# C a constant term should have been included in the

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c.) A constant term should have been included in the regression— could have tested with simple t test. One would also suspect serial correlation or even spurious regression. Could test for this by regressing ˆ u t on x t and ˆ u t 1 and calculating a t -test that the coe cient on ˆ u t 1 is zero. Would also worry about endogeneity, e.g., when productivity is unusually high ( ε t > 0 ), the fi rm probably wants to hire more labor ( c t positively correlated with ε t ) . To test for this we’d need an instrument (something correlated with c t but uncorrelated with ε t ). 3.) a.) no b.) g ( θ , Y T ) = T 1 Σ x t ( y t z 0 t θ ) , S = σ 2 Σ xx , D 0 = Σ xz c.) ˆ S σ 2 T 1 Σ x t x 0 t , ˆ σ 2 = T 1 Σ ( y t z 0 t ˆ θ ) 2 , ˆ D = T 1 Σ z t x 0 t d.) GMM FOC gives ³ T 1 Σ x t z 0 t ´ 0 ˆ σ 2 ³ T 1 Σ x t x 0 t ´ 1 ³ T 1 Σ x t ( y t z 0 t ˆ θ ´ = 0 which on rearranging is 2SLS.

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–2– Econ 220B, Winter 2010 e.) J = T ¡ T 1 Σ x t ˆ u t ¢ 0 ³ ˆ σ 2 T 1 Σ x t x 0 t ´ 1 ¡ T 1 Σ x t ˆ u t ¢ = ¡ Σ ˆ u t x 0 t ¢ ¡ Σ x t x 0 t ¢ 1 ( Σ x t ˆ u t ) ˆ σ 2 χ 2 ( r a ) f.) Use ¡ Σ u t x 0 t ¢ ¡ Σ x t x 0 t ¢ 1 ( Σ x t u t ) ˆ σ 2 for u t = y t z 0 t θ 0 . We have here conditions for m.d.s. CLT, T 1 / 2 Σ x t u t L N ( 0 , σ 2 Σ xx ) and so ¡ T 1 / 2 Σ u t x 0 t ¢ ¡ T 1 Σ x t x 0 t ¢ 1 ¡ T 1 / 2 Σ x t u t ¢ ˆ σ 2 L T 1 / 2 Σ u t x 0 t ( σ 2 Σ xx ) 1 ¡ T 1 / 2 Σ x t u t ¢ L χ 2 ( r )
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• Winter '08
• Hamilton,J

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