Estimation and testing of kink regression model with endogenous regressors

Abstract

Kink regression model which assumes continuity at the threshold point has wide applications in statistics and economics. Existing estimation methods are obtained under a rather important assumption that the errors are mean independent of the threshold variable, namely, it is exogenous. However, endogeneity can arise as a result of omitted variables, lagged dependent variable, measurement error and many other sources. In this paper, we consider the estimation and testing for the kink regression model with endogenous threshold variable and possible other endogenous regressors. We find that the conventional form of 2SLS estimator is inconsistent as the expectation of a nonlinear function is not generally the function of expectations. The continuity feature of the regression prompts us to try method based on the GMM principle using nonlinear instruments. We derive the asymptotic properties using a different way from the usual GMM estimators as the objective function is not smooth with respect to the threshold parameter. A sup-Wald test for the presence of kink effect is established and a bootstrap procedure to gain the p value is introduced. Monte Carlo simulations show that the proposed estimator and testing procedure perform well. The proposed procedures is also illustrated using an empirical application.

Appendix: Proofs

This supplementary material contains detailed proof of Theorem 1.

Let $\varvec{{\xi }}=(\varvec{{\beta }}^{'}, \gamma )^{'}$, $\varvec{{\xi }}_0=(\varvec{{\beta }}_0^{'}, \gamma _0)^{'}$, $\textbf{m}_t(\varvec{{\xi }})=\varPsi (\textbf{z}_t)[y_t - \textbf{x}_t^{'}(\gamma )\varvec{{\beta }}]$, $\bar{\textbf{m}}_n(\varvec{{\xi }})=\frac{1}{n}\sum \limits _{t=1}^n \textbf{m}_t(\varvec{{\xi }})$, $\textbf{m}(\varvec{{\xi }})= E[\textbf{m}_t(\varvec{{\xi }})]$, $Q_n(\varvec{{\xi }})=\bar{\textbf{m}}_n^{'}(\varvec{{\xi }}) W_n \bar{\textbf{m}}_n(\varvec{{\xi }})$, $Q_0(\varvec{{\xi }})=\textbf{m}^{'}(\varvec{{\xi }}) W \textbf{m}(\varvec{{\xi }})$.

Proof of Theorem 1

To establish consistency, we first show that

$$\begin{aligned} \sup _{\varvec{{\xi }}}| \bar{\textbf{m}}_n(\varvec{{\xi }})- \textbf{m}(\varvec{{\xi }})|{\mathop {\rightarrow }\limits ^{P}} 0. \end{aligned}$$

(A.1)

Since $\textbf{x}_t(\gamma )$ is continuous in $\gamma$, $\textbf{m}_t(\varvec{{\xi }})$ is continuous in $\varvec{{\xi }}$. Noting that $\Vert \textbf{x}_t(\gamma )\Vert \le \{(x_{1t}-\gamma )^2 + \Vert \textbf{x}_{2t}\Vert ^2 \}^{1/2} \le |x_{1t}- \gamma | + \Vert \textbf{x}_{2t}\Vert \le |x_{1t}| + |\gamma | + \Vert \textbf{x}_{2t}\Vert$, it follows

$$\begin{aligned} \sup _{\varvec{{\xi }}}\Vert \textbf{m}_t(\varvec{{\xi }})\Vert\le & {} \Vert \varPsi (\textbf{z}_t)\Vert \cdot \sup _{\varvec{{\xi }}} |y_t - \textbf{x}_t^{'}(\gamma )\varvec{{\beta }}|\\\le & {} \Vert \varPsi (\textbf{z}_t)\Vert \{|y_t| + \sup _{\varvec{{\beta }}}\Vert \varvec{{\beta }}\Vert \sup _{\gamma } \Vert \textbf{x}_t(\gamma )\Vert \} \\\le & {} \Vert \varPsi (\textbf{z}_t)\Vert \{|y_t| + \sup _{\varvec{{\beta }}}\Vert \varvec{{\beta }}\Vert \cdot ( |x_{1t}| + \sup _{\gamma }|\gamma |+ \Vert \textbf{x}_{2t}\Vert ) \} \\\le & {} \Vert \varPsi (\textbf{z}_t)\Vert \ \{|y_t| + C_{\beta } ( |x_{1t}| + C_{\gamma } + \Vert \textbf{x}_{2t}\Vert ) \} \end{aligned}$$

