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Analytical Finite Sample Econometrics: From A. L. Nagar to Now

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Abstract

Professor A.L. Nagar was a world-renowned econometrician and an international authority on finite sample econometrics with many path-breaking papers on the statistical properties of econometric estimators and test statistics. His contributions to applied econometrics have been also widely recognized. Nagar’s 1959 Econometrica paper on the so-called k-class estimators, together with a later one in 1962 on the double-k-class estimators, provided a very general framework of bias and mean squared error approximations for a large class of estimators and had motivated researchers to study a wide variety of issues such as many and weak instruments for many decades to follow. This paper reviews Nagar’s seminal contributions to analytical finite sample econometrics by providing historical backgrounds, discussing extensions and generalization of Nagar’s approach, and suggesting future directions of this literature.

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Notes

  1. There is a subtle connection between resampling techniques and higher-order expansions, see, for example, the classical work of Hall (1992) and the review by Horowitz (2001). We refrain from discussing the data-driven resampling methods and focus on analytical results in this paper.

  2. Some terms were missing in the formulae given by Bao and Ullah (2010) and Ullah (2004, page 186). The authors thank Raymond Kan for pointing this out.

  3. When \(K=1\), i.e., the just identified case, the 2SLS estimator (corresponding to \(k=0\)), does not have even the first moment, see Sawa (1969). For a definitive treatment of the issue of existence of moments, see Kinal (1980).

  4. Notably, Dwivedi and Srivastava (1984) presented the first and second moments in terms of double infinite series, whereas Bao and Kan (2013) presented, for any finite integer j, the j-th moment using a single infinite series.

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Acknowledgements

This paper is dedicated to Professor A.L.Nagar for this special issue. We thank Essie Maasoumi for his helpful comments.

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  1. Department of Economics, Purdue University, West Lafayette, IN, 47907, USA

    Yong Bao

  2. Department of Economics, University of California, Riverside, CA, 92521, USA

    Aman Ullah

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Bao, Y., Ullah, A. Analytical Finite Sample Econometrics: From A. L. Nagar to Now. J. Quant. Econ. 19 (Suppl 1), 17–37 (2021). https://doi.org/10.1007/s40953-021-00261-z

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