Verifiable identification condition for nonignorable nonresponse data with categorical instrumental variables

We consider a model identification problem in which an outcome variable contains nonignorable missing values. Statistical inference requires a guarantee of the model identifiability to obtain estimators enjoying theoretically reasonable properties such as consistency and asymptotic normality. Recently, instrumental or shadow variables, combined with the completeness condition in the outcome model, have been highlighted to make a model identifiable. In this paper, we elucidate the relationship between the completeness condition and model identifiability when the instrumental variable is categorical. We first show that when both the outcome and instrumental variables are categorical, the two conditions are equivalent. However, when one of the outcome and instrumental variables is continuous, the completeness condition may not necessarily hold, even for simple models. Consequently, we provide a sufficient condition that guarantees the identifiability of models exhibiting a monotone-likelihood property, a condition particularly useful in instances where establishing the completeness condition poses significant challenges. Using observed data, we demonstrate that the proposed conditions are easy to check for many practical models and outline their usefulness in numerical experiments and real data analysis.