where $C_{\beta }=\sup _{\varvec{{\beta }}}\Vert \varvec{{\beta }}\Vert$ and $C_{\gamma }= \sup _{\gamma }|\gamma |$. Hence, we obtain

$$\begin{aligned}{} & E[\Vert \varPsi (\textbf{z}_t)\Vert \ \{|y_t| + C_{\beta } ( |x_{1t}| + C_{\gamma } + \Vert \textbf{x}_{2t}\Vert ) \}] \\ & \quad \le \{E\Vert \varPsi (\textbf{z}_t)\Vert ^2\}^{1/2}\cdot \{E [ |y_t| + C_{\beta } ( |x_{1t}| + C_{\gamma } + \Vert \textbf{x}_{2t}\Vert ) ]^2\}^{1/2} \\{} & \quad \le \{E\Vert \varPsi (\textbf{z}_t)\Vert ^2\}^{1/2}\cdot \{ (Ey_t^2)^{1/2} + C_{\beta } (Ex_{1t}^2)^{1/2} + C_{\beta }C_{\gamma } \\ & \qquad + C_{\beta }(E\Vert \textbf{x}_{2t}\Vert ^2)^{1/2}\} <\infty \end{aligned}$$

by the Schwartz inequality, the Minkowski inequality and the moment assumption 2. Next it is guaranteed by Proposition 2.8 of Fan and Yao (2005) that $\bar{\textbf{m}}_n(\varvec{{\xi }})$ converges almost surely to $\textbf{m}(\varvec{{\xi }})$ for each fixed $\varvec{{\xi }}$. Therefore, we have $\sup _{\varvec{{\xi }}}\Vert \bar{\textbf{m}}_n(\varvec{{\xi }})- \textbf{m}(\varvec{{\xi }})\Vert {\mathop {\rightarrow }\limits ^{P}} 0$ based on Lemma 1 of Tauchen (1985). It implies that $\textbf{m}(\varvec{{\xi }})$ is also continuous in $\varvec{{\xi }}$.

Therefore, we have by straightforward calculations, (A.1) and Assumption 2 that

$$\begin{aligned}{} & {} \sup _{\varvec{{\xi }}}| Q_n(\varvec{{\xi }})- Q_0(\varvec{{\xi }})| \le \sup _{\varvec{{\xi }}} |\{\bar{\textbf{m}}_n(\varvec{{\xi }})- \textbf{m}(\varvec{{\xi }})\}^{'}W_n \{\bar{\textbf{m}}_n(\varvec{{\xi }})- \textbf{m}(\varvec{{\xi }})\}| \\{} & {} \ \ \ + 2 \sup _{\varvec{{\xi }}} |\textbf{m}^{'}(\varvec{{\xi }}) W_n \{\bar{\textbf{m}}_n(\varvec{{\xi }})- \textbf{m}(\varvec{{\xi }})\}| + \sup _{\varvec{{\xi }}}| \textbf{m}^{'}(\varvec{{\xi }}) ( W_n - W) \textbf{m}(\varvec{{\xi }})| {\mathop {\rightarrow }\limits ^{P}} 0. \end{aligned}$$

Next, we will show that $Q_0(\varvec{{\xi }})$ is uniquely minimized at $\varvec{{\xi }}=\varvec{{\xi }}_0$. We show that for any $\varvec{{\xi }}\ne \varvec{{\xi }}_0$, $Q_0(\varvec{{\xi }})>0=Q_0(\varvec{{\xi }}_0)$. Denote $g_t(\varvec{{\xi }})=\varPsi (\textbf{z}_t)\{\beta _{10}(x_{1t}-\gamma _0)_{-} - \beta _1 (x_{1t} - \gamma )_{-} + \beta _{20}(x_{1t}-\gamma _0)_{+} - \beta _2 (x_{1t} - \gamma )_{+} + (\varvec{{\beta }}_{30} - \varvec{{\beta }}_3)^{'}\textbf{x}_{2t}\}$, $A_1=\{ \gamma< x_{1t} \le \gamma _0\} \cup \{ \gamma _0 < x_{1t} \le \gamma \}$, $A_2=\{ x_{1t}> \gamma , \ x_{1t} >\gamma _0\}$, $A_3 = \{ x_{1t} \le \gamma , \ x_{1t} \le \gamma _0\}$. It implies by Assumption 3 that the sets $A_l\ (l=1,2,3)$ all have positive probabilities. Then

