Misspecification Methods

• Omitting relevant predictor
• Including irrelevant predictor
• Using an incorrect functional form of the model
• Measurement errors
• Incorrect specification of the error term

Omitting relevant predictor

Example

• Correct model: $Y_i={\beta }_0+{\beta }_1{X}_{1i}+\underline{{\beta }_2{X}_{2i}}+e_i$
• Misspecified: $Y_i={\hat{\beta }}_0+{\hat{\beta }}_1{X}_{1i}+e_i$
• Error term: $e_i=e_{1i}+\underline{{\beta }_2{X}_{2i}}$

Consequences

• If $X_2$ correlates with $X_1$, estimators biased and inconsistent.
• Otherwise, ${\hat{\beta }}_0$ biased; ${\hat{\beta }}_1$ unbiased.
• Error variance estimate inaccurate.
• Estimator variance biased.
• Invalid confidence intervals, hypothesis tests, and forecasts.

Including irrelevant predictor

Example

• Correct model: $Y_i={\beta }_0+{\beta }_1{X}_{1i}+{\beta }_2{X}_{2i}+e_i$
• Misspecified: $Y_i={\hat{\beta }}_0+{\hat{\beta }}_1{X}_{1i}+{\hat{\beta }}_2{X}_{2i}+\underline{{\hat{\beta }}_3{X}_{3i}}+e_{3i}$
• Error term: $e_{3i}=e_{1i}-\underline{{\hat{\beta }}_3{X}_{3i}}$

Consequences

• Estimators unbiased, error variance estimate accurate, valid confidence intervals and hypothesis tests.
• But, estimators inefficient (larger variance than correct model). Therefore, model is less accurate.

Using an incorrect functional form of the model

Occurs when estimated functional form differs from population regression function.

Consequences: Biased, inconsistent coefficient estimators.

Detection: Plot estimated function vs data.

Example: Using log-linear ($\ln Y_i={\hat{\beta }}_0+{\hat{\beta }}_1\ln X_{1i}+{\hat{\beta }}_2\ln X_{2i}+e_{4i}$) when linear is correct.

Measurement errors

Occurs from using proxies ($Y_i^{\ast }=Y+{ϵ}_Y$, $X_i^{\ast }=X+{ϵ}_X$) instead of true values.

Misspecified model: $Y_i^{\ast }={\hat{\beta }}_0^{\ast }+{\hat{\beta }}_1^{\ast }{X}_{1i}^{\ast }+{\hat{\beta }}_2^{\ast }{X}_{2i}^{\ast }+e_i^{\ast }$

Leads to biased estimates.

Incorrect specification of the error term

Example

• Correct: $Y_i=\beta X_i\cdot e_i$ (multiplicative)
• Misspecified: $Y_i=\hat{\beta }{X}_i+e_i$ (additive)

Consequences: True model has misspecified error $⟹$ estimators biased.

Detection: If estimator biased $⟹$ error term is source of misspecification.