The Annals of Applied Probability

The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices

Alexei Onatski

Full-text: Open access

Abstract

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663–714] which finds the Tracy–Widom limit for the largest eigenvalue of a nonsingular p-dimensional complex Wishart matrix Wp, n) to the case of several of the largest eigenvalues of the possibly singular (n<p) matrix Wp, n). As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643–1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.

Article information

Source
Ann. Appl. Probab., Volume 18, Number 2 (2008), 470-490.

Dates
First available in Project Euclid: 20 March 2008

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1206018194

Digital Object Identifier
doi:10.1214/07-AAP454

Mathematical Reviews number (MathSciNet)
MR2398763

Zentralblatt MATH identifier
1141.60009

Subjects
Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

Keywords
Singular Wishart matrix Tracy–Widom distribution largest eigenvalues random matrix theory approximate factor model number of factors arbitrage pricing theory

Citation

Onatski, Alexei. The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18 (2008), no. 2, 470--490. doi:10.1214/07-AAP454. https://projecteuclid.org/euclid.aoap/1206018194


Export citation

References

  • Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis. Wiley, New York.
  • Andreief, C. (1883). Note sur une relation les integrales definies des produits des fonctions. Mem. de la Soc. Sci. Bordeaux 2 1–14.
  • Bai, J. and Ng, S. (2002). Determining the number of factors in approximate factor models Econometrica 70 191–221.
  • Baik, J., Ben Arous, G. and Peche, S. (2005). Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33 1643–1697.
  • Brown, S. and Weinstein, M. (1983). A new approach to testing asset pricing models: The bilinear paradigm. J. Finance 38 711–743.
  • Cattell, R. B. (1966). The scree test for the number of factors. Multivariate Behavioral Research 1 245–276.
  • Connor, G. and Korajczyk, R. (1993). A test for the number of factors in an approximate factor model. J. Finance 58 1263–1291.
  • Chamberlain, G. and Rothschild, M. (1983). Arbitrage, factor structure, and mean-variance analysis on large asset markets. Econometrica 51 1281–1304.
  • El Karoui, N. (2007). Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices. Ann. Probab. 35 663–714.
  • Huang, R. and Jo, H. (1995). Data frequency and the number of factors in stock returns. J. Banking and Finance 19 987–1003.
  • James, A. (1954). Normal multivariate analysis and the orthogonal group. Ann. Math. Statist. 25 40–75.
  • Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. 153 259–296.
  • Makarov, I. and Papanikolaou, D. (2007). Sources of systematic risk. Unpublished manuscript, London Business School.
  • Mehta, M. (2004). Random Matrices. Academic Press, New York.
  • Olver, F. (1974). Asymptotics and Special Functions. Academic Press, New York.
  • Onatski, A. (2005). Determining the number of factors from the empirical distribution of eigenvalues. Manuscript, Columbia Univ.
  • Onatski, A. (2007). A formal statistical test for the number of factors in the approximate factor models. Unpublished manuscript, Columbia Univ.
  • Ratnarajah, T. and Vaillanourt, R. (2005). Complex singular Wishart matrices and applications. Comput. Math. Appl. 50 399–411.
  • Roll, R. and Ross, S. (1980). An empirical investigation of the arbitrage pricing theory. J. Finance 5 1073–1103.
  • Ross, S. (1976). The arbitrage theory of capital asset pricing. J. Econom. Theory 13 341–360.
  • Rudin, W. (1980). Function Theory in the Unit Ball of Cn. Springer, New York.
  • Simon, B. (2005). Trace Ideals and Their Applications, 2nd ed. Amer. Math. Soc., Providence, RI.
  • Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923–975.
  • Soshnikov, A. (2002). A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices. J. Statist. Phys. 108 1033–1056.
  • Stock, J. and Watson, M. (2002). Forecasting using principal components from a large number of predictors. J. Amer. Statist. Assoc. 97 1167–1179.
  • Tracy, C. A. and Widom, H. (1994). Level spacing distributions and the Airy kernel. Comm. Math. Phys. 159 151–174.
  • Tracy, C. A. and Widom, H. (1998). Correlation functions, cluster functions, and spacing distributions for random matrices. J. Statist. Phys. 92 809–835.
  • Trzcinka, C. (1986). On the number of factors in the arbitrage pricing model. J. Finance 41 347–368.