Parameters in the structural equations

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Unformatted text preview: κ. 9 Identification of Simultaneous Equations Can We Retrieve the Original Coefficients from the π’s? Short answer: sometimes. • • • • As well as simultaneity, we sometimes encounter another problem: identification. Consider the following demand and supply equations Supply equation (12) Demand equation (13) We cannot tell which is which! Both equations are UNIDENTIFIED or NOT IDENTIFIED, or UNDERIDENTIFIED. The problem is that we do not have enough information from the equations to estimate 4 parameters. Notice that we would not have had this problem with equations (4) and (5) since they have different exogenous variables. 10 What Determines whether an Equation is Identified or not? • We could have three possible situations: 1. An equation is unidentified · like (12) or (13) · we cannot get the structural coefficients from the reduced form estimates 2. An equation is exactly identified · e.g. (4) or (5) · can get unique structural form coefficient estimates 3. An equation is over-identified · Example given later · More than one set of structural coefficients could be obtained from the reduced form. 11 What Determines whether an Equation is Identified or not? (cont’d) • • How do we tell if an equation is identified or not? There are two conditions we could look at: - The order condition - is a necessary but not sufficient condition for an equation to be identified. - The rank condition - is a necessary and sufficient condition for identification. We specify the structural equations in a matrix form and consider the rank of a coefficient matrix. 12 Simultaneous Equations Bias (cont’d) • • Statement of the Order Condition (from Ramanathan 1995, pp.666) Let G denote the number of structural equations. An equation is just identified if the number of variables excluded from an equation is G-1. If more than G-1 are absent, it is over-identified. If less than G-1 are absent, it is not identified. Example • In the following system of equations, the Y’s are endogenous, while the X’s are exogenous. Determine whether each equation is over-, under-, or justidentified. (14)-(16) 13 Simultaneous Equations Bias (cont’d) Solution G = 3; If # excluded variables = 2, the eqn is just identified If # excluded variables > 2, the eqn is over-identified If # excluded variables < 2, the eqn is not identified Equation 14: Not identified Equation 15: Just identified Equation 16: Over-identified 14 Tests for Exogeneity • • • How do we tell whether variables really need to be treated as endogenous or not? Consider again equations (14)-(16). Equation (14) contains Y2 and Y3 - but do we really need equations for them? We can formally test this using a Hausman test, which is calculated as follows: 1. Obtain the reduced form equations corresponding to (14)-(16). The reduced forms turn out to be: (17)-(19) Estimate the reduced form equations (17)-(19) using OLS, and obtain the fitted values, 15 Tests for Exogeneity (cont’d) 2. Ru...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.

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