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A Maximum Likelihood Method for the Incidental Parameter Problem

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  • Marcelo Moreira

Abstract

This paper uses the invariance principle to solve the incidental parameter problem. We seek group actions that preserve the structural parameter and yield a maximal invariant in the parameter space with fixed dimension. M-estimation from the likelihood of the maximal invariant statistic yields the maximum invariant likelihood estimator (MILE). We apply our method to (i) a stationary autoregressive model with fixed effects; (ii) an agent-specific monotonic transformation model; (iii) an instrumental variable (IV) model; and (iv) a dynamic panel data model with fixed effects. In the first two examples, there exist group actions that completely discard the incidental parameters. In a stationary autoregressive model with fixed effects, MILE coincides with existing conditional and integrated likelihood methods. The invariance principle also gives a new perspective to the marginal likelihood approach. In an agent-specific monotonic transformation model, our approach yields an estimator that is consistent and asymptotically normal when errors are Gaussian. In an instrumental variable (IV) model, this paper unifies asymptotic results under strong instruments (SIV) and many weak instruments (MWIV) frameworks. We obtain consistency, asymptotic normality, and optimality results for the limited information maximum likelihood estimator directly from the invariant likelihood. Our approach is parallel to M-estimation in problems in which the number of parameters does not change with the sample size. In a dynamic panel data model with N individuals and T time periods, MILE is consistent as long as NT goes to infinity. We obtain a large N, fixed T bound; this bound coincides with Hahn and Kuersteiner's (2002) bound when T goes to infinity. MILE reaches (i) our bound when N is large and T is fixed; and (ii) Hahn and Kuersteiner's (2002) bound when both N and T are large.

Suggested Citation

  • Marcelo Moreira, 2008. "A Maximum Likelihood Method for the Incidental Parameter Problem," NBER Working Papers 13787, National Bureau of Economic Research, Inc.
  • Handle: RePEc:nbr:nberwo:13787
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    1. Paul A. Bekker & Jan Ploeg, 2005. "Instrumental variable estimation based on grouped data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 59(3), pages 239-267.
    2. Bekker, Paul A, 1994. "Alternative Approximations to the Distributions of Instrumental Variable Estimators," Econometrica, Econometric Society, vol. 62(3), pages 657-681, May.
    3. Jinyong Hahn & Guido Kuersteiner, 2002. "Asymptotically Unbiased Inference for a Dynamic Panel Model with Fixed Effects when Both "n" and "T" Are Large," Econometrica, Econometric Society, vol. 70(4), pages 1639-1657, July.
    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. Chioda, Laura & Jansson, Michael, 2009. "Optimal Invariant Inference When The Number Of Instruments Is Large," Econometric Theory, Cambridge University Press, vol. 25(03), pages 793-805, June.
    6. 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.
    7. Hansen, Christian & Hausman, Jerry & Newey, Whitney, 2008. "Estimation With Many Instrumental Variables," Journal of Business & Economic Statistics, American Statistical Association, vol. 26, pages 398-422.
    8. Ahn, Seung C. & Schmidt, Peter, 1995. "Efficient estimation of models for dynamic panel data," Journal of Econometrics, Elsevier, vol. 68(1), pages 5-27, July.
    9. Moreira, Marcelo J., 2009. "Tests with correct size when instruments can be arbitrarily weak," Journal of Econometrics, Elsevier, vol. 152(2), pages 131-140, October.
    10. Whitney K. Newey, 2004. "Efficient Semiparametric Estimation via Moment Restrictions," Econometrica, Econometric Society, vol. 72(6), pages 1877-1897, November.
    11. Chamberlain, Gary, 1987. "Asymptotic efficiency in estimation with conditional moment restrictions," Journal of Econometrics, Elsevier, vol. 34(3), pages 305-334, March.
    12. Gary Chamberlain & Marcelo J. Moreira, 2009. "Decision Theory Applied to a Linear Panel Data Model," Econometrica, Econometric Society, vol. 77(1), pages 107-133, January.
    13. Abrevaya, Jason, 2000. "Rank estimation of a generalized fixed-effects regression model," Journal of Econometrics, Elsevier, vol. 95(1), pages 1-23, March.
    14. Manuel Arellano & Stephen Bond, 1991. "Some Tests of Specification for Panel Data: Monte Carlo Evidence and an Application to Employment Equations," Review of Economic Studies, Oxford University Press, vol. 58(2), pages 277-297.
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    More about this item

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C23 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Models with Panel Data; Spatio-temporal Models
    • C30 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - General

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