4.1 Introduction

A simultaneuos-equations model (SEQM) consists of several interdependent equations. Typically, these equations are not standard regression equations with an endogenous variable on the left-hand side and one or several exogenous regressors on the right-hand side that are independent of the error term. Rather, endogenous variables may also appear on the right hand side of the equations that comprise the SEQM.

But SEQMs are not merely a collection of equations with endogenous regressors. They are truly systems of equations in the sense that there are cross-equation relationships between the variables.

The well-known macroeconometric model of Klein (1950) is a good example to illustrate these points. The Klein's model consists of six equations, three statistical equations and three identities. The three statistical equations look like standard regression equations:

\begin{displaymath}\begin{array}{lclllllllll} C_t & = & \alpha_0 & + & \alpha_1 ...
..._2 Y_{t-1} & + & \gamma_3 A_{t} & + & \epsilon_{3t} \end{array}\end{displaymath} (4.1)

Here, the $ \alpha$s, $ \beta$s and $ \gamma$s are unknown regression coeffcients, $ \epsilon_{1t}$, $ \epsilon_{2t}$ and $ \epsilon_{3t}$ are unobservable error terms and all capital letters denote observable variables, whose meaning will be described below as necessary.

Klein's model is completed by the following three identities:

\begin{displaymath}\begin{array}{lclclcl} Y_t & = & C_t & + & I_t & + & G_t \\ P...
...t & - & W^p_t \\ K_t & = & K_{t-1} & + & I_t & & \\ \end{array}\end{displaymath} (4.2)

These identities neither include unknown coeffcients nor error terms. They hold ``by definition". Nonetheless, they are an integral part of the model.

The first equation of (4.2), for instance, says that total spending $ Y_t$ in an economy in year $ t$ is the sum of private consumption spending $ C_t$, investment spending $ I_t$ and government spending $ G_t$. This is an accounting relationship. Similarly, the second equation of (4.2) states that we obtain private profits $ P_t$ if we subtract from total spending $ Y_t$ indirect taxes $ T_t$ and the total wage bill of private enterprises $ W^p_t.$ These identities introduce interdependencies between the variables of the statistical equations (4.1).

Note, for instance, that $ C_t$ depends on $ W^p_t$ via the first equation of (4.1) and that $ W^p_t$ depends on $ Y_t$ via the third equation of (4.1). But $ Y_t$ depends on $ C_t$ though the first identity in (4.2) which implies that $ W^p_t$ depends on $ Y_t.$ In this way, the first and third equation of (4.1) are interdependent or simultaneous. This relatedness is not a result of some relationship between the error terms $ \epsilon_{1t}$ and $ \epsilon_{3t}$ of these equations. Rather, it is a result of the equations in (4.1) and (4.2) truly being a system of equations with various cross-equation relationships between the observable variables of the system. This simulateneity has important consequences if we want to consistently estimate the unknown coeffients in (4.1).