Partial Distributional Policy Effects Under Endogeneity

Abstract

Rothe (Econometrica 80, 2269–2301 2012) introduces a new class of parameters called ‘Partial Distributional Policy Effects’ (PPE) to estimate the impact on the marginal distribution of an outcome variable due to a change in the unconditional distribution of a single covariate. Since the strict exogeneity assumption of all covariates makes this approach less applicable in empirical research, we propose the identification of the PPEs for a continuous endogenous explanatory variable using the control variable approach developed by Imbens and Newey (Econometrica 77, 1481–1512 2009). We also apply this proposed control variable PPE approach to investigate how poverty and black-white racial wage gaps contribute to the steep increase in the incarceration rate of black men over the period 1980-2010. Our control variable PPE estimates suggest that although the fall in the racial wage gap does not explain the changes in the incarceration rate of black men, changes in the poverty rate contribute about one-third of the steep increase in the incarceration rate at the upper-tail of the distribution.

Appendix

Proposition 2.

Suppose that in the triangular nonseparable model represented by Eqs. 2.1 and 2.2 , the regularity conditions stated in Imbens and Newey (2009) hold, then wehave${F_{Y}^{H}}=E\left (F_{Y|X,V}\left (y|H^{-1}(F_{X|V}(X|V)), X_{2}\right )\right )$andthe FPPE$\tilde { \alpha }(\tilde {\nu }, X_{1}| V, \tilde {H})$isidentified for any functional$\tilde {\nu }$.

Proof

For convenience, we restate all the necessaryregularity conditions we need fromImbens and Newey (2009) andRothe (2012).□

• A1: In the triangular model of Eqs. 2.3 and 3.1 introduced in Section 2, suppose (i) (independence) of the error terms of both equations, η and ε are independent of the instrumental variables Z and (ii) (monotonicity) ε is a continuously distributed scalar with CDF that is strictly increasing in the support of ε.

• A2: Let the endogenous variable X₁ have a continuous CDF, H(x₁) which is monotonically increasing in the entire support of X₁ .

• A3: Common Support: For all $X_{1}\in \mathcal {X}_{1}$, the support of V conditional on X₁ equals the support of V.

• A4: The unknown structural function $m(\overset {.}{})$ is continuously differentiable of order d and the support of the derivatives are uniformly bounded in x and z where d ≥ 2.

• A5: Let W = [X, V ] then Var(Y W) is bounded.

Using the nonseparable structural model introduced in Section 2, and the definition of cumulative distribution function, we get

$$ F_{Y}^{\tilde{H}} (y) = Pr \left( m\left( \tilde{X}_{H}, \eta \right) \leq y \right) $$

(4.1)

where $\tilde {X}_{H} = \left [\tilde {X}_{1} \hspace {0.15 cm} X_{2} \right ]$, $\tilde {X}_{1} = X_{1}| V$, V is the control variable defined as $V = F_{X_{1}|Z} (X_{1},Z) = F_{\varepsilon }(\varepsilon )$ and Z is the vector of instruments. Using the full independence and monotonicity assumptions from A1, Imbens and Newey (2009) show that there exist a control variable V such that $\tilde {X}_{1}$ and η are independent.

The counterfactual distribution function of Y, $\tilde {F}_{Y}^{H} (y)$ is defined as a scenario where ceteris paribus holding the copula of the $\tilde {X}_{1}$ and X₂ fixed, while changing the marginal distribution of $\tilde {X}_{1}$. This can be formalized using the probability integral transformation theorem which implies that $\tilde {X}_{1} = Q_{\tilde {X}_{1}} \left (U_{1}\right )$, where $Q_{\tilde {X}_{1}}$ is the quantile function of $\tilde {X}_{1}$ and U₁ ∼ U(0,1). Therefore, in this setup, Rothe (2012)’s counterfactual experiment amounts to changing $Q_{\tilde {X}_{1}}$ to another quantile function $\tilde {H}(\overset {.}{})$ while keeping the joint distribution of the rank variables U = (U₁, U₂,..., U_k) fixed.

