Skip to main content Accessibility help
Hostname: page-component-99c86f546-7mfl8 Total loading time: 0.426 Render date: 2021-12-08T10:34:13.348Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }


Published online by Cambridge University Press:  04 November 2009

Bin Chen*
University of Rochester
Yongmiao Hong*
Cornell University and Xiamen University
*Address correspondence to Bin Chen, Department of Economics, University of Rochester, Rochester, NY 14627, USA; e-mail:
Yongmiao Hong, Departments of Economics and Statistical Science, Cornell University, Ithaca, NY 14850, USA; e-mail:


We develop a nonparametric regression-based goodness-of-fit test for multifactor continuous-time Markov models using the conditional characteristic function, which often has a convenient closed form or can be approximated accurately for many popular continuous-time Markov models in economics and finance. An omnibus test fully utilizes the information in the joint conditional distribution of the underlying processes and hence has power against a vast class of continuous-time alternatives in the multifactor framework. A class of easy-to-interpret diagnostic procedures is also proposed to gauge possible sources of model misspecification. All the proposed test statistics have a convenient asymptotic N(0, 1) distribution under correct model specification, and all asymptotic results allow for some data-dependent bandwidth. Simulations show that in finite samples, our tests have reasonable size, thanks to the dimension reduction in nonparametric regression, and good power against a variety of alternatives, including misspecifications in the joint dynamics, but the dynamics of each individual component is correctly specified. This feature is not attainable by some existing tests. A parametric bootstrap improves the finite-sample performance of proposed tests but with a higher computational cost.

Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Ahn, D., Dittmar, R., & Gallant, A.R. (2002) Quadratic term structure models: Theory and evidence. Review of Financial Studies 15, 243288.CrossRefGoogle Scholar
Ahn, D. & Gao, B. (1999) A parametric nonlinear model of term structure dynamics. Review of Financial Studies 12, 721762.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996a) Testing continuous-time models of the spot interest rate. Review of Financial Studies 9, 385426.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1996b) Nonparametric pricing of interest rate derivative securities. Econometrica 64, 527560.CrossRefGoogle Scholar
Aït-Sahalia, Y. (1999) Transition densities for interest rate and other nonlinear diffusions. Journal of Finance 54, 13611395.CrossRefGoogle Scholar
Aït-Sahalia, Y. (2002) Maximum-likelihood estimation of discretely sampled diffusions: A closed-form approach. Econometrica 70, 223262.CrossRefGoogle Scholar
Aït-Sahalia, , Bickel, P.J., & Stoker, T.M. (2001) Goodness-of-fit tests for kernel regression with an application to option implied volatilities. Journal of Econometrics 105, 363412.CrossRefGoogle Scholar
Aït-Sahalia, Y., Fan, J., & Peng, H. (2009) Nonparametric transition-based tests for diffusions. Journal of the American Statistical Association, forthcoming.CrossRefGoogle Scholar
Andersen, T.G. & Lund, J. (1996) Stochastic Volatility and Mean Drift in the Short Rate Diffusion: Sources of Steepness, Level and Curvature in the Yield Curve. Working paper, Northwestern University.Google Scholar
Andrews, D.W.K. (1997) A conditional Kolmogorov test. Econometrica 65, 10971128.CrossRefGoogle Scholar
Bahadur, R.R. (1960) Stochastic comparison of tests. Annals of Mathematical Statistics 31, 276295.CrossRefGoogle Scholar
Bally, V. & Talay, D. (1996) The law of the Euler scheme for stochastic differential equations: I. Convergence rate of the distribution function. Probability Theory and Related Fields 104, 4360.CrossRefGoogle Scholar
Bates, D. (1996) Jumps and stochastic volatility: Exchange rate processes implicit in PHLX Deutschemake options. Review of Financial Studies 9, 69107.CrossRefGoogle Scholar
Bates, D. (2007) Maximum likelihood estimation of latent affine processes. Review of Financial Studies 19, 909965.CrossRefGoogle Scholar
Bhardwaj, G., Corradi, V., & Swanson, N.R. (2008) A simulation based specification test for diffusion processes. Journal of Business & Economic Statistics 26, 176193.CrossRefGoogle Scholar
