setidentifying

Set identification, also known as partial identification, is a concept in statistics and econometrics where a model and the observed data determine a set of possible values for a parameter rather than a single point. In these cases the data are insufficient to identify the parameter uniquely, so the identified set comprises all parameter values that cannot be ruled out given the model and the data.

The identified set is defined by the model’s constraints, such as moment equalities or inequalities, rather than by a strict point estimate. If the data only imply a range or region of plausible values, the model is considered set-identified. In contrast, point identification occurs when there exists exactly one parameter value consistent with the data and model. Set identification often arises from limited information, weak instruments, measurement error, or nonparametric components that prevent a unique solution.

Inference under set identification uses methods that acknowledge uncertainty about the true parameter within the identified set. This includes constructing confidence sets for the entire identified set or estimating sharp identified sets that cannot be narrowed further without changing the assumptions. Common approaches involve set-valued estimation, moment inequalities, and bootstrap or subsampling techniques to obtain coverage for the identified region.

Applications of set identification appear in econometrics, particularly in models with incomplete information, weak instruments, or partial observability. It guides researchers to report bounds on parameters and to interpret results as ranges of plausible values rather than precise estimates. Limitations include wider uncertainty and the need for careful specification of the identified set and the accompanying inference procedures.