Endogeneity (econometrics)

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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 unfortunately, oftentimes 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 $\alpha$ . Even if a variable is exogenous for parameter $\alpha$ , it might be endogenous for parameter $\beta$ .

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 is correlated with both an independent variable in the model, and with the error term. (Equivalently, the omitted variable both affects the independent variable and separately affects the dependent variable.) Assume that the "true" model to be estimated is,

$y_{i}=\alpha +\beta x_{i}+\gamma z_{i}+u_{i}$

but we omit $z_{i}$  (perhaps because we don't have a measure for it) when we run our regression. $z_{i}$  will get absorbed by the error term and we will actually estimate,

$y_{i}=\alpha +\beta x_{i}+\varepsilon _{i}$       (where $\varepsilon _{i}=\gamma z_{i}+u_{i}$ )

If the correlation of $x$  and $z$  is not 0 and $z$  separately affects $y$  (meaning $\gamma \neq 0$ ), then $x$  is correlated with the error term $\varepsilon$ .

Here, x and 1 are not exogenous for α and β, since, given x and 1, the distribution of y depends not only on α and β, but also on z and gamma.

Measurement error

Suppose that we do not get a perfect measure of one of our independent variables. Imagine that instead of observing $x_{i}^{*}$  we observe $x_{i}=x_{i}^{*}+\nu _{i}$  where $\nu _{i}$  is the measurement "noise". In this case, a model given by

$y_{i}=\alpha +\beta x_{i}^{*}+\varepsilon _{i}$

is written in terms of observables and error terms as

{\begin{aligned}y_{i}&=\alpha +\beta (x_{i}-\nu _{i})+\varepsilon _{i}\\[3pt]y_{i}&=\alpha +\beta x_{i}+(\varepsilon _{i}-\beta \nu _{i})\\[3pt]y_{i}&=\alpha +\beta x_{i}+u_{i}\quad ({\text{where }}u_{i}=\varepsilon _{i}-\beta \nu _{i})\end{aligned}}

Since both $x_{i}$  and $u_{i}$  depend on $\nu _{i}$ , they are correlated, so the OLS estimation of $\beta$  will be biased downward. Measurement error in the dependent variable, however, 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. Suppose that there are two "structural" equations,

$y_{i}=\beta _{1}x_{i}+\gamma _{1}z_{i}+u_{i}$
$z_{i}=\beta _{2}x_{i}+\gamma _{2}y_{i}+v_{i}$

Estimating either equation by itself results in endogeneity. In the case of the first structural equation, $E(z_{i}u_{i})\neq 0$ . Solving for $z_{i}$  we get (assuming that $1-\gamma _{1}\gamma _{2}\neq 0$ ),

$z_{i}={\frac {\beta _{2}+\gamma _{2}\beta _{1}}{1-\gamma _{1}\gamma _{2}}}x_{i}+{\frac {1}{1-\gamma _{1}\gamma _{2}}}v_{i}+{\frac {\gamma _{2}}{1-\gamma _{1}\gamma _{2}}}u_{i}$

Assuming that $x_{i}$  and $v_{i}$  are uncorrelated with $u_{i}$ , we have that,

$\operatorname {E} (z_{i}u_{i})={\frac {\gamma _{2}}{1-\gamma _{1}\gamma _{2}}}\operatorname {E} (u_{i}u_{i})\neq 0$

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(xz) + u. If the variable x is sequential exogenous for parameter $\alpha$ , and y does not cause x in Granger sense, then the variable x is strong/strict exogenous for the parameter $\alpha$ .

Simultaneity

Generally speaking, simultaneity occurs in the dynamic model just like in the example of static simultaneity above.