Simultaneous Equation Models class notes by Prof. Vinod all rights reserved

1. Simultaneous Equation Models class notes by Prof. Vinodall rights reserved

2. Marshallian Demand Supply • No equilibrium unless we consider both equations. Estimate simultaneously • Two equation macro equilibrium. MPC overestimated even asymptotically T • Structure has 2 equations and so does reduced form. • Prove that OLS is inconsistent • Successively weaker assumptions

3. If not OLS what? Reduced Form? • ILS, 2SLS, 3SLS,LIML, FIML, Reduced Rank regression (see T.W.Anderson, 2000) • Rewrite the 2 equation Macro model without the intercept in matrix notation. • Structure is Y +XB =U, post multiply • Y1 +XB1 =U1 • Y=X+V change notation

4. Variable Types • Jointly dependent (prices, quantities) (Y,C) • Exogenous (rainfall, GNP) (Investment) • Assumptions of SimEqModels • Included Endog mj, Excluded Endog mj* • Included Exog Kj, Excluded exog Kj* • Rewrite the structure one eq at a time • j-th eq. Is Identified if Kj* > mj

5. Identification • Demand eq. identified if it has a unique variable (GNP) excluded in the supply eq. • Supply eq. is identified it it has another unique variable (rainfall) excluded from the demand equation. • Formally identification means going from reduced form to the structure. (in general impossible since too many unknowns)

6. Proper Identification catches the imposter models • Greene Ed4 p.665 has imposter model where one simply post-multiplies the structure by a nonsingular matrix F • YF +XBF =UF. The reduced form is still the same: FF1 cancels out as identity mtx. • YFF11 +XBFF11 =UFF11 Y=X+V (rank and order conditions)

7. Algebra of Identification • We want to estimate structural parameters  and B from reduced form . Start with the definition of reduced form B 1= split them in 3 parts and derive 21 =211 Note small  and big  are different, conformable matrix multiplication is involved. Star means excluded variable, but we need to keep them with zero coefficients to do the algebra. Rank of 21 =min(K1*, m1) has to be > m1, i.e. we must exclude enough variables (rainfall absent in Demand eq. Is order condition)

8. Identification (nonsample info), Recursive Models • Instead of exclusion restriction (coeff=zero) some coefficients may be fixed at some specific and this too can help identification. • Wold recursive models y1=f(x), y2=f(y1,x) y3=f(y1,y2,x), y4=f(y1,y2,y3,x). OLS is OK on one equation at a time (this is called limited information estimation)

9. Instrumental variable estimation • Instruments must be uncorrelated with errors and correlated with the variables being instrumented out! 2SLS uses predicted Y as instrument. If the weighting matrix is (X’X)-1 then GenMethM=2SLS • Limited information methods (one eq at a time) versus full information methods (all together simultaneously in a GLS scheme)

10. Maximum Likelihood estimation • This involves least variance ratio, the smallest eigenvalue (characteristic root) in the limited info case (LIML) and if all equations are written together it is FIML. • Full info formulation often involves the Kronecker product of matrices.

11. k-class estimator • Insert a k in the 2SLS partitioned matrix in the top left corner before V’V in the 2 by 2 matrix and the same k before V’v in the top of the 2 by 1 vector [2SLS has k=1] • Let the k take different values to define a class of estimators. Even LIML becomes a special case k=eigenvalue, for OLS, k=0

12. Testing overidentifying restrictions • Hausman test of specification of x as exog • Null hyp: x is exog and both d and d* are consistent but only d* is asymptotically effi. • Under Alternative hyp x is actually endog, d is consistent and d* is inconsistent (rquires an arbitrary choice of some eq. Which does not contain x It is quadratic form in (d-d*)