$$\begin{aligned} Q_0(\varvec{{\xi }})= & {} \sum \limits _{l=1}^3 E[g_t(\varvec{{\xi }})\textbf{1}(A_l)]^{'} W E[g_t(\varvec{{\xi }})\textbf{1}(A_l)] \\\ge & \,{} E[g_t(\varvec{{\xi }})\textbf{1}(A_2)]^{'} W E[g_t(\varvec{{\xi }})\textbf{1}(A_2)] + E[g_t(\varvec{{\xi }})\textbf{1}(A_3)]^{'} W E[g_t(\varvec{{\xi }})\textbf{1}(A_3)] \end{aligned}$$

First, suppose $(\beta _1, \beta _2, \varvec{{\beta }}_3)=(\beta _{10}, \beta _{20}, \varvec{{\beta }}_{30})$ and $\gamma \ne \gamma _0$, then the right hand side of the above equation equals

$$\begin{aligned}{} & {} (\gamma -\gamma _0)^2 \{\beta _{20}^2 E[\varPsi ^{'} ({\textbf{z}}_t){\textbf{1}}(A_2)] W E[\varPsi ({\textbf{z}}_t){\textbf{1}}(A_2)] \\{} & \qquad + \beta _{10}^2 E[\varPsi ^{'} ({\textbf{z}}_t){\textbf{1}}(A_3)] W E[\varPsi ({\textbf{z}}_t){\textbf{1}}(A_3)]\} \\ & \quad = (\gamma -\gamma _0)^2 \{\beta _{20}^2 E[ E\{\varPsi ^{'} ({\textbf{z}}_t)|A_2\}] W E[E\{\varPsi ^{'} ({\textbf{z}}_t)|A_2\}] \\ & \qquad + \beta _{10}^2 E[E\{\varPsi ^{'} ({\textbf{z}}_t)| A_3\} ] W E[E\{\varPsi ^{'} ({\textbf{z}}_t)| A_3\}]\}>0. \end{aligned}$$

by Assumptions 3, 4 and 5.

Next, suppose $(\beta _1, \beta _2, \varvec{{\beta }}_3)\ne (\beta _{10}, \beta _{20}, \varvec{{\beta }}_{30})$. Then

$$\begin{aligned} Q_0(\varvec{{\xi }}) &\ge \,E[g_t(\varvec{{\xi }})\textbf{1}(A_2)]^{'} W E[g_t(\varvec{{\xi }})\textbf{1}(A_2)] \\ &=\, (\varvec{{\delta }}_0 - \varvec{{\delta }})^{'} E[E\{\textbf{x}_t \varPsi ^{'}(\textbf{z}_t) | A_2\}] W E[E\{\varPsi (\textbf{z}_t)\textbf{x}_t^{'}|A_2\}] (\varvec{{\delta }}_0 - \varvec{{\delta }})>0 \end{aligned}$$

by Assumption 4 and 5 where $\varvec{{\delta }}= (\beta _{2}, \beta _{2}\gamma (1, \textbf{0}_{1\times (p-1)}) + \varvec{{\beta }}_{3}^{'})^{'}$ with redefined intercept, as the first component of $\textbf{x}_2$ is a constant regressor, and $\varvec{{\delta }}_0 = (\beta _{20}, \beta _{20}\gamma (1, \textbf{0}_{1\times (p-1)}) + \varvec{{\beta }}_{30}^{'})^{'}$.

All the above results together with Theorem 5.6 of Su (2007) lead to $\hat{\varvec{{\xi }}}{\mathop {\rightarrow }\limits ^{P}} \varvec{{\xi }}_0$.

