In the research of economics, sociology, medicine, biology, agriculture and other fields, due to the difficulties in data acquisition, the limitations of experimental conditions, the lack of research experience and errors, researchers often omit the key explanatory variables in the setting of regression models, making the identification and processing of omitted variable models a widespread problem. In this paper, a unified identification, estimation and comparison framework is proposed, which enables the identification and estimation of the asymptotic bias of any nonparametric kernel estimator in the regression model with omitted variables. Under this framework, we investigate the exact asymptotic properties of the Nadaraya-Watson estimator, the Gasser-M\"{u}ller estimator as well as the local linear estimator with omitted variables. It is found that the asymptotic errors of the Gasser-M\"{u}ller estimator and the local linear estimator with omitted variables are the same, and smaller than that of the Nadaraya-Watson estimator. In addition, the asymptotic properties of the linear parameter estimator with omitted variables can also be derived through the proposed framework and method. On this basis, an unnoticed good property of local linear kernel estimator proposed in some references is further discussed.