Simultaneous Equation Models – Recursive and Non-Recursive | UGC NET Economics – Notes

Introduction

In econometrics, a Simultaneous Equation Model (SEM) refers to a system of equations in which the dependent variable in one equation may appear as an explanatory variable in another equation of the same system. This is fundamentally different from single equation models, where causality flows in one direction only, from the independent variables to the dependent variable. In simultaneous equation systems, there exists joint dependence or mutual dependence among variables, meaning that certain variables are determined simultaneously within the system rather than sequentially. These variables that are determined within the system are called endogenous variables, while those variables whose values are determined outside the system are called exogenous variables. The presence of mutual dependence creates a significant econometric problem known as the simultaneity bias or simultaneous equation bias, which renders the Ordinary Least Squares (OLS) estimator biased and inconsistent when applied directly to individual equations of the system.

The classic example used to illustrate simultaneous equation models is the demand-supply model, where price and quantity are jointly and simultaneously determined by the interaction of demand and supply forces. Neither price nor quantity can logically be treated as purely independent or purely dependent, because both influence each other simultaneously.

The General Structural Form

A simultaneous equation model is typically written in what is called the structural form or structural equations. The structural form expresses each endogenous variable as a function of other endogenous variables, exogenous variables, and a stochastic disturbance term. A general structural form of a system with (G) equations can be represented as:

$$\beta_{11}Y_{1t} + \beta_{12}Y_{2t} + \dots + \beta_{1G}Y_{Gt} + \gamma_{11}X_{1t} + \dots + \gamma_{1K}X_{Kt} = u_{1t}$$

where (Y) terms represent endogenous variables, (X) terms represent exogenous (or predetermined) variables, \(\beta\) and \(\gamma\) are structural parameters/coefficients, and \(u_{1t}\) is the stochastic disturbance term of the first equation.

The system as a whole can be represented in matrix notation as:

$$B Y_t + \Gamma X_t = U_t$$

where (B) is the \((G \times G)\) matrix of coefficients of endogenous variables, \(\Gamma\) is the \((G \times K)\) matrix of coefficients of exogenous variables, \(Y_t\) is the vector of endogenous variables, \(X_t\) is the vector of exogenous variables, and \(U_t\) is the vector of disturbance terms.

From the structural form, one can derive the reduced form of the model by solving the structural equations so that each endogenous variable is expressed solely as a function of the predetermined (exogenous) variables and the disturbance terms. The reduced form is written as:

$$Y_t = \Pi X_t + V_t$$

where \(\Pi = -B^{-1}\Gamma\) is the matrix of reduced form coefficients, and \(V_t = B^{-1}U_t\) is the reduced form disturbance vector. The reduced form is extremely important in econometrics because it satisfies the classical assumptions of OLS (since exogenous variables are, by definition, uncorrelated with the disturbance term), and hence reduced form equations can be consistently estimated by OLS.