a) What is simultaneity bias? Explain the conditions required for identification of parameters in a simultaneous equation model. b) In the following two-equation system check the identification status of both the equations. 𝑌1 =∝1+∝2 𝑌2 + 𝑢1 𝑌2 = 𝛽1 + 𝛽2𝑌1 + 𝛽3𝑍1 + 𝛽4𝑍2 + 𝑢2

Introduction

Simultaneity bias and identification are two fundamental concepts in econometrics, particularly when dealing with simultaneous equation models (SEMs). SEMs occur when more than one endogenous variable is determined within a system of equations, causing problems in estimation using ordinary least squares (OLS).

a) What is Simultaneity Bias?

Simultaneity bias occurs when an explanatory variable in a regression equation is simultaneously determined with the dependent variable. In such cases, the regressor is endogenous, meaning it is correlated with the error term. This violates a key OLS assumption, leading to biased and inconsistent estimates.

Example: Consider the following demand and supply equations:

Demand: Q = a – bP + u₁
Supply: Q = c + dP + u₂

Both Q and P are determined simultaneously in the market. If we try to estimate the demand equation using OLS, the price P is endogenous (correlated with u₁), leading to simultaneity bias.

Implications:

OLS becomes invalid due to endogeneity.
Estimates are not consistent or reliable.

Conditions Required for Identification

Before applying techniques like Two-Stage Least Squares (2SLS), we must check whether the structural equations are identified. This involves two key conditions:

1. Order Condition

This is a necessary condition. For an equation to be identified, the number of excluded exogenous variables from that equation must be at least equal to the number of endogenous variables minus one (K – 1).

Mathematically: M ≥ K – 1

Where:

K = number of endogenous variables in the system
M = number of excluded exogenous variables from the equation

2. Rank Condition

This is a sufficient condition. It requires that the matrix formed by the excluded variables must have full rank. In simpler terms, the excluded variables must influence the endogenous variable sufficiently to help identify the equation.

b) Check the Identification Status of the Given Two-Equation System

We are given the following system:

Equation 1: Y₁ = α₁ + α₂Y₂ + u₁
Equation 2: Y₂ = β₁ + β₂Y₁ + β₃Z₁ + β₄Z₂ + u₂

Endogenous variables: Y₁, Y₂ (K = 2)
Exogenous variables: Z₁, Z₂

Equation 1:

Includes Y₂ (endogenous)
Excludes Z₁, Z₂ (2 exogenous variables)
So, M = 2

Order condition: M ≥ K – 1 → 2 ≥ 1 → Satisfied

Conclusion: Equation 1 is over-identified.

Equation 2:

Includes Y₁ (endogenous), Z₁, Z₂ (exogenous)
No exogenous variable is excluded from Equation 2
M = 0

Order condition: M ≥ K – 1 → 0 ≥ 1 → Not satisfied

Conclusion: Equation 2 is under-identified

Conclusion

Simultaneity bias arises when explanatory variables are endogenous, leading to biased and inconsistent OLS estimators. To tackle this, structural equations must be identified, using the order and rank conditions. In the given system, the first equation is over-identified and can be estimated using instrumental variables methods. However, the second equation is under-identified, so it cannot be estimated uniquely with available data.