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Small Sample Bias in GMM Estimation of Covariance Structures

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  • Joseph G. Altonji
  • Lewis M. Segal

Abstract

We examine the small sample properties of the GMM estimator for models of covariance structures, where the technique is often referred to as the optimal minimum distance (OMD) estimator. We present a variety of Monte Carlo experiments based on simulated data and on the data used by Abowd and Card (1987, 1990) in an examination of the covariance structure of hours and earnings changes. Our main finding is that OMD is seriously biased in small samples for many distributions and in relatively large samples for poorly behaved distributions. The bias is almost always downward in absolute value. It arises because sampling errors in the second moments are correlated with sampling errors in the weighting matrix used by OMD. Furthermore, OMD usually has a larger root mean square error and median absolute error than equally weighted minimum distance (EWMD). We also propose and investigate an alternative estimator, which we call independently weighted optimal minimum distance (IWOMD). IWOMD is a split sample estimator using separate groups of observations to estimate the moments and the weights. IWOMD has identical large sample properties to the OMD estimator but is unbiased regardless of sample size. However, the Monte Carlo evidence indicates that IWOMD is usually dominated by EWMD.

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Bibliographic Info

Paper provided by National Bureau of Economic Research, Inc in its series NBER Technical Working Papers with number 0156.

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Date of creation: Jun 1994
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Publication status: published as Altonji, Joseph G. and Lewis M. Segal. "Small-Sample Bias In GMM Estimation Of Covariance Structures," Journal of Business and Economic Statistics, 1996, v14(3,Jul), 353-366.
Handle: RePEc:nbr:nberte:0156

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  1. Lehmann, Bruce N., 1990. "Residual risk revisited," Journal of Econometrics, Elsevier, Elsevier, vol. 45(1-2), pages 71-97.
  2. John M. Abowd & David Card, 1986. "Intertemporal Labor Supply and Long Term Employment Contracts," NBER Working Papers 1831, National Bureau of Economic Research, Inc.
  3. Joshua D. Angrist & Alan B. Krueger, 1995. "Split Sample Instrumental Variables," NBER Technical Working Papers, National Bureau of Economic Research, Inc 0150, National Bureau of Economic Research, Inc.
  4. repec:cup:etheor:v:10:y:1994:i:1:p:172-97 is not listed on IDEAS
  5. Altonji, Joseph G. & Martins, Ana Paula & Siow, Aloysius, 2002. "Dynamic factor models of consumption, hours and income," Research in Economics, Elsevier, Elsevier, vol. 56(1), pages 3-59, June.
  6. Chamberlain, Gary, 1982. "Multivariate regression models for panel data," Journal of Econometrics, Elsevier, Elsevier, vol. 18(1), pages 5-46, January.
  7. Arellano, Manuel & Sargan, J D, 1990. "Imhof Approximations to Econometric Estimators," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 57(4), pages 627-46, October.
  8. Bruce N. Lehmann, 1986. "Residual Risk Revisited," NBER Working Papers 1908, National Bureau of Economic Research, Inc.
  9. Koenker, Roger & Machado, José A.F. & Skeels, Christopher L. & Welsh, Alan H., 1994. "Momentary Lapses: Moment Expansions and the Robustness of Minimum Distance Estimation," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 10(01), pages 172-197, March.
  10. Shanken, Jay, 1990. "Intertemporal asset pricing : An Empirical Investigation," Journal of Econometrics, Elsevier, Elsevier, vol. 45(1-2), pages 99-120.
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