Statistical Papers

, Volume 60, Issue 1, pp 293–312 | Cite as

Local influence diagnostics for the test of mean–variance efficiency and systematic risks in the capital asset pricing model

  • Manuel GaleaEmail author
  • Patricia Giménez
Regular Article


In this paper we consider the capital asset pricing model under the multivariate normal distribution for modeling asset returns. We develop and implement local influence diagnostic techniques not based on likelihood displacement. The main interest is to consider the test of mean–variance efficiency of a given portfolio as objective function. Sensitivity of maximum likelihood estimators of slopes is also considered. For this purpose, first and second-order local influence measures are calculated under a case weights perturbation scheme. In particular, for our objective functions, which have nonzero first derivative at the critical point, the proposed second-order measures are scale invariant, unlike the normal curvature. The performance of the proposed measures is illustrated by using two real data sets as well as a simulation study and a simulated data set. Empirical results seem to indicate that the Wald test statistic is less sensitive to atypical returns than maximum likelihood estimators of the slopes. By the other hand, there is some evidence about the convenience of jointly using first and second-order influence measures, to detect months/days with outlying returns, principally in Wald test statistic.


Local influence diagnostics First and second-order approaches Capital asset pricing model Wald test statistic 

Mathematics Subject Classification

62J05 62J20 62F03 



We would like to thank the Associate Editor and two referees for their helpful comments and suggestions, leading to improvement of the paper. Also, we acknowledge the partial financial support from Projects Fondecyt 1110318 and 1150325, Chile.