Next we will use the idea of Andrews (1994), Section 3.2 to obtain the asymptotic distribution. It is clear that $S_t(\varvec{{\xi }})$ is linear in $\varvec{{\beta }}$, and continuous in $\gamma$ except at $x_{1t}=\gamma$. Note that all the terms in $S_t(\varvec{{\xi }})$ can be expressed as $\varPsi (\textbf{z}_t)(x_{1t}-\gamma )\textbf{1}(x_{1t}<\gamma )$, $\varPsi (\textbf{z}_t)(x_{1t}-\gamma )\textbf{1}(x_{1t}\ge \gamma )$, $\beta _1\varPsi (\textbf{z}_t)\textbf{1}(x_{1t}<\gamma )$, $\beta _2\varPsi (\textbf{z}_t)\textbf{1}(x_{1t}\ge \gamma )$, and $\varPsi (\textbf{z}_t)\textbf{x}_{2t}$. Then, it can be shown

$$\begin{aligned} \sup _{\varvec{{\xi }}} \Vert \frac{1}{n}\sum \limits _{t=1}^n S_t(\varvec{{\xi }}) - E[S_t(\varvec{{\xi }})]\Vert {\mathop {\rightarrow }\limits ^{P}} 0 \end{aligned}$$

based on Lemma 1 of Tauchen (1985) using the same strategy as establishing (A.1).

Further, as for any $\gamma _1> \gamma _2$

$$\begin{aligned}{} & {} \Vert E[\varPsi (\textbf{z}_t)(x_{1t}-\gamma _1)\textbf{1}(x_{1t}<\gamma _1)]- E[\varPsi (\textbf{z}_t)(x_{1t}-\gamma _2)\textbf{1}(x_{1t}<\gamma _2)]\Vert \\{} & {} \quad = \Vert E[\varPsi (\textbf{z}_t)x_{1t}\textbf{1}(\gamma _2\le x_{1t}<\gamma _1)] \\{} & {} \qquad + E\{\varPsi (\textbf{z}_t)[\gamma _1\textbf{1}(x_{1t}<\gamma _1)- \gamma _2\textbf{1}(x_{1t}<\gamma _2)]\}\Vert \\{} & {} \quad \le (E\Vert \varPsi (\textbf{z}_t)x_{1t}\Vert ^2)^{1/2}( E|\textbf{1}(\gamma _2\le x_{1t}<\gamma _1)|^2)^{1/2} \\{} & {} \qquad + (E\Vert \varPsi (\textbf{z}_t)\Vert ^2)^{1/2}( E|\gamma _1\textbf{1}(x_{1t}<\gamma _1)- \gamma _2\textbf{1}(x_{1t}<\gamma _2)|^2)^{1/2} \\{} & {} \quad \le (E\Vert \varPsi (\textbf{z}_t)\Vert ^4 Ex_{1t}^4)^{1/4} [ F(\gamma _1)- F(\gamma _2)]^{1/2} \\{} & {} \qquad + 2 (E\Vert \varPsi (\textbf{z}_t)\Vert ^2)^{1/2}( E|\gamma _1\textbf{1}(\gamma _2\le x_{1t}<\gamma _1)|^2)^{1/2}\\{} & {} \qquad + 2 (E\Vert \varPsi (\textbf{z}_t)\Vert ^2)^{1/2} (E| (\gamma _1 - \gamma _2)\textbf{1}(x_{1t}<\gamma _2)|^2)^{1/2} \\{} & {} \quad \le C_1( F(\gamma _1)- F(\gamma _2)]^{1/2} + C_2 |\gamma _1 -\gamma _2| \end{aligned}$$

using the triangular inequality, the Schwartz inequality and the Minkowski inequality where $C_1$ and $C_2$ are some positive constants, it follows that $E[S_t(\varvec{{\xi }})]$ is continuous in $\varvec{{\xi }}$.

We have by $Q_0(\varvec{{\xi }}_0)=0$ and Assumption 6 that $\textbf{m}(\varvec{{\xi }}_0)=0$. Then it follows by the mean value expansion of $\textbf{m}(\xi _0)$ at $\hat{\varvec{{\xi }}}$ that

$$\begin{aligned} \textbf{0}= \textbf{m}(\varvec{{\xi }}_0)= \textbf{m}(\hat{\varvec{{\xi }}}) + \frac{\partial \textbf{m}(\tilde{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}(\hat{\varvec{{\xi }}}- \varvec{{\xi }}_0) \end{aligned}$$

where $\tilde{\varvec{{\xi }}}$ lies between $\hat{\varvec{{\xi }}}$ and $\varvec{{\xi }}_0$ which implies $\tilde{\varvec{{\xi }}}{\mathop {\rightarrow }\limits ^{P}} \varvec{{\xi }}_0$. Therefore,

