Near exogeneity, weak identification and specification testing: Some asymptotic results

Abstract

The paper studies the asymptotic size property of various specification tests in linear structural models where instrumental variables may locally violate the exclusion restrictions. Our results provide some new insights and extensions of earlier studies. In particular, we derive an explicit formula of the asymptotic size of the tests which shows clearly the factors that influence their size under instrument endogeneity. We show that all tests have correct asymptotic size when the usual orthogonality condition holds, but their asymptotic size can be arbitrary large even if only one instrument is slightly correlated with the error term. We present a Monte Carlo experiment that confirms our theoretical findings.

Keywords:

• Asymptotic size distortions
• DWH tests
• instrument endogeneity
• weak instruments

Acknowledgements

We are grateful to the Editor Prof. N. Balakrishnan and the anonymous referees for their constructive comments and suggestions. We would like to thank Jean-Marie Dufour and Mardi Dungey for very helpful comments. The first draft of this paper was circulated as ‘Specification Testing with Weak and Invalid Instruments’.

Notes

1 See Durbin (1954), Wu (1973, 1974), and Hausman (1978).

2 For any random variable Z such that $Z\sim {\chi }^2(k;\delta ),$ its cdf is linked to Marcum’s Q-function as $\mathbb{P}[Z\le z]=1-Q_{\frac{k}{2}}[\sqrt{\delta },\sqrt{z}]$ for any $z\ge 0.$

3 See Durbin (1954), Wu (1973, 1974), Hausman (1978), Nakamura and Nakamura (1981, 1985), Engle (1982), Holly (1982, 1983b,a), Holly and Monfort (1983), Reynolds (1982), Smith (1983, 1984, 1985), Staiger and Stock (1997), Kiviet and Niemczyk (2006, 2007), Hahn, Ham, and Moon (2010), Kiviet (2013), Doko Tchatoka (2015), and Doko Tchatoka and Dufour (2016b,a).

4 Note that with this parametrization, the correlation between the structural error term ε and the instruments in X is the same and equal to $\frac{{\rho }_{z\epsilon }}{\sqrt{n}}$ for all t. The results are qualitatively the same when we vary this correlation across IVs. To ease the presentation and interpretation of the results, we also present the results with ${\rho }_n\in [0,1]$ (positive correlations) but the findings are the same for negative correlations, i.e., when ${\rho }_n\in [-1,0]$.

5 The first-stage F-statistic corresponds to the concentration parameter ${\mu }^2$ that controls identification strength of β in the notation of Staiger and Stock (1997).