multiequationgmmslides

# multiequationgmmslides - Multiple Equation Linear GMM Eric...

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

Multiple Equation Linear GMM Eric Zivot November 23, 2009 Multiple Equation Linear GMM

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

View Full Document
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
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

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

View Full Document
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
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)

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

View Full Document
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
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.

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

View Full Document
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
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

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

View Full Document
Now, E [ g i ( δ )] = E [ x i 1 y i 1 ] .
This is the end of the preview. Sign up to access the rest of the document.

## multiequationgmmslides - Multiple Equation Linear GMM Eric...

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

View Full Document
Ask a homework question - tutors are online