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. 162-198.
Download (590Kb) | Preview
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 L2-distance 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|
|Keywords:||Bootstrap; diffusion process; empirical likelihood; goodness-of-fit 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|
|Depositing User:||jiti Gao|
|Date Deposited:||09. Dec 2008 00:25|
|Last Modified:||12. Feb 2013 10:16|
 Ait-Sahalia, Y. (1996). Testing continuous-time models of the spot interest rate. Rev. Financial Studies 9 385–426.
 Ait-Sahalia, Y. (1999). Transition densities for interest rate and other nonlinear diffusions. J. Finance 54 1361–1395.
 Ait-Sahalia, Y. (2002). Maximum likelihood estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 223–262. MR1926260
 Ait-Sahalia,Y., Bickel,P.and Stoker, T. (2001). Goodness-of-fit tests for regression using kernel methods. J. Econometrics 105 363–412. MR1873358
 Ait-Sahalia,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.
 Ait-Sahalia,Y. and Kimmel, R. (2005). Estimating affine multifactor term structure models using closed-form likelihood expansions. Working paper. Available at http://www. princeton.edu/~yacine/research.htm.
 Bandi,F. and Phillips, P. C. B. (2003). Fully nonparametric estimation of scalar diffusion models. Econometrica 71 241–284. MR1956859
 Brown, B. and Chen, S. X. (1998). Combined and least squares empirical likelihood. Ann. Inst. Statist. Math. 50 697–714. MR1671990
 Cai, Z. and Hong, Y. (2003). Nonparametric methods in continuous-time finance: A selective review. In Recent Advances and Trends in Nonparametric Statistics (M. G. Akritas and D. M. Politis, eds.) 283–302.
 Chen, S.X. and Cui, H. J. (2007). On the second order properties of empirical likelihood with moment restrictions. J. Econometrics. To appear.
 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.
 Chen, S.X., Hardle,W. and Li, M. (2003). An empirical likelihood goodness-of-fit test for time series. J. Roy. Statist. Soc. Ser. B 65 663–678. MR1998627
 Cox, J.C., Ingersoll, J.E. and Ross, S. A. (1985). A theory of term structure of interest rates. Econometrica 53 385–407. MR0785475
 Fan, J. (1996). Test of significance based on wavelet thresholding and Neyman’s truncation. J. Amer. Statist. Assoc. 434 674–688. MR1395735
 Fan, J. (2005). A selective overview of nonparametric methods in financial econometrics. Statist. Sci. 20 317–357. MR2210224
 Fan,J. and Gijbels, I. (1996). Local Polynomial Modeling and Its Applications. Chapman and Hall, London. MR1383587
 Fan, J. and Huang, L. (2001). Goodness-of-fit test for parametric regression models. J. Amer. Statist. Assoc. 96 640–652. MR1946431
 Fan, J. and Yao, Q. (2003). Nonlinear Time Series: Nonparametric and Parametric Methods. Springer, New York. MR1964455
 Fan,J., Yao,Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83 189–196. MR1399164
 Fan,J. and Yim, T. H. (2004). A data-driven method for estimating conditional densities. Biometrika 91 819–834. MR2126035
 Fan,J. and Zhang, C. (2003). A re-examination of Stanton’s diffusion estimation with applications to financial model validation. J. Amer. Statist. Assoc. 457 118–134. MR1965679
 Fan,J., Zhang,C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193. MR1833962
 Fan,J. and Zhang, J. (2004). Sieved empirical likelihood ratio tests for nonparametric functions. Ann. Statist. 29 153–193. MR2102496
 Fan,Y. and Li, Q. (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64 865–890. MR1399221
 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.
 Genon-Caralot, V., Jeantheau, T. and Laredo, C. (2000). Stochastic volatility models as hidden Markov models and statistical applications. Bernoulli 6 1051–1079. MR1809735
 Gozalo,P.L. and Linton, Q. (2001). Testing additivity in generalized nonparametric regression models with estimated parameters. J. Econometrics 104 1–48. MR1862028
 Hardle,W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926–1947. MR1245774
 Hart, J. (1997). Nonparametric Smoothing and Lack-of-Fit Tests. Springer, New York. MR1461272
 Hjellvik,V., Yao,Q. and Tjostheim, D. (1998). Linearity testing using local polynomial approximation. J. Statist. Plann. Inference 68 295–321. MR1629587
 Hjort,N.L., Mckeague,I.W. and Van Keilegom, I. (2006). Extending the scope of empirical likelihood. Manuscript.
 Hong,Y. and Li, H. (2005). Nonparametric specification testing for continuous-time models with application to spot interest rates. Rev. Financial Studies 18 37–84.
 Horowitz,J.L. and Spokoiny, V. G. (2001). An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 599–632. MR1828537
 Hyndman,R.J. and Yao, Q. (2002). Nonparametric estimation and symmetry tests for conditional density functions. J. Nonparametr. Statist. 14 259–278. MR1905751
 Jiang,G. and Knight, J. (1997). A nonparametric approach to the estimation of diffusion processes with an application to a short-term interest rate model. Econometric Theory 13 615–645. MR1491253
 Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084–2102. MR1474084
 Li, G. (2003). Nonparametric likelihood ratio goodness-of-fit tests for survival data. J. Multivariate Analysis 86 166–182. MR1994727
 Li, Q. (1999). Consistent model specification tests for time series econometric models. J. Econometrics 92 101–147. MR1706996
 Lo, A. W. (1988). Maximum likelihood estimation of generalized Itô processes with discretely sampled data. Econometric Theory 4 231–247. MR0959611
 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
 Owen, A. (1988). Empirical likelihood ratio confidence regions for a single functional. Biometrika 75 237–249. MR0946049
 Pritsker, M. (1998). Nonparametric density estimation and tests of continuous time interest rate models. Rev. Financial Studies 11 449–487.
 Qin,J. and Lawless, J. (1994). Empirical likelihood and general estimating functions. Ann. Statist. 22 300–325. MR1272085
 Robinson, P. (1989). Hypothesis testing in semiparametric and nonparametric models for econometric time series. Rev. Economic Studies 56 511–534. MR1023837
 Scott, D. W. (1992). Multivariate Density Estimation. Wiley, New York. MR1191168
 Tripathi,G. and Kitamura, Y. (2003). Testing conditional moment restrictions. Ann. Statist. 31 2059–2095. MR2036400
 Vasicek, O. (1977). An equilibrium characterization of the term structure. J. Financial Economics 5 177–188.