multiequationgmmslides

multiequationgmmslides - Multiple Equation Linear GMM Eric...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Multiple Equation Linear GMM Eric Zivot November 23, 2009 Multiple Equation Linear GMM
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Notation y iM ,i = individual; M = equation There are M linear equations, y im = z 0 im (1 × L m ) δ m ( L m × 1) + ε im ,m =1 ,...,M ; i =1 ,...,n Remarks: 1. No a priori assumptions about cross equation error correlation 2. No cross equation parameter restrictions
Background image of page 2
Giant Regression Representation y 1 n × 1 . . . y M n × 1 = Z 1 n × L 1 . . . Z M n × L M δ 1 L 1 × 1 . . . δ M L M × 1 + ε 1 n × 1 . . . ε M n × 1 or y ¯ nM × 1 = Z ¯ nM × L δ L × 1 + e ¯ nM × 1 L = M X m =1 L m
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Main Issues and Questions: 1. Why not just estimate each equation separately? (a) Joint estimation may improve e ciency (b) Joint estimation is sensitive to misspeci f cation of individual equations 2. Theory may provide cross equation restrictions (a) Improve e ciency (b) test restrictions
Background image of page 4
Example: 2 equation wage equation LW i = φ 1 + β 1 S i + γ 1 IQ i + πEXPR i + ε i 1 ,L 1 =4 KWW i = φ 2 + β 2 S i + γ 2 IQ i + ε i 2 ,L 2 =3 z i 1 =( 1 ,S i ,IQ i ,EXPR i ) 0 z i 1 =( 1 ,S i ,IQ i ) 0 Note, ε i 1 and ε i 2 may be correlated (eg. due to common omitted variable ability)
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example: Panel data for wage equation LW 69 i = φ 1 + β 1 S i + γ 1 IQ i + π 1 EXPR i + ε i 1 , LW 80 i = φ 2 + β 2 S i + γ 2 IQ i + π 2 EXPR i + ε i 2 , If all coe cients do not change over time then φ 1 = φ 2 1 = β 2 1 = γ 2 1 = π 2 ε im = α i + η im α i = unobserved individual f xed e f ect
Background image of page 6
Instruments x im ( K m × 1) = instruments for m th equation E [ x im ε im ]= 0 ,m =1 ,...,M K = M X m =1 K m orthogonality conditions Note: We are not assuming cross-equation orthogonality conditions. That is, we may have E [ x im ε ik ] 6 =0 unless x im and x ik have variables in common.
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Example: 2 equation wage equation Assume 1. IQ is endogenous in both equations 2. MED is exogenous in both equations E [ MED i ε i 1 ]=0 ,E [ MED i ε i 2 ]=0 x i 1 = x i 2 =(1 ,S i ,EXPR i ,MED i ) 0
Background image of page 8
GMM Moment Conditions and Identi f cation De f ne δ ( L × 1) =( δ 0 1 ,..., δ 0 M ) 0 ,L = M X m =1 L m g i ( δ ) K × 1 = g i 1 ( δ 1 ) . . . g iM ( δ M ) = x i 1 ε i 1 . . . x iM ε iM = x i 1 ( y i 1 z 0 i 1 δ 1 ) . . . x iM ( y iM z 0 iM δ M ) Then there are K linear moment equations such that E [ g i ( δ )] = 0 E [ g i ( ˜ δ )] 6 = 0 for ˜ δ 6 = δ
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Now, E [ g i ( δ )] = E [ x i 1 y i 1 ] .
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 46

multiequationgmmslides - Multiple Equation Linear GMM Eric...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online