Simultaneous Equations Model

Abstract

Economists formulate models for consumption, production, investment, money demand and money supply, labor demand and labor supply to attempt to explain the workings of the economy. These behavioral equations are estimated equation by equation or jointly as a system of equations. These are known as simultaneous equations models. Much of today's economet-rics have been influenced and shaped by a group of economists and econometricians known as the Cowles Commission who worked together at the University of Chicago in the late 1940's, see Chapter 1. Simultaneous equations models had their genesis in economics during that period. Haavelmo's (1944) work emphasized the use of the probability approach to formulating econometric models. Koopmans and Marschak (1950) and Koopmans and Hood (1953) in two influential Cowles Commission monographs provided the appropriate statistical procedures for handling simultaneous equations models. In this chapter, we first give simple examples of simultaneous equations models and show why the least squares estimator is no longer appropriate. Next, we discuss the important problem of identification and give a simple necessary but not sufficient condition that helps check whether a specific equation is identified. Sections 11.2 and 11.3 give the estimation of a single and a system of equations using instrumental variable procedures. Section 11.4 gives a test of over-identification restrictions whereas, section 11.5 gives a Hausman specification test. Section 11.6 concludes with an empirical example. The Appendix revisits the identification problem and gives a necessary and sufficient condition for identification. 11.1.1 Simultaneous Bias Example 1: Consider a simple Keynesian model with no government C t = α + βY t + u t t = 1, 2,. .. , T (11.1) Y t = C t + I t (11.2) where C t denotes consumption, Y t denotes disposable income, and I t denotes autonomous investment. This is a system of two simultaneous equations, also known as structural equations with the second equation being an identity. The first equation can be estimated by OLS giving β OLS = T t=1 y t c t / T t=1 y 2 t and α OLS = ¯ C − β OLS ¯ Y (11.3) with y t and c t denoting Y t and C t in deviation form, i.e., y t = Y t − ¯ Y , and ¯ Y = T t=1 Y t /T. Since I t is autonomous, it is an exogenous variable determined outside the system, whereas C t and Y t are endogenous variables determined by the system. Let us solve for Y t and C t in terms of the constant and I t. The resulting two equations are known as the reduced form equations C t = α/(1 − β) + βI t (1 − β) + u t /(1 − β) (11.4) Y t = α/(1 − β) + I t /(1 − β) + u t /(1 − β) (11.5)