Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator

18 Pages Posted: 27 Jun 2007 Last revised: 10 Sep 2010

Date Written: September 1988

Abstract

New results on the exact small sample distribution of the instrumental variable estimator are presented by studying an important special case. The exact closed forms for the probability density and cumulative distribution functions are given. There are a number of surprising findings. The small sample distribution is bimodal. with a point of zero probability mass. As the asymptotic variance grows large, the true distribution becomes concentrated around this point of zero mass. The central tendency of the estimator may be closer to the biased least squares estimator than it is to the true parameter value. The first and second moments of the IV estimator are both infinite. In the case in which least squares is biased upwards, and most of the mass of the IV estimator lies to the right of the true parameter, the mean of the IV estimator is infinitely negative. The difference between the true distribution and the normal asymptotic approximation depends on the ratio of the asymptotic variance to a parameter related to the correlation between the regressor and the regression, error. In particular, when the instrument is poorly correlated with the regressor, the asymptotic approximation to the distribution of the instrumental variable estimator will not be very accurate.

Suggested Citation

Nelson, Charles R. and Startz, Richard, Some Further Results on the Exact Small Sample Properties of the Instrumental Variable Estimator (September 1988). NBER Working Paper No. t0068, Available at SSRN: https://ssrn.com/abstract=994510

Charles R. Nelson (Contact Author)

Dept of Economics ( email )

Box 353330
Seattle, WA 98195-3330
United States

Richard Startz

UCSB ( email )

Department of Economics
University of California
Santa Barbara, CA 93106-9210
United States
805-893-2895 (Phone)

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