In the evaluation of social programs, it is often difficult to conduct randomized controlled experiments due to non-compliance; therefore the local average treatment effect (LATE) is commonly applied. However, LATE identifies the average treatment effect only for a subpopulation known as compliers and requires the monotonicity assumption. Given these limitations of LATE, this paper proposes a study design and strategy to non-parametrically identify the causal effects for larger populations (such as ATT and ATE) and to remove the monotonicity assumption in the cases of non-compliance. Our strategy utilizes two types of auxiliary observations, one is an outcome before assignment and the other is a treatment before assignment. These observations do not require specially designed experiments, and are likely to be observed in baseline surveys of the standard experiment or panel data. We present the results for the random assignment and those of multiply robust representations in the case where the random assignment is violated. We then present details of the GMM estimation and testing methods which utilize overidentified restrictions. The proposed methodology is illustrated by empirical examples which revisit influential studies by Thornton (2008), Gerber et al. (2009), and Beam (2016), as well as the data set from the Oregon Health Insurance Experiment and that from an experimental data on marketing in a private sector.