In Parts I to III of this text we were concerned exclusively with single-equation models, i.e., models in which there was a single dependent variable Y and one or more explanatory variables, the X's. In such models the emphasis was on estimating and/or predicting the average value of Y conditional upon the fixed values of the X variables. The cause-and-effect relationship, if any, in such models therefore ran from the X's to the Y.

But in many situations, such a one-way or unidirectional cause-and-effect relationship is not meaningful. This occurs if Y is determined by the X's, and some of the X's are, in turn, determined by Y. In short, there is a two-way, or simultaneous, relationship between Y and (some of) the X's, which makes the distinction between dependent and explanatory variables of dubious value. It is better to lump together a set of variables that can be determined simultaneously by the remaining set of variablesâ€”precisely what is done in simultaneous-equation models. In such models there is more than one equationâ€”one for each of the mutually, or jointly, dependent or endogenous variables.1 And unlike the single-equation models, in the

'In the context of the simultaneous-equation models, the jointly dependent variables are called endogenous variables and the variables that are truly nonstochastic or can be so regarded are called the exogenous, or predetermined, variables. (More on this in Chap. 19.)

718 PART FOUR: SIMULTANEOUS-EQUATION MODELS

simultaneous-equation models one may not estimate the parameters of a single equation without taking into account information provided by other equations in the system.

What happens if the parameters of each equation are estimated by applying, say, the method of OLS, disregarding other equations in the system? Recall that one of the crucial assumptions of the method of OLS is that the explanatory X variables are either nonstochastic or, if stochastic (random), are distributed independently of the stochastic disturbance term. If neither of these conditions is met, then, as shown later, the least-squares estimators are not only biased but also inconsistent; that is, as the sample size increases indefinitely, the estimators do not converge to their true (population) values. Thus, in the following hypothetical system of equations,2

where Yi and Y2 are mutually dependent, or endogenous, variables and Xi is an exogenous variable and where ui and u2 are the stochastic disturbance terms, the variables Yi and Y2 are both stochastic. Therefore, unless it can be shown that the stochastic explanatory variable Y2 in (18.1.1) is distributed independently of u1 and the stochastic explanatory variable Y1 in (18.1.2) is distributed independently of u2, application of the classical OLS to these equations individually will lead to inconsistent estimates.

In the remainder of this chapter we give a few examples of simultaneous-equation models and show the bias involved in the direct application of the least-squares method to such models. After discussing the so-called identification problem in Chapter 19, in Chapter 20 we discuss some of the special methods developed to handle the simultaneous-equation models.

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