2.4 Endogenous explanatory variables

Endogenous explanatory variables

Introduction

One of the most common problems complicating the research of an economist is created by the inclusion of endogenous variables as an explanatory variable. The variable on the left-hand-side of a regression is an endogenous variable; its level is determined by the levels of the explanatory variables—that is, the variables on the right-hand-side of the equation. In OLS we assume that the explanatory variables are independent of the error term. However, if the level of one of these explanatory variables is determined by the levels of the other variables in the model, that explanatory variable actually is an endogenous variable. In a nutshell the problem with having endogenous explanatory variables is that these endogenous variables cause the error term in the model to be correlated with the explanatory variables thus causing the OLS estimator to be biased. This problem is also known as simultaneous equation bias and it is a problem that is subtly different from sample selection bias. See "What is the difference between 'endogeneity' and 'sample selection bias"'?" for an excellent discussion of the difference between these two econometric problems.

In this module we explore both the statistical and algebraic issues raised by the inclusion of endogenous explanatory variables in a model. This introduction is too sketchy to give you a thorough understanding of the many problems raised by simultaneous equation bias. Hopefully, by the time you finish the module along with the problem set, you will have an least an intuitive understanding of the problem and will be able to recognize it when you come across the problem in your own research. If you think the model you are estimating may have simultaneous equation bias, you should seek the advice of an econometrician.

The statistical problem

Imagine we know with certainty that the following model fully describes the true state of the supply and demand for wheat. First, the demand for wheat in any year, q _t , is a function of the price of wheat, $p_t^w,$ the income of the average individual, I _t , and the price of corn, $p_t^c.$ Second, in any year the price of wheat is a function of the amount of wheat brought to market, q _t , and a weather index, W _t , that is positively related to the amount of wheat that is harvested. Third, the error terms in the supply and demand functions are due purely to measurement errors—that is, there are no omitted variables in the model. Thus, we have the following two equation model:

and

Supply:

We assume that the error terms each are normally distributed with a mean of zero and a constant variance. Moreover, we assume that the two error terms are independent of each other—that is, we are assuming that: