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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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Suggested Citation

  • Joseph G. Altonji & Lewis M. Segal, 1994. "Small sample bias in GMM estimation of covariance structures," Working Paper Series, Macroeconomic Issues 94-8, Federal Reserve Bank of Chicago.
  • Handle: RePEc:fip:fedhma:94-8
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    References listed on IDEAS

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    1. repec:cup:etheor:v:10:y:1994:i:1:p:172-97 is not listed on IDEAS
    2. Abowd, John M & Card, David, 1987. "Intertemporal Labor Supply and Long-term Employment Contracts," American Economic Review, American Economic Association, vol. 77(1), pages 50-68, March.
    3. Shanken, Jay, 1990. "Intertemporal asset pricing : An Empirical Investigation," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 99-120.
    4. Joshua D. Angrist & Alan B. Krueger, 1993. "Split Sample Instrumental Variables," Working Papers 699, Princeton University, Department of Economics, Industrial Relations Section..
    5. Bruce N. Lehmann, 1986. "Residual Risk Revisited," NBER Working Papers 1908, National Bureau of Economic Research, Inc.
    6. 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, vol. 10(01), pages 172-197, March.
    7. Altonji, Joseph G. & Martins, Ana Paula & Siow, Aloysius, 2002. "Dynamic factor models of consumption, hours and income," Research in Economics, Elsevier, vol. 56(1), pages 3-59, June.
    8. Lehmann, Bruce N., 1990. "Residual risk revisited," Journal of Econometrics, Elsevier, vol. 45(1-2), pages 71-97.
    9. Chamberlain, Gary, 1982. "Multivariate regression models for panel data," Journal of Econometrics, Elsevier, vol. 18(1), pages 5-46, January.
    10. M. Arellano & J. D. Sargan, 1990. "Imhof Approximations to Econometric Estimators," Review of Economic Studies, Oxford University Press, vol. 57(4), pages 627-646.
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