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EIV regression with bounded errors in data: total ‘least squares’ with Chebyshev norm

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Abstract

We consider the linear regression model with stochastic regressors and stochastic errors both in regressors and the dependent variable (“structural EIV model”), where the regressors and errors are assumed to satisfy some interesting and general conditions, different from traditional assumptions on EIV models (such as Deming regression). The most interesting fact is that we need neither independence of errors, nor identical distributions, nor zero means. The first main result is that the TLS estimator, where the traditional Frobenius norm is replaced by the Chebyshev norm, yields a consistent estimator of regression parameters under the assumptions summarized below. The second main result is that we design an algorithm for computation of the estimator, reducing the computation to a family of generalized linear-fractional programming problems (which are easily computable by interior point methods). The conditions under which our estimator works are (said roughly): it is known which regressors are affected by random errors and which are observed exactly; that the regressors satisfy a certain asymptotic regularity condition; all error distributions, both in regressors and in the endogenous variable, are bounded in absolute value by a common bound (but the bound is unknown and is estimated); there is a high probability that we observe a family of data points where the errors are close to the bound. We also generalize the method to the case that the bounds of errors in the dependent variable and regressors are not the same, but their ratios are known or estimable. The assumptions, under which our estimator works, cover many settings where the traditional TLS is inconsistent.

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Notes

  1. A matrix norm \(\Vert \cdot \Vert \) is orthogonally invariant, if \(\Vert UAV\Vert =\Vert A\Vert \) for all \(A\in {\mathbb {R}}^{n\times p}\) and all unitary matrices \(U\in {\mathbb {R}}^{n\times n}\) and \(V\in {\mathbb {R}}^{p\times p}\).

  2. A preliminary version of the algorithm for CNP, presented in Theorem 2, was reported at the ICCS’15 conference and is reported in the proceedings (Hladík and Černý 2015). It was studied there from the complexity-theoretic perspective. Here we present it with a proof for the reason that the geometry on which the algorithm is based will also be necessary for the proof of Theorem 1 and cannot be avoided. This paper is a follow-up of the mentioned conference contribution.

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Acknowledgements

The work was supported by the Czech Science Foundation under grants P402/13-10660S (M. Hladík), P402/12/G097 (M. Černý) and P403/15/09663S (J. Antoch). J. Antoch also acknowledges the support from the BELSPO IAP P7/06 StUDyS network. We are also obliged to Tomáš Cipra, a senior member of DYME Research Center, for fruitful discussions.

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic, Malostranské náměstí 25, 11800, Prague, Czech Republic

    Milan Hladík

  2. Department of Econometrics, Faculty of Informatics and Statistics, University of Economics in Prague, Czech Republic, Winston Churchill Square 4, 13067, Prague, Czech Republic

    Michal Černý

  3. Department of Probability and Mathematical Statistics, Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic, Sokolovská 83, 18675, Prague, Czech Republic

    Jaromír Antoch

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  1. Milan Hladík
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  2. Michal Černý
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  3. Jaromír Antoch
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Correspondence to Michal Černý.

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Hladík, M., Černý, M. & Antoch, J. EIV regression with bounded errors in data: total ‘least squares’ with Chebyshev norm. Stat Papers 61, 279–301 (2020). https://doi.org/10.1007/s00362-017-0939-z

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