A comparative study of several bootstrap-based tests for the volatility in continuous-time diffusion models

  • Tianshun YanEmail author
  • Liping Zhang
Original Article


This article develops three bootstrap-based tests for a parametric form of volatility function in continuous-time diffusion models. The three tests are the generalized likelihood ratio test by Fan et al. (Ann Stat 29(1):153–193, 2001), the nonparametric kernel test (LWZ) by Li and Wang (J Econometrics 87(1):145–165, 1998) and Zheng (J Econ 75(2):263–289, 1996) and the nonparametric test (CHS) by Chen et al. (2017). Monte Carlo simulations are performed to evaluate the sizes and power properties of these bootstrap-based tests in finite samples over a range of bandwidth values. We find that the bootstrap-based tests are not influenced by prior restrictions on the functional form of the drift function and that the bootstrap-based CHS test has better power performance than the bootstrap-based GLR and LWZ tests in detecting a parametric form of volatility. An empirical study on weekly treasury bill rate is further conducted to demonstrate these bootstrap-based test procedures.


Continuous-time diffusion models Generalized likelihood ratio test Nonparametric kernel test Bootstrap Treasury bill rate 

JEL Classification

C12 C13 C58 



  1. Aït-Sahalia Y (1999) Transition densities for interest rate and other nonlinear diffusions. J Financ 54(4):1361–1395CrossRefGoogle Scholar
  2. Chan KC, Karolyi AG, Longstaff FA, Sanders AB (1992) An empirical comparison of alternative models of the short-term interest tate. J Financ 47(3):1209–1227CrossRefGoogle Scholar
  3. Chen Q, Hu M, Song X (2017) A nonparametric specification test for the volatility functions of diffusion processes. Economet RewGoogle Scholar
  4. Cox JC, Ingersoll JE, Ross SA (1980) An analysis of variable rate loan contracts. J Financ 35:389–403CrossRefGoogle Scholar
  5. Cox JC, Ingersoll JE, Ross SA (1985) A theory of the term structure of interest rates. Econometrica 53(2):385–407CrossRefGoogle Scholar
  6. Fan Y, Li Q (1996) Consistent model specification tests: Omitted variables, parametric and semiparametric functional forms. Econometrica 64(4):865–890CrossRefGoogle Scholar
  7. Fan J, Zhang C (2003) A reexamination of diffusion estimators with applications to financial model validation. J Am Stat Asso 98(461):118–134CrossRefGoogle Scholar
  8. Fan J, Zhang C, Zhang J (2001) Generalized likelihood ratio statistics and Wilks phenomenon. Ann Stat 29(1):153–193CrossRefGoogle Scholar
  9. Gallant AR, Long JR (1997) Estimating stochastic differential equations efficiently by minimum chi-Ssquared. Biometrika 84:125–141CrossRefGoogle Scholar
  10. Hsiao C, Li Q (2001) A consistent test for conditional heteroskedasticity in time-series regression models. Economet Theor 17(1):188–221CrossRefGoogle Scholar
  11. Kim MS, Wang S (2006) Sizes of two bootstrap-based nonparametric specification tests for the drift function in continuous time models. Comput Stat Data An 50:1793–1806CrossRefGoogle Scholar
  12. Li Q, Wang S (1998) A simple consistent bootstrap test for a parametric regression function. J Econometrics 87(1):145–165CrossRefGoogle Scholar
  13. Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5(2):177–188CrossRefGoogle Scholar
  14. Zheng JX (1996) A consistent test of functional form via nonparametric estimation technique. J Econ 75(2):263–289CrossRefGoogle Scholar
  15. Zheng X (2009) Testing heteroscedasticity in nonlinear and nonparametric regressions. Can J Stat 37(2):282–300CrossRefGoogle Scholar

Copyright information

© ISEG – Instituto Superior de Economia e Gestão 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsXi’an Jiaotong UniversityXi’anChina
  2. 2.School of FinanceChongqing Technology and Business UniversityChongqingChina

Personalised recommendations