### Application of Simultaneous Equation in Finance Research [chapter]

Carl R. Chen, Cheng Few Lee
2010 Handbook of Quantitative Finance and Risk Management
A. Introduction Empirical finance research often employs a single equation for estimation and testing. However, single equation rarely happens in the economic or financial theory. Using OLS method to estimate equation(s) which should otherwise be treated as a simultaneous equation system is likely to produce biased and inconsistent parameter estimators. To illustrate, let's start with a simple Keynesian consumption function specified as the follow: Where t C is the consumption expenditure at
more » ... e t, t Y is the national income at time t, t I is the investment expenditure at time t, which is assumed fixed at 0 I , and t µ is the stochastic disturbance term at time t. Equation (1.1) is the consumption function; Equation (1.2) is the equilibrium condition (national income accounting identity); and equation (1.3) is the investment function. Some of the variables in the model are endogenous, others are exogenous. For example, t C and t Y are endogenous, meaning they are determined within the model. On the other hand, t I is the exogenous variable, which is not determined in the model, hence not correlated with t µ . A simultaneous equation bias arises when OLS is applied to estimate the consumption function because t Y is correlated with the disturbance term t µ , which violates the OLS assumption that independent variables are orthogonal to the disturbance term. To see this, through a series of substitutions, we can obtain the reduced form equations from the structural equations (1.