Chen, Song Xi and Gao, Jiti and Tang, Chenghong (2005): A test for model specification of diffusion processes. Published in: Annals of Statistics , Vol. 36, No. 1 (February 2008): pp. 162198.

PDF
MPRA_paper_11976.pdf Download (604kB)  Preview 
Abstract
We propose a test for model specification of a parametric diffusion process based on a kernel estimation of the transitional density of the process. The empirical likelihood is used to formulate a statistic, for each kernel smoothing bandwidth, which is effectively a Studentized L2distance between the kernel transitional density estimator and the parametric transitional density implied by the parametric process. To reduce the sensitivity of the test on smoothing bandwidth choice, the final test statistic is constructed by combining the empirical likelihood statistics over a set of smoothing bandwidths. To better capture the finite sample distribution of the test statistic and data dependence, the critical value of the test is obtained by a parametric bootstrap procedure. Properties of the test are evaluated asymptotically and numerically by simulation and by a real data example.
Item Type:  MPRA Paper 

Original Title:  A test for model specification of diffusion processes 
Language:  English 
Keywords:  Bootstrap; diffusion process; empirical likelihood; goodnessoffit test; time series; transitional density 
Subjects:  C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General 
Item ID:  11976 
Depositing User:  jiti Gao 
Date Deposited:  09. Dec 2008 00:25 
Last Modified:  24. Apr 2015 21:54 
References:  [1] AitSahalia, Y. (1996). Testing continuoustime models of the spot interest rate. Rev. Financial Studies 9 385–426. [2] AitSahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. J. Finance 54 1361–1395. [3] AitSahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closedform approximation approach. Econometrica 70 223–262. MR1926260 [4] AitSahalia,Y., Bickel,P.and Stoker, T. (2001). Goodnessoffit tests for regression using kernel methods. J. Econometrics 105 363–412. MR1873358 [5] AitSahalia,Y., Fan,J. and Peng, H. (2006). Nonparametric transition based tests for jump diffusions. Working paper. Available at http://www.princeton.edu/~yacine/research. htm. [6] AitSahalia,Y. and Kimmel, R. (2005). Estimating affine multifactor term structure models using closedform likelihood expansions. Working paper. Available at http://www. princeton.edu/~yacine/research.htm. [7] Bandi,F. and Phillips, P. C. B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241–284. MR1956859 [8] Brown, B. and Chen, S. X. (1998). Combined and least squares empirical likelihood. Ann. Inst. Statist. Math. 50 697–714. MR1671990 [9] Cai, Z. and Hong, Y. (2003). Nonparametric methods in continuoustime finance: A selective review. In Recent Advances and Trends in Nonparametric Statistics (M. G. Akritas and D. M. Politis, eds.) 283–302. [10] Chen, S.X. and Cui, H. J. (2007). On the second order properties of empirical likelihood with moment restrictions. J. Econometrics. To appear. [11] Chen, S.X., Gao,J. and Tang, C. Y. (2007). A test for model specification of diffusion processes. Technical report, Dept. Statistics, Iowa State Univ. [12] Chen, S.X., Hardle,W. and Li, M. (2003). An empirical likelihood goodnessoffit test for time series. J. Roy. Statist. Soc. Ser. B 65 663–678. MR1998627 [13] Cox, J.C., Ingersoll, J.E. and Ross, S. A. (1985). A theory of term structure of interest rates. Econometrica 53 385–407. MR0785475 [14] Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 434 674–688. MR1395735 [15] Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics. Statist. Sci. 20 317–357. MR2210224 [16] Fan,J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications. Chapman and Hall, London. MR1383587 [17] Fan, J. and Huang, L. (2001). Goodnessoffit test for parametric regression models. J. Amer. Statist. Assoc. 