$$\begin{aligned} \textbf{0}= & {} \sqrt{n}\left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \textbf{m}(\varvec{{\xi }}_0) =\left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \sqrt{n}\textbf{m}(\hat{\varvec{{\xi }}}) \nonumber \\{} & {} \ \ + \left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \frac{\partial \textbf{m}(\tilde{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}} \sqrt{n}(\hat{\varvec{{\xi }}}- \varvec{{\xi }}_0)\nonumber \\= & {} - \left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \{ \sqrt{n}\bar{\textbf{m}}_n (\hat{\varvec{{\xi }}}) -\sqrt{n}\textbf{m}(\hat{\varvec{{\xi }}}) - \sqrt{n}\bar{\textbf{m}}_n(\hat{\varvec{{\xi }}}) \} \nonumber \\{} & {} + \left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \frac{\partial \textbf{m}(\tilde{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}} \sqrt{n}(\hat{\varvec{{\xi }}}- \varvec{{\xi }}_0) \nonumber \\= & {} - \left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \{ \sqrt{n}\bar{\textbf{m}}_n(\hat{\varvec{{\xi }}}) - \sqrt{n}\textbf{m}(\hat{\varvec{{\xi }}}) \} \nonumber \\{} & {} + \left( \frac{\partial \bar{\textbf{m}}_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}}\right) ^{'}W_n \frac{\partial \textbf{m}(\tilde{\varvec{{\xi }}})}{\partial \varvec{{\xi }}^{'}} \sqrt{n}(\hat{\varvec{{\xi }}}- \varvec{{\xi }}_0) \end{aligned}$$

(A.2)

where the last second equality follows by $\frac{\partial Q_n(\hat{\varvec{{\xi }}})}{\partial \varvec{{\xi }}} =0$ using the first order condition.

Denote $\textbf{v}_n(\varvec{{\xi }})= \sqrt{n}\{\bar{\textbf{m}}_n(\varvec{{\xi }}) - \textbf{m}(\varvec{{\xi }})\} = \frac{1}{\sqrt{n}}\sum \nolimits _{t=1}^n \{\textbf{m}_t(\varvec{{\xi }}) - E[\textbf{m}_t(\varvec{{\xi }})]\}$. We have by Theorem 2.21 of Fan and Yao (2005) that $\textbf{v}_n(\varvec{{\xi }}_0){\mathop {\rightarrow }\limits ^{D}} \textbf{N}(\textbf{0}, \varOmega )$. Then we have

$$\begin{aligned} \textbf{v}_n(\hat{\varvec{{\xi }}}) = \textbf{v}_n(\hat{\varvec{{\xi }}}) - \textbf{v}_n(\varvec{{\xi }}_0) + \textbf{v}_n(\varvec{{\xi }}_0) {\mathop {\rightarrow }\limits ^{d}} \textbf{N}(\textbf{0}, \varOmega ) \end{aligned}$$

(A.3)

provided that $\textbf{v}_n(\varvec{{\xi }})$ is stochastically equicontinuous by Theorem 5.7 of Su (2007) and $\hat{\varvec{{\xi }}}{\mathop {\rightarrow }\limits ^{P}} \varvec{{\xi }}_0$.

Combing the above results, it follows by (A.2), (A.3), assumptions 5-6 and Slutsky Theorem that

$$\begin{aligned} \sqrt{n}(\hat{\varvec{{\xi }}}- \varvec{{\xi }}_0)&= \left( \Big \{\frac{1}{n}\sum \limits _{t=1}^n S_t(\hat{\varvec{{\xi }}})\Big \}^{'} W_n \Big \{\frac{1}{n} \sum \limits _{t=1}^nS_t(\hat{\varvec{{\xi }}})\Big \}\right) ^{-1} \Big \{\frac{1}{n}\sum \limits _{t=1}^n S_t(\hat{\varvec{{\xi }}})\Big \}^{'} W_n \textbf{v}_n(\hat{\varvec{{\xi }}})\\&{\mathop {\rightarrow }\limits ^{d}} \textbf{N}\Big ( \textbf{0}, \ (G^{'}WG)^{-1}G^{'}W\varOmega W G (G^{'}WG)^{-1} \Big ). \end{aligned}$$

$\square$