Endogeneity (econometrics)

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. (Equivalently, the omitted variable affects the independent variable and separately affects the dependent variable.)

Assume that the "true" model to be estimated is

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

but ${\displaystyle z_{i}}$ is omitted from the regression model (perhaps because there is no way to measure it directly). Then the model that is actually estimated is

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

where ${\displaystyle \varepsilon _{i}=\gamma z_{i}+u_{i}}$ (thus, the ${\displaystyle z_{i}}$ term has been absorbed into the error term).

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

Here, ${\displaystyle x}$ is not exogenous for ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, since, given ${\displaystyle x}$, the distribution of ${\displaystyle y}$ depends not only on ${\displaystyle \alpha }$ and ${\displaystyle \beta }$, but also on ${\displaystyle z}$ and ${\displaystyle \gamma }$.

Suppose that a perfect measure of an independent variable is impossible. That is, instead of observing ${\displaystyle x_{i}^{*}}$, what is actually observed is ${\displaystyle x_{i}=x_{i}^{*}+\nu _{i}}$ where ${\displaystyle \nu _{i}}$ is the measurement error or "noise". In this case, a model given by

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

can be written in terms of observables and error terms as

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

Since both ${\displaystyle x_{i}}$ and ${\displaystyle u_{i}}$ depend on ${\displaystyle \nu _{i}}$, they are correlated, so the OLS estimation of ${\displaystyle \beta }$ will be biased downward.

Measurement error in the dependent variable, ${\displaystyle y_{i}}$, 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:

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

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

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

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

Assuming that ${\displaystyle x_{i}}$ and ${\displaystyle \gamma _{i}}$ are uncorrelated with ${\displaystyle u_{i}}$,

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

Therefore, attempts at estimating either structural equation will be hampered by endogeneity.

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 ${\displaystyle \alpha }$, and y does not cause x in the Granger sense, then the variable x is strongly/strictly exogenous for the parameter ${\displaystyle \alpha }$.

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