2.4 Endogenous explanatory variables  (Page 2/10)

Finally, we assume that income, the price of corn, and the weather index are non-stochastic variables—i.e., these variables are independent of the two error terms. Clearly, the price of wheat and the quantity of wheat are stochastic variables. A stochastic variable is a random variable —i.e., a variable whose value is determined as a result of a process involving an uncertain outcome.

What we have here is an ideal model in the sense that we know and can measure all of the variables in the model. The model as written has two endogenous variables— q _t and $p_t^w$ —and three exogenous variables— $I_t,$ $p_t^c,$ and $W_t.$ Equations (1) and (2) are known as structural equations . What makes this model useful for our purposes is that there is an endogenous explanatory variable in each of the two structural equations.

What we ultimately want to know is if we can use ordinary least squares (OLS) to obtain unbiased estimates of the parameters in Equations (1) and (2). One of the assumptions of OLS is that each of the explanatory variables are independent of the error term, ${\epsilon }_t;$ if this assumption is violated, OLS will produce biased estimates of the slope parameters. Thus, what we need to do is see if the error term in each equation is independent of the endogenous variable on the right-hand-side of that equation. That is, we want to see if $E\left({\epsilon }_t{p}_t^w\right)=0$ and $E\left({\eta }_t{q}_t\right)=0.$

It is convenient in answering our question to use the two structural equations to find what are known as the reduced form equations —that is, one equation for each endogenous variable in which the endogenous variable is written as a function solely of exogenous variables and error terms. We can find the reduce form equations by solving the structural equations simultaneously for the endogenous variables. Substituting (2) into (1), we get:

$$q_t={\alpha }_0+{\alpha }_1\left({\beta }_0+{\beta }_1{q}_t+{\beta }_2{W}_t+{\eta }_t\right)+{\alpha }_2{I}_t+{\alpha }_3{p}_t^c+{\epsilon }_t$$

$$q_t={\alpha }_0+{\alpha }_1{\beta }_0+{\alpha }_1{\beta }_1{q}_t+{\alpha }_1{\beta }_2{W}_t+{\alpha }_1{\eta }_t+{\alpha }_2{I}_t+{\alpha }_3{p}_t^c+{\epsilon }_t$$

$$q_t-{\alpha }_1{\beta }_1{q}_t=\left({\alpha }_0+{\alpha }_1{\beta }_0\right)+{\alpha }_1{\beta }_2{W}_t+{\alpha }_2{I}_t+{\alpha }_3{p}_t^c+\left({\epsilon }_t+{\alpha }_1{\eta }_t\right)$$

or

Substituting (1) into (2) yields:

$$p_t^w={\beta }_0+{\beta }_1\left({\alpha }_0+{\alpha }_1{p}_t^w+{\alpha }_2{I}_t+{\alpha }_3{p}_t^c+{\epsilon }_t\right)+{\beta }_2{W}_t+{\eta }_t$$

$$p_t^w={\beta }_0+{\beta }_1{\alpha }_0+{\alpha }_1{\beta }_1{p}_t^w+{\alpha }_2{\beta }_1{I}_t+{\alpha }_3{\beta }_1{p}_t^c+{\beta }_1{\epsilon }_t+{\beta }_2{W}_t+{\eta }_t$$

or

Equations (4) and (5) are the reduced form equations for this model. We can use them to calculate $E\left({\epsilon }_t{p}_t^w\right)=0$ and $E\left({\eta }_t{q}_t\right)=0.$ In particular,

$$E\left({\epsilon }_t{p}_t^w\right)=E\left[{\epsilon }_t\left(\frac{{\beta }_0+{\beta }_1{\alpha }_0}{1-{\alpha }_1{\beta }_1}+\frac{{\alpha }_2{\beta }_1}{1-{\alpha }_1{\beta }_1}{I}_t+\frac{{\alpha }_3{\beta }_1}{1-{\alpha }_1{\beta }_1}{p}_t^c+\frac{{\beta }_2}{1-{\alpha }_1{\beta }_1}{W}_t+\frac{{\beta }_1{\epsilon }_t+{\eta }_t}{1-{\alpha }_1{\beta }_1}\right)\right]$$

$$E\left({\epsilon }_t{p}_t^w\right)=E\left[{\epsilon }_t\left(\frac{{\beta }_0+{\beta }_1{\alpha }_0}{1-{\alpha }_1{\beta }_1}+\frac{{\alpha }_2{\beta }_1}{1-{\alpha }_1{\beta }_1}{I}_t+\frac{{\alpha }_3{\beta }_1}{1-{\alpha }_1{\beta }_1}{p}_t^c+\frac{{\beta }_2}{1-{\alpha }_1{\beta }_1}{W}_t\right)+{\epsilon }_t\left(\frac{{\beta }_1{\epsilon }_t+{\eta }_t}{1-{\alpha }_1{\beta }_1}\right)\right]$$

or

Factoring out the non-stochastic terms from the expected value operators gives:

$$E\left({\epsilon }_t{p}_t^w\right)=\left(\frac{{\beta }_0+{\beta }_1{\alpha }_0}{1-{\alpha }_1{\beta }_1}+\frac{{\alpha }_2{\beta }_1}{1-{\alpha }_1{\beta }_1}{I}_t+\frac{{\alpha }_3{\beta }_1}{1-{\alpha }_1{\beta }_1}{p}_t^c+\frac{{\beta }_2}{1-{\alpha }_1{\beta }_1}{W}_t\right)E\left[{\epsilon }_t\right]+\frac{{\beta }_{1}E\left({\epsilon }_t^2\right)}{1-{\alpha }_1{\beta }_1}+\frac{E\left({\eta }_t{\epsilon }_t\right)}{1-{\alpha }_1{\beta }_1}.$$

Moreover, by assumption $E\left({\epsilon }_t\right)=0,$ $E\left({\eta }_t{\epsilon }_t\right)=0,$ and $E\left({\epsilon }_t^2\right)={\sigma }_{\epsilon }^2.$ Thus, we get:

A similar analysis yields:

Equations (6) and (7) are what create the endogeneity problem (or simultaneous equation bias )—using OLS to estimate the parameters of equations that have an endogenous variable as an explanatory variable yields biased estimates of the unknown parameters. Figure 1 illustrates the endogeneity problem. In this figure we have demand and supply equations that have both risen due to changes in exogenous variables. What the researcher observes are two (red) points: (1) the intersection of the old demand and supply curves and (2) the intersection of the new demand and supply curves.