Endogeneity

Definition

A situation in which an explanatory variable is correlated with the error term. Instrumental Variable techniques are commonly used to reduce this problem

Omitted Variable Bias

Assume that the true model to be estimated is $y_i=\alpha +\beta x_i\gamma z_i+u_i$ but $z_i$ is omitted from the model. Then the model estimated is $y_i=\alpha +\beta x_i+{ϵ}_i$ where ${ϵ}_i=\gamma z_i+u_i$ Thus, the $z_i$ term has absorbed into the error term.

If $\mathrm{Cov}(x,z)\ne 0$, then the explanatory variable $x$ is correlated with the error term $ϵ$. Therefore, estimator will be inconsistent $\hat{\beta }\stackrel{P}{\nrightarrow }\beta$

Measurement Error

Suppose we can not observe accurate data $x_i$, but can only observe data with measurement error $x_i=x_i^{\ast }+{\nu }_i$. Then, the estimated $\hat{\beta }$ of the regression $y_i=\alpha +\beta x_i^{\ast }+{ϵ}_i$ does not converge to true $\beta$, and it has attenuation bias $\hat{\beta }\stackrel{P}{\to }\beta \frac{\mathrm{Var}(x^{\ast })}{\mathrm{Var}(x^{\ast })+\mathrm{Var}(ϵ)}$

Simultaneous Equations Bias

Suppose that two variables affect each other according to the following structural equation: $\{\begin{array}{l}{y}_i={\beta }_1{x}_i+{\gamma }_1{z}_i+u_i\\ z_i={\beta }_2{x}_i+{\gamma }_2{y}_i+v_i\end{array}$

Then, estimating either equation results in endogeneity. In the case of the first equation, $E(z_i,u_i)\ne 0$. Because $z_i=\frac{{\beta }_2+{\gamma }_2{\beta }_1}{1-{\gamma }_1{\gamma }_2}{x}_i+\frac{1}{1-{\gamma }_1{\gamma }_2}{v}_i+\frac{{\gamma }_2}{1-{\gamma }_1{\gamma }_2}{u}_i$ Therefore, estimator will be inconsistent $\hat{\beta }\stackrel{P}{\nrightarrow }\beta$

Lagged-Dependent Variable Models with Serial Correlation

Consider a model with lagged dependent variable $y_t={\beta }_1+{\beta }_2{y}_{t-1}+{\beta }_3{x}_t+e_t$ If model’s error $e_t$ follow the AR(1) process $e_t=\rho e_{t-1}+v_t$, the lagged dependent variable $y_{t-1}$ must be correlated with the error term $e_t$. Therefore, if $\rho \ne 0$, there will be a correlation between $y_{t-1}$ and $e_t$