Endogeneity (econometrics)

In econometrics, endogeneity broadly refers to situations in which an explanatory variable is correlated with the error term. 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; 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, unfortunately, ignored by researchers conducting non-experimental research and doing so precludes making policy recommendations. Instrumental variable techniques are commonly used to address 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.
Exogeneity versus endogeneity.
In a stochastic model, the notion of the "usual exogeneity", "sequential exogeneity", "strong/strict exogeneity" can be defined. Exogeneity is articulated in such a way that a variable or variables is exogenous for parameter formula_1. Even if a variable is exogenous for parameter formula_1, it might be endogenous for parameter formula_3.
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.
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 formula_5 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 formula_7 (thus, the formula_5 term has been absorbed into the error term).
If the correlation of formula_9 and formula_10 is not 0 and formula_10 separately affects formula_12 (meaning formula_13), then formula_9 is correlated with the error term formula_15.
Here, formula_9 is not exogenous for formula_1 and formula_3, since, given formula_9, the distribution of formula_12 depends not only on formula_1 and formula_3, but also on formula_10 and formula_24.
Measurement error.
Suppose that a perfect measure of an independent variable is impossible. That is, instead of observing formula_25, what is actually observed is formula_26 where formula_27 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 formula_30 and formula_31 depend on formula_27, they are correlated, so the OLS estimation of formula_3 will be biased downward.
Measurement error in the dependent variable, formula_34, 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, formula_37. Solving for formula_5 while assuming that formula_39 results in
Assuming that formula_30 and formula_42 are uncorrelated with formula_31,
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 formula_1, and "y" does not cause "x" in the Granger sense, then the variable "x" is strongly/strictly exogenous for the parameter formula_1.
Simultaneity.
Generally speaking, simultaneity occurs in the dynamic model just like in the example of static simultaneity above.