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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:
Consider the following demand and supply equations
We cannot tell which is which!
Both equations are UNIDENTIFIED or NOT IDENTIFIED, or
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
• 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
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.
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
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
1. Obtain the reduced form equations corresponding to (14)-(16). The
reduced forms turn out to be:
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.
- Summer '13
- Financial Markets