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Multiple Equation GMM with Common Coe cients: Panel Data Eric Zivot November 25, 2009
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Multi-equation GMM with common coe cients Example (panel wage equation) LW 69 i = φ + βS 69 i + γIQ i + πEXPR 69 i + ε i 1 LW 80 i = φ + 80 i + i + 80 i + ε i 2 Note: common coe cients across two equations and cov ( ε i 1 i 2 ) 6 =0
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The general multi-equation model with common coe cients δ is y im = z 0 im δ + ε im ,i =1 ,...,n ; m ,...,M E [ z im ε im ] 6 = 0 fo rsomee lemento f z im Themode lfo rthe m th equation is y m n × 1 = Z m δ + ε m ,m
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The stacked system (across equations) is y 1 = Z 1 δ + ε 1 . . . y M = Z M δ + ε M y 1 . . . y M = Z 1 . . . Z M δ + ε 1 . . . ε M This is in the form of a giant regression y Mn × 1 = Z δ + ε
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Assume each equation m has k m instruments x im satisfying E [ x im ε im ]=0 Very often the same instruments x i are used for each equation. The moment conditions are g i ( δ )= x i 1 ( y i 1 z 0 i 1 δ ) . . . x iM ( y iM z 0 iM δ ) There are a total of K = P M m =1 k m moment conditions.
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Identi f cation E [ g i ( δ )] = 0 for δ = δ 0 6 = 0 for δ 6 = δ 0 Identi f cation implies that E [ x i 1 y i 1 ] . . . E [ x iM y iM ] E [ x i 1 z 0 i 1 ] . . . E [ x iM z 0 iM ] δ 0 = 0 or σ xy Σ xz δ 0 = 0 For δ 0 to be the unique solution requires the rank condition rank( Σ xz )= L
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Rema rk :Inthemode lw ithcommoncoe cients, some equations may be indi- vidually unidenti f ed but identi f ed withing the system.
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The moments g i ( δ ) are a MDS with E [ g i ( δ 0 ) g i ( δ 0 ) 0 ]= S = S 11 ··· S 1 M . . . . . . . . . S M 1 S MM E cient GMM estimation min δ J ( δ , ˆ S 1 )= n g n ( δ ) 0 ˆ S 1 g n ( δ ) g n ( δ S xy S xz δ S xy = S x 1 y 1 . . . S x M y M , S xz = S x 1 z 1 . . . S x M z M Straightforward algebra gives δ ( ˆ S 1 )=( S 0 xz ˆ S 1 S xz ) 1 S 0 xz ˆ S 1 S xy
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Special cases 1. 3SLS with common coe cients conditional homoskedasticity x i 1 = x i 2 = ··· = x iM = x i S = S 3 SLS = Σ E [ x i x 0 i ] 2. SUR with common coe cients (Random e f ects) conditional homoskedasticity x i 1 = x i 2 = = x iM = x i x i = union ( z i 1 ,..., z iM ) S = S SUR = Σ E [ z i z 0 i ]
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Random E f ects Estimator The SUR model with common coe cients is the giant regression y Mn × 1 = Z δ + ε E [ εε 0 ]= Σ I n Σ = σ 11 ··· σ 1 M . . . . . . . . . σ M 1 σ MM The FGLS estimator is the random e f ects (RE) estimator and has the form ˆ δ RE =( Z 0 ( ˆ Σ 1 I n ) Z ) 1 Z 0 ( ˆ Σ 1 I n ) y
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The elements of ˆ Σ are usually estimated using the pooled OLS estimator of the giant regression ˆ δ OLS =( Z 0 Z ) 1 Z 0 y and forming ˆ σ mh y m Z m ˆ δ OLS ) 0 ( y h Z h ˆ δ OLS ) /n
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Simplifying the RE estimator It turns out that the RE estimator may also be derived from an alternative representation of the giant regression.
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panelslides - Multiple Equation GMM with Common Coecients:...

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