13.2 Types of Specification Errors 1. Omitting a Relevant Variable: Underfitting or "Underspecifying"a Model True model:Yt=B1+B2×2+B3×3t+pt(13.1) Misspecified model:Y =A1+A2X2t+ut(13.2)
13.2 Types of Specification Errors 1.Omitting a Relevant Variable: “Underfitting” or “Underspecifying” a Model True model: Yt=B1+B2X2t+B3X3t+μt (13.1) Misspecified model: Yt=A1+A2X2t+μt (13.2)
The consequences of omitting variable bias(X3) (1 If X, X, are correlated: Oa, and a, are biased ar, a, can have an upward or downward bias E(a1)≠B1E(a1)=B1+B3(X3-b2X2)(13.4) E(a2)+B2 E(a2)=B2+B3 b32 o ar and a, are inconsistent (2)f×2and×3 are not correlated a2 is unbiased, consistent, b32 will be zero a, biased, unless X, is zero in the model(13. 4)
（1）If X2 ,X3 are correlated: ◎a1 and a2 are biased, a1 , a2 can have an upward or downward bias E(a1 )≠B1 E(a1 )= B1 +B3（ (13.4) E(a2 ) ≠B2 E(a2 )= B2 +B3b32 ◎ a1 and a2 are inconsistent. （2）If X2 and X3 are not correlated a2 is unbiased, consistent, b32 will be zero a1 biased, unless is zero in the model(13.4) X b X ) 3 − 32 2 The consequences of omitting variable bias (X3 ) X3
(3) The error variance estimated from the misspecified model is a biased estimator of the true error variance g 2 The conventionally estimated variance of a2 is a biased estimator of the variance of the true estimator b2 B 2 .E[var(a2)]=var(b2)+ 2 21 ' Var(a2)will overestimate the true variance of b2, that is, it will have a positive bias. (4) The usual confidence interval and hypothesis-testing procedures are unreliable. The confidence interval will be wider
（3）The error variance estimated from the misspecified model is a biased estimator of the true error variance σ2 ——The conventionally estimated variance of a2 is a biased estimator of the variance of the true estimator b2 ∵ E[var(a2 )]=var(b2 )+ ∴Var(a2 ) will overestimate the true variance of b2 , that is, it will have a positive bias. （4）The usual confidence interval and hypothesis-testing procedures are unreliable. The confidence interval will be wider. 2 2i 2 3i 2 3 (n - 2) x B x