Simultaneous equations

The Structure and the Reduced Form: Identification

• Consider following structural equations:
• (p) = + p + + +
• (p) = + p + +
• Estimating reduced form equation we can not recover these structural coefficients. Why?

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The Structure and the Reduced Form: Identification

• This system is under-identified, knowledge of the reduced form does not reveal the structure. Just show this.
• Derive reduced form for and . Then

+ +

= + +

Estimating these reduced equations we have 6 parameters. But in structural equations, we have eight parameters: Estimating reduced form equation we can not recover these structural coefficients. Then we say this system is under-identified; knowledge of the reduced form does not reveal the structure

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Estimating Simultaneous Equations Models

• OLS on the structural equations doesn’t work. What does?
• Indirect least squares (ILS), instrumental variables (IV), and two-stage least squares (2SLS)

We have introduced Indirect Least Squares(ILS) estimation in the previous slides.

Estimating Simultaneous Equations Models

• Two-Stage Least Squares (2SLS)

Consider a simple system with one exogenous variable:

(p) = + + (5)

• (p) = + + +

Let’s derive reduced form for .

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Estimating Simultaneous Equations Models

• +

….> + +

Regressing on = + ……(6)

Equation 5: (p) = + +

Use equation (5) to write + + [ ] ….(6)’

Note that is a linear function of , which is uncorrelated with . Also,

is necessarily uncorrelated with (− ). Thus, OLS estimates of in (6)’ are consistent – OLS regression on fitted values from a first stage is the 2SLS second stage.

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Estimating Simultaneous Equations Models

Over-Identified Models