Simultaneous equations

Simultaneous equation Model

Suppose we have two equations of demand and supply for some good such as

(p) = + p + + (1)

(p) = + p + + (2)

and are additional observed determinants of demand and supply (e.g., income, other prices). Ignoring constants, and in equation (1) and (2), we impose equilibrium condition of (p)=(p)=: we rewrite (1) and (2) as

= p + (1)

= (2)

 

Simultaneous equation Model

• = p + (1)
• = (2)

This is a simultaneous equation model (SEM) since and are determined simultaneously. Both variables are determined within the model, so are endogenous.

• Is demand function, while (2) is the supply function: So is negative, while is positive; is the demand shock and is the supply shock.

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Structural Form

• (1) and (2) are structural in the sense that they are directly implied by economics theory.

• We assume E() = 0 (3)

So the structural errors are uncorrelated (orthogonal).

• Our goal is to estimate the structural coefficient β that measures the causal effect of one endogenous variable on the other endogenous variable

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Simultaneity Bias

• Plugging (1) into the right hand side of (2) leads to

=

……>

This indicates that ----cov

So structural model (1) suffers endogeneity issue (simultaneity bias)—the regresser in (1) is correlated with the error term. Consequently, OLS applied to (1) gives inconsistent and biased estimate. So does (2).

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Endogeneity

• Typically an economic theory implies SEM, so several variables are determined simultaneously within the model. Those variables are endogenous from the economics perspective

• We just show SEM suffers simultaneity bias. Therefore those variables are correlated with the error, so are endogenous from the econometrics perspective