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Local rank tests in a multivariate nonparametric relationship

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  • Natercia Fortuna

Abstract

Consider a multivariate nonparametric model where the unknown vector of functions depends on two sets of explanatory variables. For a fixed level of one set of explanatory variables, we provide consistent statistical tests, called local rank tests, to determine whether the multivariate relationship can be explained by a smaller number of functions. We also provide estimators for the smallest number of functions, called local rank, explaining the relationship. The local rank tests and the estimators of the local rank are based on the asymptotics of the eigenvalues of some matrix. This matrix is estimated by using kernel-based methods and the asymptotics of its eigenvalues is established by using the so-called Fujikoshi expansions along with some techniques of the theory of U-statistics. We present a simulation study which examines small sample properties of local rank tests. We also apply the local rank tests and the local rank estimators of the paper to a demand system given by a newly constructed data set. Our results can be viewed as localized counterparts of tests for a number of factors in a nonparametric relationship introduced by Donald

Suggested Citation

  • Natercia Fortuna, 2004. "Local rank tests in a multivariate nonparametric relationship," Econometric Society 2004 North American Summer Meetings 446, Econometric Society.
  • Handle: RePEc:ecm:nasm04:446
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    References listed on IDEAS

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    1. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
    2. Lwebel Arthur & Perraudin William, 1995. "A Theorem on Portfolio Separation with General Preferences," Journal of Economic Theory, Elsevier, vol. 65(2), pages 624-626, April.
    3. Lewbel, Arthur, 1989. "A Demand System Rank Theorem," Econometrica, Econometric Society, vol. 57(3), pages 701-705, May.
    4. Kleibergen, Frank & Paap, Richard, 2006. "Generalized reduced rank tests using the singular value decomposition," Journal of Econometrics, Elsevier, vol. 133(1), pages 97-126, July.
    5. Robin, Jean-Marc & Smith, Richard J., 2000. "Tests Of Rank," Econometric Theory, Cambridge University Press, vol. 16(02), pages 151-175, April.
    6. Pagan,Adrian & Ullah,Aman, 1999. "Nonparametric Econometrics," Cambridge Books, Cambridge University Press, number 9780521355643, Fall.
    7. White, Halbert & Hong, Yongmiao, 1999. "M-Testing Using Finite and Infinite Dimensional Parameter Estimators," University of California at San Diego, Economics Working Paper Series qt9qz123ng, Department of Economics, UC San Diego.
    8. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-1430, November.
    9. Fortuna, Natercia, 2008. "Local rank tests in a multivariate nonparametric relationship," Journal of Econometrics, Elsevier, vol. 142(1), pages 162-182, January.
    10. Stephen G. Donald, 1997. "Inference Concerning the Number of Factors in a Multivariate Nonparametric Relationship," Econometrica, Econometric Society, vol. 65(1), pages 103-132, January.
    11. Donald, Stephen G. & Fortuna, Natércia & Pipiras, Vladas, 2011. "Local and Global Rank Tests for Multivariate Varying-Coefficient Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(2), pages 295-306.
    12. Cragg, John G. & Donald, Stephen G., 1993. "Testing Identifiability and Specification in Instrumental Variable Models," Econometric Theory, Cambridge University Press, vol. 9(02), pages 222-240, April.
    13. repec:cup:etheor:v:9:y:1993:i:2:p:222-40 is not listed on IDEAS
    14. Russell, Thomas & Farris, Frank, 1993. "The geometric structure of some systems of demand equations," Journal of Mathematical Economics, Elsevier, vol. 22(4), pages 309-325.
    15. Christopher J. Nicol, 2001. "The rank and model specification of demand systems: an empirical analysis using United States microdata," Canadian Journal of Economics, Canadian Economics Association, vol. 34(1), pages 259-289, February.
    16. Lewbel, Arthur, 1991. "The Rank of Demand Systems: Theory and Nonparametric Estimation," Econometrica, Econometric Society, vol. 59(3), pages 711-730, May.
    17. Hausman, J. A. & Newey, W. K. & Powell, J. L., 1995. "Nonlinear errors in variables Estimation of some Engel curves," Journal of Econometrics, Elsevier, vol. 65(1), pages 205-233, January.
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    1. Fortuna, Natercia, 2008. "Local rank tests in a multivariate nonparametric relationship," Journal of Econometrics, Elsevier, vol. 142(1), pages 162-182, January.
    2. Donald, Stephen G. & Fortuna, Natércia & Pipiras, Vladas, 2011. "Local and Global Rank Tests for Multivariate Varying-Coefficient Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(2), pages 295-306.
    3. Donald, Stephen G. & Fortuna, Nat rcia & Pipiras, Vladas, 2007. "On Rank Estimation In Symmetric Matrices: The Case Of Indefinite Matrix Estimators," Econometric Theory, Cambridge University Press, vol. 23(06), pages 1217-1232, December.

    More about this item

    Keywords

    Nonparametric relationship; local rank; local rank estimation; kernel smoothing; consistent tests; demand systems.;

    JEL classification:

    • D12 - Microeconomics - - Household Behavior - - - Consumer Economics: Empirical Analysis

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