xbeta
Simultaneous equations model

Model

Let $y_{i}$ be a $M \times 1$ vector of endogenous variables and $x_{i}$ be a $K \times 1$ vector of exogenous covariates.

The structural model can be written as

\underset{M \times M}{Γ_{0}} \underset{M \times 1}{y_{i}} + \underset{M \times K}{B_{0}} \underset{K \times 1}{x_{i}} = \underset{M \times 1}{ε_{i}},

$E [x_{i} ε_{im}] = 0$ for all $m$ . $Γ_{0}$ and $B_{0}$ are the structural parameters.

If we define $ε_{i} \equiv (ε_{i 1}, ε_{i 2}, \dots, ε_{iM})^{⊤}$ , then we have

E [x_{i} ε_{i}^{⊤}] = \underset{K \times M}{0}

If $Γ_{0}$ is nonsingular, the corresponding reduced form model is

\begin{aligned} y_{i} & = - Γ_{0}^{- 1} B_{0} x_{i} + Γ_{0}^{- 1} ε_{i} \\ = Π_{0}^{⊤} x_{i} + ν_{i} \end{aligned}

The relationship between the structural and reduced form parameters is thus

\underset{M \times K}{Π_{0}^{⊤}} = - {\underset{M \times M}{Γ_{0}}}^{- 1} \underset{M \times K}{B_{0}}

and the error terms are related by

ν_{i} = \underset{M \times M}{Γ_{0}^{- 1}} \underset{M \times 1}{ε_{i}} .

Note that we still have $E [x_{i} ν_{i}^{⊤}] = 0$ .

Examples

Supply and Demand

\begin{matrix} q_{i} & = γ_{11} p_{i} + B_{11} + B_{12} a_{i} + ε_{i 1} \\ q_{i} & = γ_{21} p_{i} + B_{21} + B_{22} w_{i} + ε_{i 2} \end{matrix}

The first equation is the supply equation and the second is the demand demand equation. The exogenous variables are the wage $w_{i}$ , a supply shifter, and $a_{i}$ , a demand shifter. Price is endogenous to both equations.

category: Statistics, Econometrics

Created on March 27, 2009 16:47:47 by Jason Blevins (207.192.72.34)

xbeta Simultaneous equations model

Model

Examples

Supply and Demand

xbeta
Simultaneous equations model