Random Coefficient. Models
Random coefficient models (RCMs) grew out of Zellner’s path-breaking article (1969) on aggregation and have undergone considerable modification over time.1 Initially, RCMs were primarily concerned with relaxing an assumption typically made by researchers who use classical models. This assumption is that there is a constant vector of coefficients relating the dependent and independent variables. Unfortunately, as Keynes (Moggridge, 1973, p. 286) long ago observed, the assumption of constant coefficients is unlikely to be a reasonable one. Recent work on RCMs has focused on also relaxing the following assumptions frequently made by researchers in econometrics: (i) the true functional forms of the systematic components of economic relationships (whether linear or nonlinear) are known; (ii) excluded variables are proxied through the use of an additive error term and, therefore, have means equal to zero and are independent of the included explanatory variables; and (iii) variables are not subject to measurement error.
The purpose of this chapter is to provide an accessible description of RCMs. The chapter is divided into six sections, including this introduction. Section 2 discusses some characteristics of what we characterize as first-generation models. Essentially, these models attempt to deal with the problem that arises because aggregate time series data and cross-section data on micro units are unlikely to have constant coefficients. Thus, the focus of this literature is on relaxing the constant coefficient assumption. Section 3 describes a generalized RCM whose origins are in work by Swamy and Mehta (1975) and Swamy and Tinsley (1980). The generalized RCM, which relaxes the constant coefficient assumption and all three of the restrictions mentioned in the preceding paragraph, is referred to as a second-generation model. It is based on the assumptions that
1. any variable that is not mismeasured is true;
2. any economic equation with the correct functional form, without any omitted explanatory variable, and without mismeasured variables is true.
Economic theories are true if and only if they deal with the true economic relationships. They cannot be tested unless we know how to estimate the true economic relationships. The generalized RCM corresponds to the underlying true economic relationship if each of its coefficients is interpreted as the sum of three parts: (i) a direct effect of the true value of an explanatory variable on the true value of an explained variable; (ii) an indirect effect (or omitted-variable bias) due to the fact that the true value of the explanatory variable affects the true values of excluded variables and the latter values, in turn, affect the true value of the explained variable; and (iii) an effect of mismeasuring the explanatory variable. A necessary condition that a specified model coincides with the underlying true economic relationship is that each of its coefficients has this interpretation. Importantly, the second-generation models satisfy the conditions for observability of stochastic laws, defined by Pratt and Schlaifer (1988), whenever they coincide with the underlying true economic relationships. In order to enhance the relevance of the following discussion, the presentation of RCMs is made in the context of a money-demand model that has been extensively applied in the literature. Also, to heighten accessibility, the section does not attempt rigorous exposition of some technical concepts (e. g. stochastic laws), but refers the interested reader to the relevant literature. Section 4 applies five criteria to validate RCMs. Section 5 uses an example of a money demand model to illustrate application of the second-generation RCMs. Section 6 concludes.