The Application of a System of Simultaneous Equations to an Innovation Diffusion Model

The use of systems of simultaneous eqtuations to describe social behavior, extensive in econometrics, appears to have promise for explaining behavior in other social sciences as well. Two problems that may arise with the use of such equations are (1) identification, and (2) estimation. This paper considers the problem of identification and extends the methodology to include a description of the Theil-Basmann method of estimation. A specific model concerning the diffusion of technical innovations is formulated and tested. The results suggest that variables in the model represent an interdependeint system. S ysteiis of interdependent simultaneous equations have been used extensively to describe social behavior in the field of economics, or, more specifically, in econometrics. A system of equations in the social sciences is usually considered a nonexperimental model, i.e., a model for which data came from a nonexperimental setting such as timeseries or cross-sectional studies. Two problems which arise when using a system of simultaneous equations in a nonexperimental setting are (1) identification and (2) estimaThe authors wish to thankl Dr. Norman McKowni, Director of Institutional Research at Oregon State University, for his critical reading and comments on earlier drafts of this paper. Published as Technical Paper 2459, Oregon Agricultural Experiment Stationi. This content downloaded from 157.55.39.18 on Sat, 06 Aug 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms AN INNOVATION DIFFUSION MODEL 183 tion. The problem of identification has been given extensive treatment in the econometric literature and has recently been introduced into the literature of sociology by Blalock' and Boudon.2 A logical next step is to consider the estimation problem in the context of building belhavioral theory. Since econometrics has pioneered in the development of estimation procedures for determining coefficients in systems of simultaneous equations, a reasonable starting place for other behavioral sciences is to consider these developments and their applicability to their own problems. The purpose of this paper is to show the results of applying the Theil-Basmann method (two-stage leastsquares) to a system of simultaneous equations involving both sociological and economiic variables. In this paper, first, the identificatio-n problem is reviewed. Then the logic of the twostage estimation procedure is presented. Finally, a specific model concerning the diffusion of technical innovations is formulated and tested. REVIEW OF IDENTIFICATION PROBLEM Blalock3 has shown how the identificationi problem arises in attempting to specify a system of equations (model) to describe status inconsistency. One of the models had the following three equations: Y = b,Xj + bX2? + b3T + ce W = b4Z + e10 Z =f(Xl kX2) where Y is votes for liberal candidates; X1 is income; X2 is education; W represents the strain factor; Z stands for status inconsistency; bl, b2, b3, and b4 are coefficients to be estimated from some data; and k is a known parameter. The error terms, ey and e,, are assumed to account for variables which are excluded from the equations and are assumed uncorrelated with one another. In order for the coefficients in a given equation to be called identifiable, Blalock states that "the number of variables excluded from this equation must be at least equal to one less thanl the number of equations."4 Applying this counting rule to each of the equations in the above systemii, we find: (1) the first equation has only the variable Z excluded, thus it would need another variable such as a V in the second equation to identify it; (2) the second equation is identified since it has two variables, X1 and X2 excluded, which is equal to one less than the number of equations; (3) the third equation is an exact mathematical relation without an error term and hence the identification question is not relevant. The first equation is said to be underidentified and the secon(d, exactly identified. Coefficients in individual equations in a system can be estimated by ordinary leastsquares regression techniques, provided the equation is just identified. If a model containis one or more underidentified equations, theni certain alterations must be made before the coefficients can be estimated. Blalock ha.s noted that by bringing into the second equation another exogenous variable (i.e., one that is not to be explained within the system), one that would not logically appear in the first equation, the equation for Y can be identified. This adding of a variable would not affect the idenitification of the second equation. Another case of identification is where an equation is overidentified.5 For example, if the first equation contained another variable such as age, X3, which did not appear in the equation for W, then the second equation would be overidentified. By the counting rule the second equation would have three variables excluded, which is equal to the number of equations. The case of overidentification would seem to be the most prevalent case in behavioral science theory construction; that is, one is more likely to think of more variables to include in 1 Hubert M. Blalock, Jr., "The Identification Problem and Theory Building: The Case of Status Inconsistency," Ai;i-erican Sociological Review, 31 (February 1966), pp. 52-61. 2 Raymond Boudon, "A Method of Linear Causal Analysis: Dependence Analysis," Amiierican Sociological Reviezw, 30 (June 1965), pp. 365-374. 3 Blalock, loc. cit. 4 An equivalent form of this counting rule can be expressed as "the number of endogenous variables appearing in any given equation cannot be greater than one more than the number of exogenous variables left out of this equation." 5 Identified means that the equation is either exactly or overidentified. Not identified means the equation is underidentified. This content downloaded from 157.55.39.18 on Sat, 06 Aug 2016 05:41:48 UTC All use subject to http://about.jstor.org/terms