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Local rank tests in a multivariate nonparametric relationship

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  • Natércia Fortuna

    ()
    (CEMPRE, Faculdade de Economia da Universidade do Porto)

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 local rank are defined in terms of the eigenvalues of a kernel-based estimator of some matrix. The asymptotics of the eigenvalues is established by using the so-called Fujikoshi expansion along with some techniques of the theory of U-statistics. We present a simulation study which examines the small sample properties of local rank tests. We also apply the local rank tests and the local rank estimators to a demand system given by a newly constructed data set. This work can be viewed as a “local” extension of the tests for a number of factors in a nonparametric relationship introduced by Stephen Donald.

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Bibliographic Info

Paper provided by Universidade do Porto, Faculdade de Economia do Porto in its series FEP Working Papers with number 137.

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Length: 46 pages
Date of creation: Feb 2004
Date of revision:
Handle: RePEc:por:fepwps:137

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Keywords: nonparametric relationship; local rank; local rank estimation; kernel smoothing; consistent tests; demand systems.;

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References

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  1. Powell, James L & Stock, James H & Stoker, Thomas M, 1989. "Semiparametric Estimation of Index Coefficients," Econometrica, Econometric Society, vol. 57(6), pages 1403-30, November.
  2. repec:cup:etheor:v:9:y:1993:i:2:p:222-40 is not listed on IDEAS
  3. Cragg, John G. & Donald, Stephen G., 1997. "Inferring the rank of a matrix," Journal of Econometrics, Elsevier, vol. 76(1-2), pages 223-250.
  4. J. A. Hausman & W. K. Newey & J. L. Powel, 1988. "Nonlinear Errors in Variables: Estimation of Some Engel Curves," Working papers 504, Massachusetts Institute of Technology (MIT), Department of Economics.
  5. Stephen G. Donald & Natércia Fortuna & Vladas Pipiras, 2005. "Local and global rank tests for multivariate varying-coefficient models," FEP Working Papers 196, Universidade do Porto, Faculdade de Economia do Porto.
  6. Kleibergen, F.R. & Paap, R., 2003. "Generalized Reduced Rank Tests using the Singular Value Decomposition," Econometric Institute Research Papers EI 2003-01, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
  7. Fortuna, Natercia, 2008. "Local rank tests in a multivariate nonparametric relationship," Journal of Econometrics, Elsevier, vol. 142(1), pages 162-182, January.
  8. 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.
  9. Lewbel, Arthur, 1991. "The Rank of Demand Systems: Theory and Nonparametric Estimation," Econometrica, Econometric Society, vol. 59(3), pages 711-30, May.
  10. Pagan,Adrian & Ullah,Aman, 1999. "Nonparametric Econometrics," Cambridge Books, Cambridge University Press, number 9780521355643, October.
  11. 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.
  12. 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.
  13. 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.
  14. Lewbel, Arthur, 1989. "A Demand System Rank Theorem," Econometrica, Econometric Society, vol. 57(3), pages 701-05, May.
  15. 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.
  16. Robin, J.M. & Smith, R.J., 1995. "Tests of Rank," Cambridge Working Papers in Economics 9521, Faculty of Economics, University of Cambridge.
  17. 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.
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Citations

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Cited by:
  1. 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.
  2. Stephen G. Donald & Natércia Fortuna & Vladas Pipiras, 2005. "On rank estimation in symmetric matrices: the case of indefinite matrix estimators," FEP Working Papers 167, Universidade do Porto, Faculdade de Economia do Porto.
  3. Natercia Fortuna, 2004. "Local rank tests in a multivariate nonparametric relationship," Econometric Society 2004 North American Summer Meetings 446, Econometric Society.

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