameters in the first and second moments of the true distribution. The paper discusses the properties of these estimators, their asymptotic normality, and the conditions under which they are consistent. It also extends the method to cases where second-order moments can be consistently estimated. Theoretical results are derived, including the asymptotic covariance matrix of the pseudo maximum likelihood estimator. The paper also introduces the quasi-generalized pseudo maximum likelihood (QGPML) estimator, which incorporates a nuisance parameter and is shown to be asymptotically normal and efficient under certain conditions. The paper concludes that exponential families of type I and II are essential for achieving consistent and asymptotically normal estimators, and that the pseudo maximum likelihood method is a robust and general approach applicable to various econometric models.