Simultaneous Equations Models

Abstract

The estimation of simultaneous equations models (SEMs) is of great importance in econometrics [34, 35, 62, 64, 96, 124, 130, 132, 149]. The most commonly used estimation procedures are the Three Stage Least-Squares (3SLS) procedure and the computationally expensive maximum likelihood procedure [33, 60, 97, 106, 107, 119, 120, 143, 153]. Here the methods used for solving SURE models will be extended to 3SLS estimation of SEMs. The ith structural equation of the SEM can be written as

$$ y_i = X_i \beta _i + Y_i \gamma _i + u_i ,\,i = 1, \ldots ,G, $$

((6.1))

where, for the ith structural equation, y _i ∈ ℜ^T is the dependent vector, X _i is the T x k _i matrix of full column rank of exogenous variables, y _i is the T x g _i matrix of other included endogenous variables, β _i and γ _i are the structural parameters to be estimated, and U _i ∈ ℜ^T are the disturbance terms. for $W_i \equiv \left( {X_i \,Y_i } \right),\,\delta _i^T \equiv \left( {\beta _i^T \,\gamma _i^T } \right)\,and U = \left( {u_1 \ldots u_G } \right)$ the stacked system of the structural equations can be written as

$$ \left( {\begin{array}{*{20}{c}} {{{y}_{1}}} \\ {{{y}_{2}}} \\ \vdots \\ {{{y}_{G}}} \\ \end{array} } \right) = \left( {\begin{array}{*{20}{c}} {{{W}_{1}}} & \, & \, & \, \\ \, & {{{W}_{2}}} & \, & \, \\ \, & \, & \ddots & \, \\ \, & \, & \, & {{{W}_{G}}} \\ \end{array} } \right)\left( {\begin{array}{*{20}{c}} {{{\delta }_{1}}} \\ {{{\delta }_{2}}} \\ \vdots \\ {{{\delta }_{G}}} \\ \end{array} } \right) + \left( {\begin{array}{*{20}{c}} {{{u}_{1}}} \\ {{{u}_{2}}} \\ \vdots \\ {{{u}_{G}}} \\ \end{array} } \right) $$

((6.2))

or as

$$ vec\left( Y \right) = \left( {\mathop {\mathop \oplus \limits_{i = 1} }\limits^G WS_i } \right)vec\left( {\left\{ {\delta _i } \right\}_G } \right) + vec\left( U \right), $$

((6.3))

where $W \equiv \left( {X\,Y} \right) \in \Re ^{T \times \left( {K + G} \right)}$, X is a T x K matrix of all predetermined variables, $Y \equiv \left( {y_1 \ldots y_G } \right)$, S _i is a (K + G) x (K _i + g _i ) selector matrix such that WS _i = W _i (i = 1,…, G), and $vec\left( U \right) \equiv \left( {u_1^T \ldots u_G^T } \right)^T$. The disturbance vector vec(U) has zero mean and variance-covariance matrix ∑⊗I _T where ∑ is a G x G non-negative definite matrix. It is assumed that e _i = k _i + g _i ≤ K, that is, all structural equations are identifiable. The notation used here is consistent with that employed in the previous chapter and, similarly, the direct sum $\oplus _{i = 1}^G$ and set operator {•} _G will be abbreviated by ⊕ _i and {•}, respectively.