A robust test of exogeneity based on quantile regressions

ABSTRACT

In this paper, we propose a robust test of exogeneity. The test statistics is constructed from quantile regression estimators, which are robust to heavy tails of errors. We derive the asymptotic distribution of the test statistic under the null hypothesis of exogeneity at a given quantile. The finite sample properties of the test are investigated through Monte Carlo simulations that exhibit not only good size and power properties, but also good robustness to outliers.

KEYWORDS:

• Regression quantile
• endogeneity
• two-stage estimation
• Hausman test

JEL CODES:

• C21

Acknowledgements

We are grateful for comments by participants in the 2013 French Econometrics Conference and the 2014 EEA conference in Toulouse.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1. Hahn and Hausman [9], Butler [10], Chmelarova and Hill [11], Lavergne and Nguimkeu [12], and Lee and Okui [13].

2. To name just a few, see Amemiya [7], Powell [14], Chen and Portnoy [15], Kemp [16], Sakata [17], Arias et al. [18], Garcia et al. [19], Chen et al. [20], Hong and Tamer [21], Kim and Muller [5,22], Chernozhukov and Hansen [6,23,24], Ma and Koenker [25], Horowitz and Lee [26], and Lee [27].

3. Note that in the iid case, the term $f\left(F^{-1}\right(\theta ){)}^{-1}$ typically appears in the variance formula of a quantile estimator [28]. However, due to Assumption 3(iv), $F^{-1}\left(\theta \right)$ is now zero so that in this case, we instead have $f(0{)}^{-1}$.

4. However, they may be efficient is some particular cases. For example, LAD regressions are efficient under errors following a Laplace distribution law.

5. Readers are referred to Kim and White [29] for how the value of m is determined.

6. In the Hausman test specification, one would have instead: σ₁₁ = var(u_t), σ₂₂ = var(v_t), and σ₁₂ = cov(u_t, v_t).

Additional information

Funding

The first author acknowledges financial support from the National Research Foundation of Korea – a grant funded by the Korean Government [NRF-2009-327-B00088] and from the Aix-Marseille School of Economics. The second author acknowledges financial support from the A*MIDEX project [no. ANR-11-IDEX-0001-02] funded by the  Investissements d'Avenir  French Government program, managed by the French National Research Agency (ANR).