Under assumptions A2 and A3, $\tilde {H}(x_{1}|v)$ is also monotonically increasing. Therefore, $\tilde {H}^{-1}(\overset {.}{})$ also exists and is identified in the support of (X₁, V). Again, by using the probability integral transformation theorem, we rewrite $\tilde {X}_{1}$ in terms of their unconditional quantile function:

$$ \tilde{X}_{1}=\tilde{H}^{-1}\left( U_{1}\right). $$

(4.2)

Thus, the covariate vector $\tilde {X}_{H}$ can be rewritten as $\tilde {X}_{H}=\left [\tilde {H}^{-1}\left (U_{1}\right )\hspace {0.25cm}X_{2}\right ]$ and hence we get,

$$\begin{array}{@{}rcl@{}} F_{Y}^{\tilde{H}}(y)&=&Pr\left( m\left( \tilde{X}_{H},\eta \right)\leq y\hspace{ 0.15cm}|\hspace{0.15cm}\tilde{X}_{1}=x_{1}|v,X_{2}=x_{2}\right) \\ &=&\int Pr\left( m\left( \tilde{H}^{-1}\left( U_{1}\right) ,x_{2},\eta \right)\right.\\ &\leq&\left. \vphantom{\tilde{H}^{-1}}y\hspace{0.15cm}|\hspace{0.15cm}U_{1}=u_{1},X_{2}=x_{2} \right)dF_{U_{1}X_{2}}(u_{1},x_{2}) \end{array} $$

(4.3)

Under the assumption A2 and A3, Eq. 4.2 shows that there exists a one to one correspondence between $\tilde {X_{1}}$ and U₁ over the range of $\tilde {H}^{-1}(\overset {.}{})$. Hence, following Rothe (2012) we get,

$$\begin{array}{@{}rcl@{}} F_{Y}^{\tilde{H}} (y) &=& \int Pr \left( m\left( \tilde{x}_{1}, x_{2}, \eta \right) \leq y \hspace{0.15 cm} | \hspace{0.15 cm} Q_{\tilde{X}_{1}} (U_{1})\right.\\ &=&\left. \tilde{x}_{1}, X_{2} = x_{2} \right) dF_{U_{1}X_{2}} \left( \tilde{H}_{1} (\tilde{x}_{1}), x_{2} \right) \\ &=& \int Pr \left( m\left( \tilde{x}_{1}, x_{2}, \eta \right) \leq y \hspace{0.15 cm} | \hspace{0.15 cm} \tilde{X}_{1}\right.\\ &=&\left. \vphantom{\tilde{X}_{1}}\tilde{x}_{1}, X_{2} = x_{2} \right) dF_{U_{1}X_{2}} \left( \tilde{H}_{1} (\tilde{x}_{1}), x_{2} \right) \\ &=& \int F_{Y | \tilde{X}} \left( y, \tilde{X}_{1} , X_{2} \right) dF_{U_{1}X_{2}} \left( \tilde{H}_{1} (\tilde{x}_{1}), x_{2} \right) \end{array} $$

(4.4)

The regularity conditions A4 and A5 imply that $m\left (\tilde {X}_{1}, X_{2}, \eta \right ) = m\left (X_{1}|V,\right .$X₂, η) is contained in the support of (X, V). Then by Imbens and Newey (2009) Theorem 9, $F_{Y | X, V}(\overset {.}{})$ is identified. Therefore, from Eq. 3.1 we get,

$$\begin{array}{@{}rcl@{}} && F_{Y}^{\tilde{H}} (y) = \int F_{Y | X, V} \left( y, \tilde{H}_{1}^{-1} (U_{1}) , X_{2} \right) dF_{U_{1}X_{2}} (u_{1},x_{2}) \\ && \hspace{1.25 cm} = E\left( F_{Y|X, V} \left( y, \tilde{H}^{-1}(U_{1}), X_{2} \right) \right) \end{array} $$

(4.5)

Again, under the assumption A2 and A3, Eq. 2 shows that there exists a one to one correspondence between $\tilde {X_{1}}$ and U₁ over the range of $\tilde {H}^{-1}(\overset {.}{})$. Hence we get,

$$ F_{Y}^{\tilde{H}} (y) = E\left( F_{Y|X, V} \left( y, \tilde{H}^{-1}\left( F_{X_{1} | V}(X_{1}| V) \right), X_{2} \right) \right) $$

(4.6)

The proof of the second part is similar to part 1. Using the above steps it can be easily shown that F_{Y |X, V} is identified over the area of integration as shown in the right hand side of Eq. 5. Hence, both $F_{Y}^{\tilde {H}} (y)$ and $\tilde {\nu }_{H} = \nu \left (F_{Y}^{\tilde {H}}(y)\right )$ are identified.

Appendix Fig. 1 scatter plot of observed and predicted black-white racial wage gap and observed and predicted poverty rate