96 640–652. MR1946431 [18] Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York. MR1964455 [19] Fan,J., Yao,Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 189–196. MR1399164 [20] Fan,J. and Yim, T. H. (2004). A datadriven method for estimating conditional densities. Biometrika 91 819–834. MR2126035 [21] Fan,J. and Zhang, C. (2003). A reexamination of Stanton’s diffusion estimation with applications to financial model validation. J. Amer. Statist. Assoc. 457 118–134. MR1965679 [22] Fan,J., Zhang,C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193. MR1833962 [23] Fan,J. and Zhang, J. (2004). Sieved empirical likelihood ratio tests for nonparametric functions. Ann. Statist. 29 153–193. MR2102496 [24] Fan,Y. and Li, Q. (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64 865–890. MR1399221 [25] Gao,J. and King, M. L. (2005). Estimation and model specification testing in nonparametric and semiparametric regression models. Unpublished paper. Available at www.maths.uwa. edu.au/~jiti/jems.pdf. [26] GenonCaralot, V., Jeantheau, T. and Laredo, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051–1079. MR1809735 [27] Gozalo,P.L. and Linton, Q. (2001). Testing additivity in generalized nonparametric regression models with estimated parameters. J. Econometrics 104 1–48. MR1862028 [28] Hardle,W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926–1947. MR1245774 [29] Hart, J. (1997). Nonparametric Smoothing and LackofFit Tests. Springer, New York. MR1461272 [30] Hjellvik,V., Yao,Q. and Tjostheim, D. (1998). Linearity testing using local polynomial approximation. J. Statist. Plann. Inference 68 295–321. MR1629587 [31] Hjort,N.L., Mckeague,I.W. and Van Keilegom, I. (2006). Extending the scope of empirical likelihood. Manuscript. [32] Hong,Y. and Li, H. (2005). Nonparametric specification testing for continuoustime models with application to spot interest rates. Rev. Financial Studies 18 37–84. [33] Horowitz,J.L. and Spokoiny, V. G. (2001). An adaptive, rateoptimal test of a parametric meanregression model against a nonparametric alternative. Econometrica 69 599–632. MR1828537 [34] Hyndman,R.J. and Yao, Q. (2002). Nonparametric estimation and symmetry tests for conditional density functions. J. Nonparametr. Statist. 14 259–278. MR1905751 [35] Jiang,G. and Knight, J. (1997). A nonparametric approach to the estimation of diffusion processes with an application to a shortterm interest rate model. Econometric Theory 13 615–645. MR1491253 [36] Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084–2102. MR1474084 [37] Li, G. (2003). Nonparametric likelihood ratio goodnessoffit tests for survival data. J. Multivariate Analysis 86 166–182. MR1994727 [38] Li, Q. (1999). Consistent model specification tests for time series econometric models. J. Econometrics 92 101–147. MR1706996 [39] Lo, A. W. (1988). Maximum likelihood estimation of generalized Itô processes with discretely sampled data. Econometric Theory 4 231–247. MR0959611 [40] Müller, H.G. and Stadtmüller, U. (1999). Multivariate boundary kernels and a continuous least squares principle. J. Roy. Statist. Soc. Ser. B 61 439–458. MR1680306 [41] Owen, A. (1988). Empirical likelihood ratio confidence regions for a single functional. Biometrika 75 237–249. MR0946049 [42] Pritsker, M. (1998). Nonparametric density estimation and tests of continuous time interest rate models. Rev. Financial Studies 11 449–487. [43] Qin,J. and Lawless, J. (1994). Empirical likelihood and general estimating functions. Ann. Statist. 22 300–325. MR1272085 [44] Robinson, P. (1989). Hypothesis testing in semiparametric and nonparametric models for econometric time series. Rev. Economic Studies 56 511–534. MR1023837 [45] Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York. MR1191168 [46] Tripathi,G. and Kitamura, Y. (2003). Testing conditional moment restrictions. Ann. Statist. 31 2059–2095. MR2036400 [47] Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5 177–188. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/11976 