## Models

A casual look at the published empirical work in business and economics will reveal that many economic relationships are of the single-equation type. That is why we devoted the first three parts of this book to the discussion of single-equation regression models. In such models, one variable (the dependent variable Y) is expressed as a linear function of one or more other variables (the explanatory variables, the X's). In such models an implicit assumption is that the cause-and-effect relationship, if any, between Y and the X's is unidirectional: The explanatory variables are the cause and the dependent variable is the effect.

However, there are situations where there is a two-way flow of influence among economic variables; that is, one economic variable affects another economic variable(s) and is, in turn, affected by it (them). Thus, in the regression of money M on the rate of interest r, the single-equation methodology assumes implicitly that the rate of interest is fixed (say, by the Federal Reserve System) and tries to find out the response of money demanded to the changes in the level of the interest rate. But what happens if the rate of interest depends on the demand for money? In this case, the conditional regression analysis made in this book thus far may not be appropriate because now M depends on r and r depends on M. Thus, we need to consider two equations, one relating M to r and another relating r to M. And this leads us to consider simultaneous-equation models, models in which there is more than one regression equation, one for each interdependent variable.

In Part IV we present a very elementary and often heuristic introduction to the complex subject of simultaneous-equation models, the details being left for the references.

In Chapter 18, we provide several examples of simultaneous-equation models and show why the method of ordinary least squares considered previously is generally inapplicable to estimate the parameters of each of the equations in the model.

In Chapter 19, we consider the so-called identification problem. If in a system of simultaneous equations containing two or more equations it is not possible to obtain numerical values of each parameter in each equation because the equations are observationally indistinguishable, or look too much like one another, then we have the identification problem. Thus, in the regression of quantity Q on price P, is the resulting equation a demand function or a supply function, for Q and P enter into both functions? Therefore, if we have data on Q and P only and no other information, it will be difficult if not impossible to identify the regression as the demand or supply function. It is essential to resolve the identification problem before we proceed to estimation because if we do not know what we are estimating, estimation per se is meaningless. In Chapter 19 we offer various methods of solving the identification problem.

In Chapter 20, we consider several estimation methods that are designed specifically for estimating the simultaneous-equation models and consider their merits and limitations.