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Simultaneous Equation Models class notes by Prof. Vinod all rights reserved. Marshallian Demand Supply . No equilibrium unless we consider both equations. Estimate simultaneously Two equation macro equilibrium. MPC overestimated even asymptotically T  

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Marshallian demand supply l.jpg
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


If not ols what reduced form l.jpg
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


Variable types l.jpg
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


Identification l.jpg
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)


Proper identification catches the imposter models l.jpg
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)


Algebra of identification l.jpg
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)


Identification nonsample info recursive models l.jpg
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)


Instrumental variable estimation l.jpg
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)


Maximum likelihood estimation l.jpg
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.


K class estimator l.jpg
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


Testing overidentifying restrictions l.jpg
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*)