Misspecification

Definition

Model misspecification occurs when the true data-generating process (DGP) differs from the assumed statistical model. Misspecification can affect consistency of estimators, validity of standard errors and test statistics, and the accuracy of forecasts. The severity depends on which aspects of the model are wrong: error distribution misspecification leaves Ordinary Least Squares (OLS) consistent (if the mean is correctly specified) but invalidates classical inference; mean-function misspecification biases all coefficient estimates; omitted-variable misspecification induces endogeneity.

Key Ideas

Consequences by type:
- Distributional misspecification: OLS/Maximum Likelihood Estimation (MLE) may remain consistent but standard errors are invalid — corrected by White (1980) heteroskedasticity-robust standard errors or Quasi-MLE (QMLE) sandwich estimator.
- Mean-function misspecification: OLS converges to the best linear projection, which may differ substantially from the conditional mean. Nonlinear models (probit, GARCH) converge to pseudo-true parameters minimising the Kullback-Leibler (KL) divergence from the true density (White 1982).
- Dynamic misspecification: omitted lags or incorrect lag length inflate residual variance and induce spurious dynamics.
Pseudo-true parameters: even a misspecified model converges (under regularity conditions) to well-defined population values — but those values may not equal the structural parameters of interest (White 1982).
Specification tests:
- Hausman test: compares two consistent estimators that differ under misspecification (e.g., OLS vs. Instrumental Variables (IV)); asymptotically $\chi^2$ under the null of correct specification.
- White (1980) information matrix test: tests whether the Information Matrix equality holds; rejection indicates distributional misspecification.
- RESET test (Regression Specification Error Test; Ramsey 1965): augments the regression with powers of fitted values; rejection indicates omitted nonlinearities.
- Ljung-Box test: tests for autocorrelation in residuals; rejection indicates dynamic misspecification.
Fair (1980) misspecification component: in stochastic simulation, the misspecification component is the gap between the model's one-step-ahead squared residual and the simulation variance — estimated by rolling-window re-estimation. See Forecast Uncertainty Decomposition.
Bayesian perspective: in Bayesian inference, misspecification manifests as posterior predictive failure — the realized data fall outside the predictive distribution. See Posterior Predictive Check.

How It Works

Detecting misspecification:

Examine residuals for autocorrelation, heteroskedasticity, non-normality.
Apply specification tests (Hausman, RESET, Ljung-Box, ARCH test).
Compare information criteria (AIC, BIC) across competing specifications.
Posterior predictive checks in the Bayesian setting.

Robust inference under misspecification:

White (1980) sandwich standard errors: $\hat V(\hat\beta) = (X'X)^{-1} \hat\Omega (X'X)^{-1}$ where $\hat\Omega = \sum_i \hat\varepsilon_i^2 x_i x_i'$ .
Newey-West standard errors extend to autocorrelation.
QMLE: maximise a possibly misspecified log-likelihood; sandwich variance gives valid inference even when the log-likelihood is wrong.

Why It Matters

All econometric models are misspecified to some degree ("all models are wrong but some are useful"). Understanding the direction and magnitude of misspecification guides model refinement and robust inference.
In forecasting, misspecification is often the dominant source of forecast error at medium-to-long horizons, exceeding coefficient uncertainty and error-term variance.
Latent-class and mixture models face a particularly severe form: multiple equally-fitting misspecified models can yield substantially different parameter estimates (Albert-Dodd (2004); see Latent Class Model).

Open Questions