Brandt, M. & Santa-Clara, P. (2002) Simulated likelihood estimation of diffusions with an application to exchange rate dynamics in incomplete markets. Journal of Financial Economics 63, 161210.CrossRefGoogle Scholar
Carr, P. & Wu, L. (2003) The finite moment log stable process and option pricing. Journal of Finance 58, 753777.CrossRefGoogle Scholar
Carr, P. & Wu, L. (2004) Time-changed Lévy processes and option pricing. Journal of Financial Economics 71, 113141.CrossRefGoogle Scholar
Carrasco, M., Chernov, M., Florens, J.P., & Ghysels, E. (2007) Efficient estimation of general dynamic models with a continuum of moment conditions. Journal of Econometrics, 140, 529573.CrossRefGoogle Scholar
Chacko, G. & Viceira, L. (2003) Spectral GMM estimation of continuous-time processes. Journal of Econometrics 116, 259292.CrossRefGoogle Scholar
Chan, K.C., Karolyi, G.A., Longstaff, F.A., & Sanders, A.B. (1992) An empirical comparison of alternative models of the short-term interest rate. Journal of Finance 47, 12091227.CrossRefGoogle Scholar
Chen, B. & Hong, Y. (2005) Diagnosing Multivariate Continuous-Time Models with Application to Affine Term Structure Models. Working paper, Cornell University.Google Scholar
Chen, S.X., Gao, J., & Tang, C.Y. (2008) A test for model specification of diffusion processes. Annals of Statistics 36, 162198.CrossRefGoogle Scholar
Chernov, M., Gallant, A.R., Ghysels, E., & Tauchen, G. (1999) A New Class of Stochastic Volatility Models with Jumps: Theory and Estimation. Working paper, Columbia University, University of North Carolina at Chapel Hill, and Duke University.Google Scholar
Chib, S., Pitt, M.K., & Shephard, N. (2004) Likelihood Based Inference for Diffusion Driven Models. Working paper, University of Oxford.Google Scholar
Christoffersen, P., Jacobs, K., & Mimouni, K. (2009) Models for S&P 500 Dynamics: Evidence from Realized Volatility, Daily Returns, and Option Prices. Working paper, McGill University.Google Scholar
Cleveland, W.S. (1979) Robust locally weighted regression and smoothing scatterplots. Journal of the American Statistical Association 74, 829836.CrossRefGoogle Scholar
Cox, J.C., Ingersoll, J.E., & Ross, S.A. (1985) A theory of the term structure of interest rates. Econometrica 53, 385407.CrossRefGoogle Scholar
Dai, Q. & Singleton, K. (2000) Specification analysis of affine term structure models. Journal of Finance 55, 19431978.CrossRefGoogle Scholar
Del Moral, P. & Jacod, J. (2001) Interacting particle filtering with discrete observations. In Doucet, A., de Freitas, N., & Gordon, N. (eds.), Sequential Monte Carlo Methods in Practice, pp. 4377. Springer.CrossRefGoogle Scholar
Del Moral, P., Jacod, J., & Protter, P. (2001) The Monte-Carlo method for filtering with discrete-time observations. Probability Theory and Related Fields 120, 346368.CrossRefGoogle Scholar
Doucet, A., de Freitas, N., & Gordon, N., eds. (2001) Sequential Monte Carlo Medhods in Practice. Springer.CrossRefGoogle Scholar
Duffee, G. (2002) Term premia and interest rate forecasts in affine models. Journal of Finance 57, 405443.CrossRefGoogle Scholar
Duffie, D. & Kan, R. (1996) A yield-factor model of interest rates. Mathematical Finance 6, 379406.CrossRefGoogle Scholar
Duffie, D., Pan, J., & Singleton, K. (2000) Transform analysis and asset pricing for affine jump-diffusions. Econometrica 68, 13431376.CrossRefGoogle Scholar
Duffie, D., Pedersen, L., & Singleton, K. (2003) Modeling credit spreads on sovereign debt: A case study of Russian bonds. Journal of Finance 55, 119159.CrossRefGoogle Scholar
Engle, R. (2002) Dynamic conditional correlation — a simple class of multivariate GARCH models. Journal of Business & Economic Statistics 20, 339350.CrossRefGoogle Scholar
Epps, T.W. & Pulley, L.B. (1983) A test for normality based on the empirical characteristic function. Biometrika 70, 723726.CrossRefGoogle Scholar
Fan, J. (1993) Local linear regression smoothers and their minimax efficiency. Annals of Statistics 21, 196216.CrossRefGoogle Scholar
Fan, J. & Gijbels, I. (1996) Local Polynomial Modelling and Its Applications. Chapman and Hall.Google Scholar
Fan, J. & Yao, Q. (2003) Nonlinear Time Series:Nonparametric and Parametric Methods. Springer-Verlag.CrossRefGoogle Scholar
Fan, Y., Li, Q., & Min, I. (2006) A nonparametric bootstrap test of conditional distributions. Econometric Theory 22, 587612.CrossRefGoogle Scholar
Fenton, V. & Gallant, A.R. (1996) Convergence rates of SNP density estimators. Econometrica 64, 719727.CrossRefGoogle Scholar
Feuerverger, A. & McDunnough, P. (1981) On the efficiency of empirical characteristic function procedures. Journal of the Royal Statistical Society, Series B 43, 2027.Google Scholar
