Econometric Methods and Applications II Chapter 2: Simultaneous equations Econometric Methods and Applications II, Chapter 2, Slide 1
2.1 Introduction An example motivating the problem of simultaneous equations: Econometric Methods and Applications II, Chapter 2, Slide 2
2.1 Introduction Econometric Methods and Applications II, Chapter 2, Slide 3
2.1 Introduction Econometric Methods and Applications II, Chapter 2, Slide 4
2.1 Introduction Econometric Methods and Applications II, Chapter 2, Slide 5
2.2 IV assumptions IV assumptions: (2.1) Econometric Methods and Applications II, Chapter 2, Slide 6
2.2 IV assumptions Moment conditions and matrix notation: K Econometric Methods and Applications II, Chapter 2, Slide 7
2.2 IV assumptions Matrix notation (continued): Econometric Methods and Applications II, Chapter 2, Slide 8
2.3 IV estimation based on GMM Generalized Methods of Moments (GMM-particular estimation principle): (2.2) (2.2) (2.2) (2.3) This GMM estimator is consistent, but not efficient! Econometric Methods and Applications II, Chapter 2, Slide 9
2.3 IV estimation based on GMM Efficient GMM: (2.4) Econometric Methods and Applications II, Chapter 2, Slide 10
2.3 IV estimation based on GMM Consistency: 2.1: Econometric Methods and Applications II, Chapter 2, Slide 11
2.3 IV estimation based on GMM Asymptotic normality: 2.2: s Econometric Methods and Applications II, Chapter 2, Slide 12
2.3 IV estimation based on GMM The system 2SLS estimator: 2.4 Econometric Methods and Applications II, Chapter 2, Slide 13
2.3 IV estimation based on GMM GMM with asymptotically optimal weighting matrix: 2.3 Econometric Methods and Applications II, Chapter 2, Slide 14
2.3 IV estimation based on GMM Practical implementation of GMM with optimal weighting matrix: Econometric Methods and Applications II, Chapter 2, Slide 15
2.3 IV estimation based on GMM Three stage least squares estimation (3SLS): 2.1 2.2 Econometric Methods and Applications II, Chapter 2, Slide 16
2.3 IV estimation based on GMM Equivalence of optimal GMM and 3SLS: Econometric Methods and Applications II, Chapter 2, Slide 17
2.3 IV estimation based on GMM Conditions for the efficiency of 3SLS: 2.4 Econometric Methods and Applications II, Chapter 2, Slide 18
2.4 Identification in the classical simultaneous equation model Notation and exclusion restriction: Econometric Methods and Applications II, Chapter 2, Slide 19
2.4 Identification in the classical simultaneous equation model Identification: A B A A Econometric Methods and Applications II, Chapter 2, Slide 20
2.4 Identification in the classical simultaneous equation model Structural and reduced form: Econometric Methods and Applications II, Chapter 2, Slide 21
2.4 Identification in the classical simultaneous equation model General system: Let M denote the total number of exogenous variables z in the system. Econometric Methods and Applications II, Chapter 2, Slide 22
2.4 Identification in the classical simultaneous equation model Condition for identification: 2.5 Econometric Methods and Applications II, Chapter 2, Slide 23
2.4 Identification in the classical simultaneous equation model Example with graphical illustration: q: quantity, p: price, i: income 0 1 pt 2it ut 0 1 pt vt ( 1 1 ) pt 0 0 2it ( vt ut ) pt 0 0 2 v ut, it t 1 1 1 1 1 1 0 0 2 E ( pt qt, i ) pt it, 1 1 1 1 qt 0 1 pt vt. Econometric Methods and Applications II, Chapter 2, Slide 24
2.5 Identification of triangular systems based on control functions Triangular system: y1 1 y2 z1 1 u1, y2 z 2 u2. y1 is G1 1, y2 is G2 1. - A triangular system is similar to a recursive system with simultaneity, with the exception that at least one or several of the endogenous variables are only determined by exogenous variables (but not by other endogenous variables). - Still, they appear in sets of equations determining other endogenous variables. - E.g., y2 is only determined by the exogenous variables z, but determines y1. Assumptions: E (u1 z ) 0, E (u2 z ) 0, E (u1 u2, z ) E (u1 u2 ). - A sufficient condition for E(u1 u2, z) E(u1 u2 ) is that (u1, u2 ) is independent of z. - Then, E ( y u, z ) y z E (u u, z ) 1 2 1 2 1 1 1 2 1 y2 z1 1 E (u1 u2 ) 1 y2 z1 1 H1 (u2 ) if we knew H1 (u2 ) and u2, we could estimate 1 by a system estimation method, such as system OLS or FGLS. Econometric Methods and Applications II, Chapter 2, Slide 25
2.5 Identification of triangular systems based on control functions Estimation: - Assume that H1 (u2 ) E (u1 u2 ) 1u2 (linearity) - Then, in a first stage, we can estimate y2 z 2 u2 by a standard system method to obtain ˆ2 and the estimated residuals, uˆi 2. - In a second stage, one estimates the structural equation of interest by plugging in the estimated first stage residuals: yi1 1 yi 2 zi1 1 1uˆi 2 ei, with ei being the estimation error due to the estimation of ˆ2. - System OLS or FGLS might be used for estimation, but the asymptotic variance has to be adjusted due to the first stage estimation (similar to two stage least squares). Econometric Methods and Applications II, Chapter 2, Slide 26