Endogeneity

Paul Schrimpf

2022-11-16

Reading

• Required: Song (2021) chapter 11

\[ \def\Er{{\mathrm{E}}} \def\En{{\mathbb{En}}} \def\cov{{\mathrm{Cov}}} \def\var{{\mathrm{Var}}} \def\R{{\mathbb{R}}} \newcommand\norm[1]{\left\lVert#1\right\rVert} \def\rank{{\mathrm{rank}}} \newcommand{\inpr}{ \overset{p^*_{\scriptscriptstyle n}}{\longrightarrow}} \def\inprob{{\,{\buildrel p \over \rightarrow}\,}} \def\indist{\,{\buildrel d \over \rightarrow}\,} \DeclareMathOperator*{\plim}{plim} \]

Endogeneity

Omitted Variables

• Desired model \[ Y_i = X_i'\beta_0 + W_i'\gamma_0 + u_i \] Assume \(\Er[u] = \Er[Xu] = \Er[Wu] = 0\)
• Estimated model \[ Y_i = X_i'\beta + u_i \]
• What is \(\plim \hat{\beta}\)?

Omitted Variables

• \(\plim \hat{\beta} \inprob \beta_0 + \Er[X_i X_i']^{-1} \Er[X_i W_i'] \gamma_0\)

Omitted Variables

• If \(\gamma_0 = 0\), what is variance of \(\hat{\beta}\) when \(W\) is and is not included in the model?

Errors in Variables

• See problem set 6

Simultaneity Bias

• Equilibrium conditions often lead to variables that are simultaneously determined
• Demand and supply: \[ \begin{align*} Q_i^D & = P_i \beta_D + X_D'\gamma_D + u_{D,i} \\ Q_i^S & = P_i \beta_S + X_S'\gamma_S + u_{S,i} \\ Q_i^S & = Q_i^D \end{align*} \]

Simultaneity Bias

• Structural equations: (demand and inverse supply): \[ \begin{align*} Q_i & = P_i \beta_D + X_D'\gamma_D + u_{D,i} \\ P_i & = Q_i \frac{1}{\beta_S} - X_S'\gamma_D\frac{1}{\beta_S} - u_{S,i}\frac{1}{\beta_S} \\ \end{align*} \]
• Reduced form: \[ \begin{align*} Q_i = & \frac{\beta_D}{\beta_D - \beta_S} \left( -X_{D,i}' \gamma_D + X_{S,i}'\gamma_S - u_{D,i} + u_{S,i} \right) + X_{D,i}'\gamma_D + u_{D,i} \\ P_i = & \frac{1}{\beta_D - \beta_S}\left(-X_{D,i}' \gamma_D + X_{S,i}'\gamma_S - u_{D,i} + u_{S,i} \right) \end{align*} \]

References

Song, Kyunchul. 2021. “Introduction to Econometrics.”