When the explanatory variables are not stochastic, then they are strong exogenous for all the parameters.
Static models
The following are some common sources of endogeneity.
Omitted variable
In this case, the endogeneity comes from an uncontrolled confounding variable, a variable that is correlated with both the independent variable in the model and with the error term. (Equivalently, the omitted variable affects the independent variable and separately affects the dependent variable.)
Assume that the "true" model to be estimated is
but is omitted from the regression model (perhaps because there is no way to measure it directly).
Then the model that is actually estimated is
where (thus, the term has been absorbed into the error term).
If the correlation of and is not 0 and separately affects (meaning ), then is correlated with the error term .
Here, is not exogenous for and , since, given , the distribution of depends not only on and , but also on and .
Dynamic models
The endogeneity problem is particularly relevant in the context of time series analysis of causal processes. It is common for some factors within a causal system to be dependent for their value in period t on the values of other factors in the causal system in period t − 1. Suppose that the level of pest infestation is independent of all other factors within a given period, but is influenced by the level of rainfall and fertilizer in the preceding period. In this instance it would be correct to say that infestation is exogenous within the period, but endogenous over time.
Let the model be y = f(x, z) + u. If the variable x is sequential exogenous for parameter , and y does not cause x in the Granger sense, then the variable x is strongly/strictly exogenous for the parameter .
Simultaneity
Generally speaking, simultaneity occurs in the dynamic model just like in the example of static simultaneity above.