The OLS and other single equation models assume that variables treated as independent variables are exogenous. This implies a one-way relationship between independent and dependent variables. It is assumed that exogenous independent variables determine the dependent variable and its one-way causation. However, if the independent variables are also affected by the dependent variable, then, they are not truly exogenous. It means that there exists a two-way causation. In such a case, the OLS estimates will be biased and inconsistent which is referred to as the Simultaneous Equation Bias.
system of equations and Simultaneous equation bias
In the presence of two-way causation, we cannot apply a single equation approach. We need to use a system of equations to determine the multiple causations between variables. Furthermore, a multiple equations model includes separate equations for all the endogenous variables and these are known as simultaneous equation models. Moreover, these models can include both endogenous and exogenous variables, used to define various causal relationships.
Suppose, we have 2 variables X and Y such that they are jointly dependent, i.e. X=f(Y) and Y=f(X). Then, we have to use a system of equations instead of a single-equation approach like OLS. If we apply OLS to estimate such a relationship, the estimates will be biased as well as inconsistent because the error term will be related to the independent variables.
Let us assume the following simultaneous equation model:
Finally, we can see here that the X and error term μ are related to each other. This is because the error term appears on the right side of the equation where X is the dependent variable. This means that the covariance of X and μ is not zero which violates one of the assumptions of OLS because E(μX)≠0.
It can be proved that the covariance of X and μ is not zero in this case. If we estimate the first equation using OLS, the coefficients will be biased. If B2 lies somewhere between 0 and 1, the coefficient B2 will be overestimated and the constant B1 will be underestimated.
Hence, if we use OLS to estimate an equation or relationship that belongs to a system of equations, the estimates are biased. This is known as the simultaneous equation bias. To estimate such models, we need to employ other techniques like Indirect Least Squares (ILS), Two-Stage Least Squares (2SLS), Three-Stage Least Squares (3SLS) or Maximum Likelihood techniques (LIML and FIML).