  1. Amenc N, Le Sourd V (2003) Portfolio theory and performance analysis. Wiley, New YorkGoogle Scholar
  2. Bartholdy J, Peare P (2003) Unbiased estimation of expected return using. Int Rev Financ Anal 12:69–81CrossRefGoogle Scholar
  3. Broquet C, Cobbaut R, Gillet R, van den Berg A (2004) Gestion de Portefeuille, 4th edn. De Boeck Université, BruxellesGoogle Scholar
  4. Cademartori D, Romo C, Campos R, Galea M (2003) Robust estimation of systematic risk using the t distribution in the Chilean stock markets. Appl Econ Lett 10:447–453CrossRefGoogle Scholar
  5. Cadigan NG, Farrell PJ (2002) Generalized local influence with applications to fish stock cohort analysis. Appl Stat 51:469–483zbMATHGoogle Scholar
  6. Campbell J, Lo A, MacKinlay A (1997) Econometrics of financial markets. Princeton University Press, PrincetonzbMATHGoogle Scholar
  7. Chatterjee S, Hadi AS (1988) Sensitivity analysis in linear regression. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  8. Chen F, Zhu H, Lee S (2009) Perturbation selection and local influence analysis for nonlinear structural equation model. Psychometrika 74:493–516MathSciNetCrossRefzbMATHGoogle Scholar
  9. Chen F, Zhu H, Song X, Lee S (2010) Perturbation selection and local influence analysis for generalized linear mixed models. J Comput Gr Stat 19:826–842MathSciNetCrossRefGoogle Scholar
  10. Cook RD (1986) Assessment of local influence. J R Stat Soc B 48:133–169MathSciNetzbMATHGoogle Scholar
  11. Cook RD, Weisberg S (1982) Residuals and influence in regression. Chapman and Hall, LondonzbMATHGoogle Scholar
  12. Díaz-García JA, Galea M, Leiva-Sánchez V (2003) Influence diagnostics for elliptical multivariate regression models. Commun Stat Theory Methods 32:625–641MathSciNetCrossRefzbMATHGoogle Scholar
  13. Elsas R, El-Shaer M, Theissen E (2003) Beta and returns revisited evidence from the German stock market. J Int Financ Mark Inst Money 13:1–18CrossRefGoogle Scholar
  14. Escobar E, Meeker W (1992) Assessing influence in regression analysis with censored data. Biometrics 48:507–528MathSciNetCrossRefzbMATHGoogle Scholar
  15. Fama E, French K (1992) The cross-section of expected stock returns. J Financ 47:427–465CrossRefGoogle Scholar
  16. Fung WK, Kwan CW (1997) A note on local influence based on normal curvature. J R Stat Soc Ser B 59:839–843MathSciNetCrossRefzbMATHGoogle Scholar
  17. Galea M, Bolfarine H, de Castro M (2002) Local influence in comparative calibration models. Biom J 44:59–81MathSciNetCrossRefzbMATHGoogle Scholar
  18. Galea M, Díaz-García J, Vilca F (2008) Influence diagnostics in the capital asset pricing model under elliptical distributions. J Appl Stat 35(2):179–192MathSciNetCrossRefGoogle Scholar
  19. Galea M, Paula GA, Bolfarine H (1997) Local influence in elliptical linear regression models. Statistician 46:71–79Google Scholar
  20. Gibbons M, Ross S, Shanken J (1989) A test of the efficiency of a given portfolio. Econometrica 57:1121–1153MathSciNetCrossRefzbMATHGoogle Scholar
  21. Giménez P, Galea M (2013) Influence measures on corrected score estimators in functional heteroscedastic measurement error models. J Multivar Anal 114:1–15MathSciNetCrossRefzbMATHGoogle Scholar
  22. Hadi AS, Nyquist H (1999) Frechet distance as a tool for diagnosing multivariate data. Linear Algebra Appl 289:183–201MathSciNetCrossRefzbMATHGoogle Scholar
  23. Lawrance AJ (1988) Regression transformation diagnostics using local influence. J Am Stat Assoc 83:1067–1072MathSciNetCrossRefGoogle Scholar
  24. Lee SY, Wang SJ (1996) Sensitivity analysis of structural equation models. Psychometrika 61:93–108MathSciNetCrossRefzbMATHGoogle Scholar
  25. Lesaffre E, Verbeke G (1998) Local influence in linear mixed models. Biometrics 54:570–582CrossRefzbMATHGoogle Scholar
  26. Levy H (2012) The capital asset pricing model in the 21st century: analytical, empirical, and behavioral perspectives. Cambridge University Press, New YorkGoogle Scholar
  27. Lintner J (1965) The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev Econ Stat 41:13–37CrossRefGoogle Scholar
  28. Liu S (2000) On local influence for elliptical linear models. Stat Pap 41:211–224MathSciNetCrossRefzbMATHGoogle Scholar
  29. Liu S (2002) Local influence in multivariate elliptical linear regression models. Linear Algebra Appl 354:159–174MathSciNetCrossRefzbMATHGoogle Scholar
  30. Magnus JR, Neudecker H (1979) The commutation matrix: some properties and applications. Ann Stat 7:381–394MathSciNetCrossRefzbMATHGoogle Scholar
  31. Mossin J (1966) Equilibrium in capital asset market. Econometrica 35:768–783CrossRefGoogle Scholar
  32. Murray MK, Rice JW (1993) Differential geometry and statistics. Chapman and Hall, LondonCrossRefzbMATHGoogle Scholar
  33. Poon WY, Poon YS (1999) Conformal normal curvature and assessment of local influence. J R Stat Soc Ser B 61:51–61MathSciNetCrossRefzbMATHGoogle Scholar
  34. R Development Core Team (2013) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna.
  35. Ruppert D, Matteson D (2015) Statistics and data analysis for financial rngineering, with R examples, 2nd edn. Springer, New YorkzbMATHGoogle Scholar
  36. Sharpe W (1964) Capital asset prices: a theory of markets equilibrium under conditions of risk. J Financ 19:425–442Google Scholar
  37. Shi L (1997) Local influence in principal components analysis. Biometrika 84:175–186MathSciNetCrossRefzbMATHGoogle Scholar
  38. Shi X, Zhu H, Ibrahim JG (2009) Local influence for generalized linear models with missing covariates. Biometrics 65:1164–1174MathSciNetCrossRefzbMATHGoogle Scholar
  39. Thomas W, Cook RD (1990) Assessing influence on predictions from generalized linear models. Technometrics 32:59–65MathSciNetCrossRefGoogle Scholar
  40. Uribe-Opazo M, Borssoi J, Galea M (2012) Influence diagnostics in Gaussian spatial linear models. J Appl Stat 39(3):615–630MathSciNetCrossRefGoogle Scholar
  41. van der Hart J, Slagter E, van Dijk D (2003) Stock selection strategies in emerging markets. J Empir Financ 10:105–132CrossRefGoogle Scholar
  42. Wu X, Luo Z (1993) Second-order approach to local influence. J R Stat Soc B 55:929–939zbMATHGoogle Scholar
  43. Zhao Y, Lee AH (1998) Influence diagnostics for simultaneous equations models. Aust N Z J Stat 40:345–357CrossRefzbMATHGoogle Scholar
  44. Zhu HT, Lee SY (2001) Local influence for incomplete data models. J R Stat Soc B 63:111–126MathSciNetCrossRefzbMATHGoogle Scholar
  45. Zhu HT, Ibrahim JG, Lee S, Zhang H (2007) Perturbation selection and influence measures in local influence analysis. Ann Stat 35:2565–2588MathSciNetCrossRefzbMATHGoogle Scholar
  46. Zhu F, Shi L, Liu S (2015) Influence diagnostics in log-linear integer-valued GARCH models. AStA Adv Stat Anal 99:311–335MathSciNetCrossRefzbMATHGoogle Scholar
  47. Zhu F, Liu S, Shi L (2016) Local influence analysis for Poisson autoregression with an application to stock transaction data. Stat Neerl 70:4–25MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Departamento de EstadísticaPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaUniversidad Nacional de Mar del PlataMar del PlataArgentina

Personalised recommendations