Consider a system of simultaneous equations where each observation gives one point as its solution and which is shifted by error terms. Even if there are many observations, the cloud of points does not allow us to understand the model. In order to identify the relationship between
(both endogenous variables), we need to have at least one exogenous variable (
in our case), so we simulate several values of
. If there are no error terms
in the model, we would get the blue points that reflect the true relationship between
. If the error terms
do not suffer from endogeneity, we could use ordinary least squares to fit the model of the form
The problem is that such is not the case. To simplify matters and make the visualization more appealing, we keep the error term in the unobserved equation
by default (the bottom checkbox changes this setting).
We may express
algebraically in reduced form from the system of simultaneous equations and see that it depends on
As a result, each observation shifts
to the value of
horizontally. At the very same time,
shifts to the value of
changes from observation to observation randomly, we see such a pattern on the plot:
We may divide the vertical to horizontal shift and find the slope of shifts, which is the same for all observations as
cancels out. Indeed:
This effect is shown on the plot. Use the
slider to see the shift effect. Use the checkboxes to turn on and off different plot elements and study their behavior.
In order to estimate the model, you can use either instrumental variable (IV) or control function (CF) approaches. For instance,
can be regressed on
, which is a natural instrument for this toy example. In practice, finding and substantiating relevant instruments is a challenging task.
 C. Dougherty, Introduction to Econometrics
, 5th ed., Oxford, UK: Oxford University Press, 2016 pp. 345–350.