In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term.
[1] The distinction between endogenous and exogenous variables originated in simultaneous equations models, where one separates variables whose values are determined by the model from variables which are predetermined.
[a][2] Ignoring simultaneity in the estimation leads to biased estimates as it violates the exogeneity assumption of the Gauss–Markov theorem.
The problem of endogeneity is often ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations.
[3] Instrumental variable techniques are commonly used to mitigate this problem.
Besides simultaneity, correlation between explanatory variables and the error term can arise when an unobserved or omitted variable is confounding both independent and dependent variables, or when independent variables are measured with error.
Even if a variable is exogenous for parameter
When the explanatory variables are not stochastic, then they are strong exogenous for all the parameters.
If the independent variable is correlated with the error term in a regression model then the estimate of the regression coefficient in an ordinary least squares (OLS) regression is biased; however if the correlation is not contemporaneous, then the coefficient estimate may still be consistent.
There are many methods of correcting the bias, including instrumental variable regression and Heckman selection correction.
The following are some common sources of endogeneity.
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.
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).
is correlated with the error term
Suppose that a perfect measure of an independent variable is impossible.
is the measurement error or "noise".
, they are correlated, so the OLS estimation of
Measurement error in the dependent variable,
, does not cause endogeneity, though it does increase the variance of the error term.
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,
, Therefore, attempts at estimating either structural equation will be hampered by endogeneity.
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.
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
Generally speaking, simultaneity occurs in the dynamic model just like in the example of static simultaneity above.