specification-error-tests

Detecting unimportant variables via likelihood ratio test

Restricted model (if $β_{2} = 0$ ): $Y_{i} = β_{0} + β_{1} X_{1 i} + e_{i}$

Unrestricted: $Y_{i} = β_{0} + β_{1} X_{1 i} + β_{2} X_{2 i} + e_{i}$

$L R = 2 (ULL F - R LL F)$

Unrestricted (add $X_{3}$ ): $Y_{i} = β_{0} + β_{1} X_{1 i} + β_{2} X_{2 i} + β_{3} X_{3 i} + e_{i}$

Restricted: $Y_{i} = β_{0} + β_{1} X_{1 i} + β_{2} X_{2 i} + e_{i}$

$L R = 2 (ULL F - R LL F)$

Model: $Y_{i} = β_{0} + β_{1} X_{1 i} + β_{2} X_{2 i} + e_{i}$

Steps:

Regress and get $\hat{Y}$ .
Regress again with powers: $Y_{i} = β_{0} + β_{1} X_{1 i} + β_{2} X_{2 i} + β_{3} \hat{Y}^{2} + β_{4} \hat{Y}^{3} + β_{5} \hat{Y}^{4} + e_{i}$

F-stat: $F = \frac{( R _{b}^{2} - R _{l}^{2} ) / k _{1}}{( 1 - R _{b}^{2} ) / ( n - k _{2} )}$

If $F > F_{t ab l e}$ , form incorrect.

If $F < F_{t ab l e}$ , form correct.

Advantage: No alternative model needed.

Disadvantage: No alternative if incorrect.

Example: True $Y_{i} = β X_{i} \cdot e_{i}$ ; estimated $Y_{i} = \hat{β} X_{i} + e_{i}$

Check if $\hat{β}$ unbiased for $β$ ; if not, error term misspecified.