misspecified

Misspecified is an adjective used in statistics, econometrics, and related fields to describe a model, specification, or assumption that does not accurately capture the underlying data-generating process. A misspecified model can arise from incorrect functional form, omitted variables, incorrect distributional assumptions about errors, measurement error, or endogeneity among regressors.

Common forms include functional form misspecification, such as imposing a linear relationship when the true relationship is nonlinear, or using the wrong link function in a generalized linear model. Omitted-variable misspecification occurs when an important predictor is left out, leading to biased estimates. Incorrect distributional assumptions about errors, such as assuming homoscedastic and normal errors when they are heteroskedastic or nonnormal, can also be misspecification. Endogeneity or simultaneity, measurement error, or incorrect treatment of error structure are additional sources.

Consequences of misspecification include biased and inconsistent parameter estimates, biased standard errors, and unreliable predictions or inferences. Hypothesis tests and confidence intervals may be invalid, and model-based conclusions may be misleading.

Detection and remedies involve specification tests, residual diagnostics, and model comparison. Tests such as the RESET test, tests for endogeneity (e.g., Hausman), and information criteria (AIC, BIC) can help diagnose misspecification. Remedies include revising the model to include relevant variables, transforming variables, adding nonlinear terms or interactions, using alternative model families, or adopting semi-parametric or nonparametric approaches. Addressing endogeneity with instrumental variables, fixed effects, or causal methods is also common.

Misspecification is a central concern in model building and causal inference, and recognizing its possibility is important for credible analysis.