underidentification

Underidentification is a condition in statistical modeling where the observed data do not provide enough information to uniquely determine the model’s structural parameters under the given restrictions. In econometrics and related fields, it means there is no unique mapping from the parameters to the probability distribution of the observed data, so multiple parameter values are consistent with the data. This is a fundamental identification problem, distinct from standard estimation error due to sampling variability.

Causes of underidentification include having too few valid instruments relative to the number of endogenous variables, weak or irrelevant instruments, and overly flexible models with more parameters than the data can support. Misspecification that violates the assumed identifying restrictions can also produce underidentification. In instrumental variables and simultaneous-equations settings, it typically arises when the instrument set fails to satisfy the rank condition: not enough instruments, or instruments that do not sufficiently correlate with the endogenous regressors to isolate the structural effects.

Consequences include non-unique or unstable parameter estimates, invalid standard errors and hypothesis tests, and compromised causal interpretation. Detection relies on identification checks and rank conditions; standard estimation software may report underidentification or produce flat likelihoods and multiple near-equivalent solutions when the model is unidentified.

Remedies focus on increasing information and strengthening restrictions. Potential approaches are adding valid and relevant instruments, imposing theory-based restrictions to fix certain parameters, simplifying or reparameterizing the model, or collecting more data. In some contexts, partial identification or Bayesian methods with informative priors can provide bounds or distributions for the parameters when full identification is not possible.

Example: a two-equation system with two endogenous variables and only one exogenous instrument that affects only one equation can be underidentified.