An endogenous variable is something that "is determined within the system" (often called y1, y2, y3...)

An exogenous variable is a variable that "is determined outside of the system" (often called x1, x2, x3...)

(But the statistical description for exogeneity is that the x-variable is statistically independent of the error term (and endogeneity that it is dependent of the error term.))

Example:

y1 = b12*y2 + g10 + g11*x1 + err1 (1)

y2 = b21*y1 + g20 + g21*x2 + err1 (2)

You can notice that in equation (1) y1 is influenced by y2 and in equation (2) y2 is influenced by y1. So they are influencing each other. They are determined within the system.

Notice that err1 influences y1, and y1 influences y2 (from equation 2), so y2 in equation (1) is not statistically independent of the error term err1. If one tries to estimate each of the two equations with ordinary least square (OLS) then that will give biased and inconsistent results.

The above is called a structural model, or a structural equations model (SEM).

If you solve for the y:s so that:

y1 = d10 + d11*x1 + d12*x2 + err3 (3)

y2 = d20 + d21*x1 + d22*x2 + err4 (4)

Such a system is called a reduced system. Now the difficulty is how to, from the estimates of the reduced model (the "d:s"), get some estimates for the structural form. That is called the identification problem.

Sometimes the parameter in the structural form is using instrumental variables. That is variables that are independent of the error term but strongly correlated to the explanatory endogenous variable.

The funny thing is that in the attached file there is only one explanatory endogenous variable in one equation (like b21=0 in eqation (2)) (LOS explaines AE, but AE does not explain LOS). Such a system is a "Wold causal chain" and can be estimated with OLS. So the instrumental method estimation in in the paper is completely unnecessary, and I believe that it has caused some inefficiency.