Abstract
The extreme sensitivity of instrumental variables (IV) estimators to outliers is a crucial problem too often neglected or poorly dealt with. We address this issue by making the practitioner aware of the existence, usefulness, and inferential implications of robust-to-outliers instrumental variables estimators. We describe how the standard IV estimator can be made robust to outliers, provide a brief description of alternative robust IV estimators, simulate the behaviour of both the standard IV estimator and each robust IV estimator in presence of different types of outliers, and conclude by replicating a celebrated study on the deep determinants of development in order to establish the danger of ignoring outliers in an IV model.
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- 1.
For the sake of clarity, vectors and matrices are in bold font and scalars are in normal font.
- 2.
Note that these authors were more interested in the advantages offered by quantile regressions to study the distribution of a given variable than in the resistance of a median regression-type estimator to certain types of outliers.
- 3.
- 4.
Ready-to-use Stata code for the Stahel and Donoho estimator is available upon request to the authors.
- 5.
We have implemented this estimator using the -cqiv- Stata code written by Chernozhukov et al. (2010), given that other interested practitioners are likely to turn to the same source. However, note that we obtain similar results when we adopt a more flexible function of the first-stage residuals in the second stage. Results are available upon request.
- 6.
For the simulation, we define \(\omega _{i} = I\left (\delta _{i}(\mathbf{X}_{C}.\mathbf{Z_{C}^{E}})\vdots_{(\mathbf{X}_{ D},\mathbf{Z}_{D}^{E})} < \sqrt{\chi _{p_{1 } +m_{3 },0.95 }^{2}}\right )\), \(\phi _{i} =\sigma \frac{\psi _{i}(\frac{r_{i}} {\sigma } )} {r_{i}}\), \(\psi _{i}(u) = u\left [1 -\left ( \frac{u} {1.546}\right )^{2}\right ]^{2}I\left (\left \vert u\right \vert \leq 1.546\right )\) where I is the indicator function.
- 7.
The loss function is defined as \(\rho (u) = \frac{u^{2}} {2} I\left (\left \vert u\right \vert \leq 4.685\right ) + \left (4.685\left \vert u\right \vert -\frac{u^{2}} {2} \right )I\left (\left \vert u\right \vert > 4.685\right )\).
- 8.
Obviously, in the absence of contamination, the standard IV estimator should be adopted.
- 9.
- 10.
As in Rodrik et al. (2004), regressors have been scaled by expressing them as deviations from their mean divided by their standard deviation.
- 11.
The first-stage F-statistics measure the correlation of the excluded instruments with the endogenous regressors, adjusted for the presence of two endogenous regressors. With weak identification, the classical IV estimator can be severely biased.
- 12.
The DFBETA statistics measure how each regression coefficient changes when each observation is deleted in turn.
- 13.
The value for the degrees of freedom of the Chi-Square distribution corresponds to the presence in the IV model of one dependent variable, three explanatory variables and two excluded instruments with 1 + 3 + 2 = 6.
- 14.
Note that these dummies take the value of one for all countries in a specific group. They are not restricted to the identified outliers. These dummies capture therefore common group (regional) effects.
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Acknowledgements
Catherine Dehon gratefully acknowledges research support from the IAP research network grant nr. P7/06 of the Belgian government (Belgian Science Policy) and from the ARC contract of the Communauté Française de Belgique. Catherine Dehon is also member of ECORE, the association between CORE and ECARES.
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Dehon, C., Desbordes, R., Verardi, V. (2015). The Pitfalls of Ignoring Outliers in Instrumental Variables Estimations: An Application to the Deep Determinants of Development. In: Beran, J., Feng, Y., Hebbel, H. (eds) Empirical Economic and Financial Research. Advanced Studies in Theoretical and Applied Econometrics, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-319-03122-4_12
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