Introduction
In econometrics, simultaneous equation models are used when two or more endogenous variables are determined together, influencing each other. Unlike single-equation models where one variable is dependent and the others are independent, simultaneous equation models have multiple equations with interdependent variables. One crucial concept in dealing with such models is identification.
What is Meant by Identification?
Identification refers to the ability to estimate the structural parameters of an equation in a simultaneous equation system uniquely and consistently using available data. If an equation is identified, it means we can uniquely estimate its parameters from the reduced form equations. If it is not identified, we cannot obtain unique estimates, and no inference can be made about the parameters.
Why Identification is Needed?
In simultaneous equation models, endogenous variables appear on both sides of equations. Hence, the ordinary least squares (OLS) method fails due to simultaneity bias. Before applying techniques like Two-Stage Least Squares (2SLS), we must determine whether an equation is identified.
Identification Rules
1. Order Condition
Let:
- K = total number of endogenous variables in the model
- M = number of exogenous variables excluded from the equation under consideration
The order condition for identification is:
M โฅ K โ 1
If this condition is:
- Satisfied with equality (M = K โ 1) โ Just-identified
- Satisfied with strict inequality (M > K โ 1) โ Over-identified
- Not satisfied (M < K โ 1) โ Not identified (under-identified)
2. Rank Condition
This is a more technical and sufficient condition for identification. It checks whether a unique solution exists by ensuring that the matrix of excluded variables has full rank. However, if the order condition fails, rank condition will also fail, so we primarily use the order condition for basic identification checks.
Given Simultaneous Equation Model
We are given:
Equation 1 (Demand): Qt = ฮฑ0 + ฮฑ1Pt + ฮฑ2Xt + u1t
Equation 2 (Supply): Qt = ฮฒ0 + ฮฒ1Pt + u2t
Where:
- Qt and Pt are endogenous (determined within the system)
- Xt is exogenous (e.g., income, weather, policy, etc.)
Total number of endogenous variables = K = 2 (Q and P)
Identification of the Demand Function
To identify the demand function, we check how many exogenous variables are excluded from this equation:
- Demand function includes Xt โ none excluded
- M = 0
Order condition: M โฅ K โ 1 โ 0 โฅ 1 โ Not satisfied
Result: Demand function is under-identified
Identification of the Supply Function
Supply function does not include Xt โ Xt is excluded from this equation
- M = 1
Order condition: M โฅ K โ 1 โ 1 โฅ 1 โ Satisfied
Result: Supply function is just-identified
Conclusion
In a simultaneous equation system, identification is critical before estimating parameters. It determines whether an equationโs parameters can be estimated uniquely. In the given model:
- The demand function is under-identified because no exogenous variable is excluded from it.
- The supply function is just-identified as it excludes one exogenous variable (Xt) and satisfies the order condition.
This identification status guides which equations can be estimated and which cannot using techniques like 2SLS or LIML.