Gallant, A.R. & Long, J.R. (1997) Estimating stochastic differential equations efficiently by minimum chi-square. Biometrika 125141.CrossRefGoogle Scholar
Gallant, A.R. & Tauchen, G. (1996) Which moments to match? Econometric Theory 12, 657681.CrossRefGoogle Scholar
Gallant, A.R. & Tauchen, G. (1998) Reprojecting partially observed systems with application to interest rate diffusions. Journal of the American Statistical Association 93, 1024.CrossRefGoogle Scholar
Gao, J. & King, M. (2004) Adaptive testing in continuous-time diffusion models. Econometric Theory 20, 844882.CrossRefGoogle Scholar
Golightly, A. & Wilkinson, D. (2006) Bayesian sequential inference for nonlinear multivariate diffusions. Statistics and Computing 16, 323338.CrossRefGoogle Scholar
Gordon, N., Salmond, D., & Smith, A. (1993) Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proceedings F140, 107113.Google Scholar
Hall, R. (1978) Stochastic implications of the life cycle–permanent income hypothesis: Theory and practice. Journal of Political Economy 86, 971987.CrossRefGoogle Scholar
Hansen, B. (2008) Uniform convergence rates for kernel estimation with dependent data. Econometric Theory 24, 726748.CrossRefGoogle Scholar
Hansen, L.P. & Scheinkman, J.A. (1995) Back to the future: Generating moment implications for continuous-time Markov processes. Econometrica 63, 767804.CrossRefGoogle Scholar
Hausenblas, E. (2002) Error analysis for approximation of stochastic differential equations driven by Poisson random measures. SIAM Journal of Numerical Analysis 40, 87113.CrossRefGoogle Scholar
Heston, S. (1993) A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6, 327344.CrossRefGoogle Scholar
Hjellvik, V., Yao, Q., & Tjøhstheim, D. (1998) Linearity testing using local polynomial approximation. Journal of Statistical Planning and Inference 68, 295321.CrossRefGoogle Scholar
Hong, Y. & Li, H. (2005) Nonparametric specification testing for continuous-time models with applications to term structure of interest rates. Review of Financial Studies 18, 3784.CrossRefGoogle Scholar
Jiang, G. & Knight, J. (2002) Estimation of continuous-time processes via the empirical characteristic function. Journal of Business & Economic Statistics 20, 198212.CrossRefGoogle Scholar
Johannes, M., Polson, N., & Stroud, J. (2009) Optimal filtering of jump diffusions: Extracting latent states from asset prices. Review of Economic Studies, forthcoming.Google Scholar
Kloeden, P., Platen, E., & Schurz, H. (1994) Numerical Solution of SDE through Computer Experiments. Springer-Verlag.CrossRefGoogle Scholar
Kydland, F.E. & Prescott, E. (1982) Time to build and aggregate fluctuations. Econometrica 50, 13451370.CrossRefGoogle Scholar
Li, F. (2007) Testing the parametric specification of the diffusion function in a diffusion process. Econometric Theory 23, 221250.CrossRefGoogle Scholar
Li, F. & Tkacz, G. (2006) A consistent bootstrap test for conditional density functions with time-series data. Journal of Econometrics 133, 841862.CrossRefGoogle Scholar
Lucas, R. (1988) On the mechanics of economic development. Journal of Monetary Economics 22, 342.CrossRefGoogle Scholar
Lucas, R. & Prescott, E. (1971) Investment under uncertainty. Econometrica 39, 659681.CrossRefGoogle Scholar
Lucas, R. & Stokey, N.L. (1983) Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 5594.CrossRefGoogle Scholar
Masry, E. (1996) Multivariate local polynomial regression for time series: Uniform strong consistency and rates. Journal of Time Series Analysis 17, 571599.CrossRefGoogle Scholar
Müller, H. (1984) Smooth optimum kernel estimators of densities, regression curves and modes. Annals of Statistics 12, 766774.CrossRefGoogle Scholar
Pedersen, A.R. (1995) A new approach to maximum likelihood estimation for stochastic differential equations based on discrete observations. Scandinavian Journal of Statistics 22, 5571.Google Scholar
Pitt, M.K. & Shephard, N. (1999) Filtering via simulation: Auxiliary particle filters. Journal of the American Statistical Association 94, 590599.CrossRefGoogle Scholar
Pritsker, M. (1998) Nonparametric density estimation and tests of continuous time interest rate models. Review of Financial Studies 11, 449487.CrossRefGoogle Scholar
Rust, J. (1994) Structural estimation of Markov decision processes. In Engle, R. & McFadden, D. (eds.), Handbook of Econometrics, vol. 4, pp. 30813143. North-Holland.Google Scholar
Singleton, K. (2001) