1. Simultaneous Equation Models class notes by Prof. Vinod all 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 m j, Excluded Endog m j * Included Exog K j, Excluded exog K j * Rewrite the structure one eq at a time j-th eq. Is Identified if K j * > m j

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(K 1 *, m 1 ) has to be > m 1, 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 y 1 =f(x), y 2 =f(y 1,x) y 3 =f(y 1,y 2,x), y 4 =f(y 1,y 2,y 3,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*)