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Asymptotic Properties of Quasi-Maximum Likelihood Estimators and Test Statistics

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  • Thomas E. MaCurdy

Abstract

We examine the implications of arbitrage in a market with many assets. The absence of arbitrage opportunities implies that the linear functionals that give the mean and cost of a portfolio are continuous; hence there exist unique portfolios that represent these functionals. The mean variance efficient set is a cone generated by these portfolios. Ross [16, 18J showed that if there is a factor structure, then the distance between the vector or mean returns and the space spanned by the factor loadings is bounded as the number of assets increases. We show that if the covariance matrix of asset returns has only K unbounded eigenvalues, then the corresponding K eigenvectors converge and play the role of factor loadings in Ross' result. Hence only a principal components analysis is needed to test the arbitrage pricing theory. Our eigenvalue conditional can hold even though conventional measures of the approximation error in a K factor model are unbounded. We also resolve the question of when a market with many assets permits so much diversification that risk-free investment opportunities are available.

Suggested Citation

  • Thomas E. MaCurdy, 1981. "Asymptotic Properties of Quasi-Maximum Likelihood Estimators and Test Statistics," NBER Technical Working Papers 0014, National Bureau of Economic Research, Inc.
    • Handle: RePEc:nbr:nberte:0014
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    1. Amemiya, Takeshi, 1973. "Regression Analysis when the Dependent Variable is Truncated Normal," Econometrica, Econometric Society, vol. 41(6), pages 997-1016, November.
    2. White, Halbert, 1980. "Nonlinear Regression on Cross-Section Data," Econometrica, Econometric Society, vol. 48(3), pages 721-746, April.
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    Cited by:

    1. Emma Tominey, 2010. "The Timing of Parental Income and Child Outcomes: The Role of Permanent and Transitory Shocks," CEE Discussion Papers 0120, Centre for the Economics of Education, LSE.
    2. Pedro Carneiro & Kjell Salvanes & Emma Tominey, 2024. "Insurance against Income Shocks, Parental Investments, and Child Development," CEPEO Working Paper Series 24-04, UCL Centre for Education Policy and Equalising Opportunities, revised Jun 2024.
    3. Jongmoo Jay Choi & Elyas Elyasiani, 1996. "Derivative Exposure and the Interest Rate and Exchange Rate Risks of U.S. Banks," Center for Financial Institutions Working Papers 96-53, Wharton School Center for Financial Institutions, University of Pennsylvania.
    4. Hugo Kruiniger, 2002. "On the Estimation of Panel Regression Models with Fixed Effects," Working Papers 450, Queen Mary University of London, School of Economics and Finance.
    5. Emma Tominey, 2010. "The Timing of Parental Income and Child Outcomes: The Role of Permanent and Transitory Shocks," Discussion Papers 10/21, Department of Economics, University of York.
    6. Hall, Bronwyn H, 1987. "The Relationship between Firm Size and Firm Growth in the U.S. Manufacturing Sector," Journal of Industrial Economics, Wiley Blackwell, vol. 35(4), pages 583-606, June.
    7. Jongmoo Choi & Elyas Elyasiani, 1997. "Derivative Exposure and the Interest Rate and Exchange Rate Risks of U.S. Banks," Journal of Financial Services Research, Springer;Western Finance Association, vol. 12(2), pages 267-286, October.
    8. Songhua Tan & Qianqian Zhu, 2022. "Asymmetric linear double autoregression," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(3), pages 371-388, May.
    9. Zvi Griliches & Bronwyn H. Hall & Ariel Pakes, 1988. "R&D, Patents, and Market Value Revisited: Is There Evidence of A SecondTechnological Opportunity Related Factor?," NBER Working Papers 2624, National Bureau of Economic Research, Inc.
    10. Choi, Jongmoo Jay & Jiang, Cao, 2009. "Does multinationality matter? Implications of operational hedging for the exchange risk exposure," Journal of Banking & Finance, Elsevier, vol. 33(11), pages 1973-1982, November.

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