and . Measurement error Suppose that a perfect measure of an independent variable is impossible. That is, instead of observing , what is actually observed is where is the measurement error or "noise". In this case, a model given by can be written in terms of observables and error terms as Since both and depend on , they are correlated, so the OLS estimation of will be biased downward. Measurement error in the dependent variable, , does not cause endogeneity, though it does increase the variance of the error term. Simultaneity Suppose that two variables are codetermined, with each affecting the other according to the following "structural" equations: Estimating either equation by itself results in endogeneity. In the case of the first structural equation, . Solving for while assuming that results in . Assuming that and are uncorrelated with , . Therefore, attempts at estimating either structural equation will be hampered by endogeneity. 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. See also Virtuous circle and vicious circle Heterogeneity Dependent and independent variables References Further reading External links by Mark Thoma Seth Godin's simple views on endogeneity Causality Estimation theory Econometric models