Transcript Simultaneous Equation Models

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 
• Structure has 2 equations and so does
reduced form.
• Prove that OLS is inconsistent
• Successively weaker assumptions
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
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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
• 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
• 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
• 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
• 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
• 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
• 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
• 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
• 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*)