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Approximate Distributions of the Likelihood Ratio Statistic in a Structural Equation with Many Instruments


  • Yukitoshi Matsushita

    (CIRJE, Faculty of Economics, University of Tokyo)


This paper studies the properties of Likelihood Ratio (LR) tests associated with the limited information maximum likelihood (LIML) estimators in a structural form estimation when the number of instrumental variables is large. Two types of asymptotic theories are developed to approximate the distribution of the likelihood ratio (LR) statistics under the null hypothesis H0 : ƒÀ = ƒÀ0: the (large sample) asymptotic expansion and the large-Kn asymptotic theory. The size comparison of two modified LR tests based on these two asymptotics is made with Moreira's conditional likelihood ratio (CLR) test and the large K t-test.

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  • Yukitoshi Matsushita, 2007. "Approximate Distributions of the Likelihood Ratio Statistic in a Structural Equation with Many Instruments," CIRJE F-Series CIRJE-F-466, CIRJE, Faculty of Economics, University of Tokyo.
  • Handle: RePEc:tky:fseres:2007cf466

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    References listed on IDEAS

    1. Fujikoshi, Yasunori & Morimune, Kimio & Kunitomo, Naoto & Taniguchi, Masanobu, 1982. "Asymptotic expansions of the distributions of the estimates of coefficients in a simultaneous equation system," Journal of Econometrics, Elsevier, vol. 18(2), pages 191-205, February.
    2. Marcelo J. Moreira, 2003. "A Conditional Likelihood Ratio Test for Structural Models," Econometrica, Econometric Society, vol. 71(4), pages 1027-1048, July.
    3. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-681, May.
    4. Donald W. K. Andrews & Marcelo J. Moreira & James H. Stock, 2006. "Optimal Two-Sided Invariant Similar Tests for Instrumental Variables Regression," Econometrica, Econometric Society, vol. 74(3), pages 715-752, May.
    5. Morimune, Kimio, 1983. "Approximate Distributions of k-Class Estimators When the Degree of Overidentifiability Is Large Compared with the Sample Size," Econometrica, Econometric Society, vol. 51(3), pages 821-841, May.
    6. Theodore W. Anderson & Naoto Kunijtomo & Yukitoshi Matsushita, 2005. "A New Light from Old Wisdoms : Alternative Estimation Methods of Simultaneous Equations and Microeconometric Models," CIRJE F-Series CIRJE-F-321, CIRJE, Faculty of Economics, University of Tokyo.
    7. Yukitoshi Matsushita, 2007. "t-Tests in a Structural Equation with Many Instruments," CIRJE F-Series CIRJE-F-467, CIRJE, Faculty of Economics, University of Tokyo.
    8. Naoto Kunitomo & Yukitoshi Matsushita, 2003. "On Finite Sample Distributions of the Empirical Likelihood Estimator and the GMM Estimator," CIRJE F-Series CIRJE-F-200, CIRJE, Faculty of Economics, University of Tokyo.
    9. Naoto Kunitomo & Yukitoshi Matsushita, 2003. "Asymptotic Expansions of the Distributions of Semi-Parametric Estimators in a Linear Simultaneous Equations System," CIRJE F-Series CIRJE-F-237, CIRJE, Faculty of Economics, University of Tokyo.
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    Cited by:

    1. Naoto Kunitomo & T. W. Anderson, 2007. "On Likelihood Ratio Tests of Structural Coefficients: Anderson-Rubin (1949) revisited," CIRJE F-Series CIRJE-F-499, CIRJE, Faculty of Economics, University of Tokyo.

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