31_SimultB

# 31_SimultB - Simultaneous equations II Wooldridge 16...

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Simultaneous equations - II Wooldridge 16, Gujarati 19-20 73-261 Y. Kryukov CMU, Tepper School of Business November 10, 2010 Y. Kryukov (CMU) 73-261: Simult. eq-s Nov-10 1 / 18

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Overview Topic 3.1 Simultaneous equations More than 2 equations GRETL Y. Kryukov (CMU) 73-261: Simult. eq-s Nov-10 2 / 18
it excludes an exogenous variable that is used in the other equation Example : ( D ) : q = α 1 p + β 10 + u 1 ( S ) : q = α 2 p + β 20 + β 21 Cost + u 2 Demand equation excludes Cost Y. Kryukov (CMU) 73-261: Simult. eq-s Nov-10 3 / 18

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ILS: structural and reduced form Structural form ( D ) : q = α 1 p + β 10 + u 1 ( S ) : q = α 2 p + β 20 + β 21 Cost + u 2 Reduced form: q = π 11 + π 12 Cost + v 1 p = π 21 + π 22 Cost + v 2 where: π 11 = α 1 β 20 α 2 β 10 α 1 α 2 , π 12 = α 1 β 21 α 1 α 2 , π 21 = β 20 β 10 α 1 α 2 , π 22 = β 21 α 1 α 2 . (*) Y. Kryukov (CMU) 73-261: Simult. eq-s Nov-10 4 / 18
ILS: estimation Estimate of reduced coe¢ cients is unbiased: ˆ π 11 , ˆ π 12 , ˆ π 21 , ˆ π 22 Want to back out structural coe¢ cients, i.e. solve (*) for: α 1 , β 10 , α 2 , β 20 , β 21 ) Cannot pin down all unknowns ) Will have in±nitely many solutions This is an "under-identi±ed" system of equations Y. Kryukov (CMU) 73-261: Simult. eq-s Nov-10 5 / 18

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Can solve for demand coe¢ cients: ˆ α 1 = ˆ π 12 / ˆ π 22 , ˆ β 10 = ˆ π 11 ˆ α 1 ˆ π 21 Cannot solve for supply coe¢ cients:
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31_SimultB - Simultaneous equations II Wooldridge 16...

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