Composite quantile regression estimation of linear error-in-variable models using instrumental variables

Abstract

In this paper, we develop a composite quantile regression estimator of linear error-in-variable models based on instrumental variables. The proposed estimator is consistent and asymptotically normal under fairly general assumptions. It neither requires the measurement errors and the regression errors to have the same variance nor to belong to the same location-scale symmetric distribution. The simulation results show that the proposed method generates unbiased and efficient estimates for different types of the distributions of the regression errors in finite samples. An application to real data collected from the survey of identical twins to study the economic returns to schooling is also provided.

Appendix

Proof of Theorem 1

Let $Q_n=\sqrt{n}(\hat{\beta }^\mathrm{CQR}-\beta )$ and $R_{nk}=\sqrt{n}(\hat{b}_{\tau _k}-b_{\tau _k})$. Then $(R_{n1}, \ldots , R_{nK}, Q_n)$ is the minimizer of the following criterion

$$\begin{aligned} \mathcal {L}_n=\sum _{k=1}^{K}\sum _{i=1}^{n}\left[ \rho _{\tau _k}\left( \epsilon _i^*-b_{\tau _k}-r_i- \frac{R_{n,k}+(\Gamma Z_i)^TQ_n}{\sqrt{n}}\right) -\rho _{\tau _k}\left( \epsilon _i^*-b_{\tau _k}-r_i\right) \right] . \end{aligned}$$

where $\epsilon _i^*=Y_i-(\Gamma Z_i)^T\beta =\epsilon _i+\delta _i^T\beta$ and $r_i=(\hat{\Gamma }Z_i-\Gamma Z_i)^T\beta$.

By applying the identity in Knight (1998), we have

$$\begin{aligned} \mathcal {L}_n =\sum _{k=1}^{K}\sum _{i=1}^{n}\frac{R_{nk}+(\Gamma Z_i)^TQ_n}{\sqrt{n}}\eta _{ik}+B_n, \end{aligned}$$

where $B_n=\sum \nolimits _{k=1}^{K}\sum \nolimits _{i=1}^{n}\int _{r_i}^{r_i+\left\{ R_{n,k}+(\Gamma Z_i)^TQ_n\right\} /\sqrt{n}}\left[ I(\epsilon _i^*\le b_{\tau _k}+t)-I(\epsilon _i^*\le b_{\tau _k})\right] dt$.

First, we consider $B_n$.

$$\begin{aligned} \text {E}(B_n|Z_i)= & {} \sum \limits _{k=1}^K \sum _{i=1}^{n}\int _{r_i}^{r_i+\left\{ R_{nk}+(\Gamma Z_i)^TQ_n\right\} /\sqrt{n}}t f(b_{\tau _k})\left\{ 1+o(1)\right\} dt \\= & {} \frac{1}{2}f(b_{\tau _k})R_{nk}^2+\frac{1}{2}Q_n^T \left\{ \frac{1}{n}\sum \limits _{k=1}^{K}\sum \limits _{i=1}^{n}f(b_{\tau _k})\Gamma Z_iZ_i^T\Gamma ^T\right\} Q_n\\&\quad +\frac{1}{\sqrt{n}}\sum \limits _{k=1}^K \sum _{i=1}^{n}f(b_{\tau _k})r_i\left\{ R_{nk}+(\Gamma Z_i)^TQ_n\right\} +o(1). \end{aligned}$$

Define $R_n=B_n-E(B_n|Z_i)$. It can be shown that $R_n=o_p(1)$. Therefore, we have

$$\begin{aligned} \mathcal {L}_n= & {} \sum _{k=1}^{K}\sum _{i=1}^{n}\frac{R_{nk}+(\Gamma Z_i)^TQ_n}{\sqrt{n}}\eta _{ik}+E(B_n|Z_i)+R_n\\= & {} \frac{1}{2}f(b_{\tau _k})R_{nk}^2+\frac{1}{\sqrt{n}}\sum \limits _{k=1}^K \sum _{i=1}^{n}\left\{ \eta _{ik}+f(b_{\tau _k})r_i\right\} R_{nk}+\frac{1}{2}Q_n^TS_nQ_n\\&\quad +\frac{1}{\sqrt{n}}\sum \limits _{k=1}^K \sum _{i=1}^{n}\left\{ \eta _{ik}+f(b_{\tau _k})r_i\right\} (\Gamma Z_i)^TQ_n+o_p(1), \end{aligned}$$

where $S_n=\frac{1}{n}\sum \nolimits _{k=1}^{K}\sum \nolimits _{i=1}^{n}f(b_{\tau _k})\Gamma Z_iZ_i^T\Gamma ^T$. It can been shown that $S_n=E(S_n)+o_p(1)=\sum \nolimits _{k=1}^{K}f(b_{\tau _k})\Gamma \Sigma _{ZZ^T}\Gamma ^T=o_p(1)$.

Since $\mathcal {L}_n$ is a convex function, we have

$$\begin{aligned} \sqrt{n}(\hat{\beta }^\mathrm{CQR}-\beta )= & {} -\frac{\left( \Gamma \Sigma _{ZZ^T}\Gamma ^T\right) ^{-1}}{\sum \nolimits _{k=1}^{K}f(b_{\tau _k})}\sum _{k=1}^{K}\sum _{i=1}^{n}n^{-\frac{1}{2}} (\Gamma Z_i)\left\{ \eta _{ik}+f(b_{\tau _k})r_i\right\} +o_p(1)\\= & {} -\frac{\left( \Gamma \Sigma _{ZZ^T}\Gamma ^T\right) ^{-1}}{\sum \nolimits _{k=1}^{K}f(b_{\tau _k})}\sum _{k=1}^{K}\sum _{i=1}^{n}n^{-\frac{1}{2}} (\Gamma Z_i)\left\{ \eta _{ik}{+}f(b_{\tau _k})\upsilon _i^T\beta \right\} {+}o_p(1). \end{aligned}$$

It is easy to obtain that

$$\begin{aligned} \text {E}\left( \sum _{k=1}^{K}\sum _{i=1}^{n}n^{-\frac{1}{2}} (\Gamma Z_i)\left\{ \eta _{ik}+f(b_{\tau _k})\upsilon _i^T\beta \right\} \right) =0 \end{aligned}$$

and

$$\begin{aligned} \text {Var}\left( \sum _{k=1}^{K}\sum _{i=1}^{n}n^{-\frac{1}{2}} (\Gamma Z_i)\left\{ \eta _{ik}+f(b_{\tau _k})\upsilon _i^T\beta \right\} \right) =\Gamma \Sigma _{ZZ^T}\Gamma ^T\sum \limits _{k=1}^K\sum \limits _{k'=1}^K\psi (\tau _k,\tau _{k'}). \end{aligned}$$

Then the proof of Theorem 1 is concluded by the Central Limit Theorem.