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The Limited Information Maximum Likelihood Estimator as an Angle

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  • T. W. Anderson

    (Department of Statistics and Department of Economics, Stanford University)

  • Naoto Kunitomo

    (Faculty of Economics, University of Tokyo)

  • Yukitoshi Matsushita

    (JSPS and Graduate School of Economics, University of Tokyo)

Abstract

When an econometric structural equation includes two endogenous variables and their coefficients are normalized so that their sum of squares is 1, it is natural to express them as the sine and cosine of an angle. The Limited Information Maximum Likelihood (LIML) estimator of this angle when the error covariance matrix is known has constant variance. Of all estimators with constant variance the LIML estimator minimizes the variance. Competing estimators, such as the Two-Stage Least Squares estimator, has much larger variance for some values of the parameter. The effect of weak instruments is studied.

Suggested Citation

  • T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2009. "The Limited Information Maximum Likelihood Estimator as an Angle," CIRJE F-Series CIRJE-F-619, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2009cf619
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    File URL: http://www.cirje.e.u-tokyo.ac.jp/research/dp/2009/2009cf619.pdf
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    References listed on IDEAS

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    1. Phillips, Peter C B, 1985. "The Exact Distribution of LIML: II," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 26(1), pages 21-36, February.
    2. Hillier, Grant H, 1990. "On the Normalization of Structural Equations: Properties of Direct Estimators," Econometrica, Econometric Society, vol. 58(5), pages 1181-1194, September.
    3. Gary Chamberlain, 2007. "Decision Theory Applied to an Instrumental Variables Model," Econometrica, Econometric Society, vol. 75(3), pages 609-652, May.
    4. T. W. Anderson & Naoto Kunitomo & Yukitoshi Matsushita, 2008. "On Finite Sample Properties of Alternative Estimators of Coefficients in a Structural Equation with Many Instruments," CIRJE F-Series CIRJE-F-577, CIRJE, Faculty of Economics, University of Tokyo.
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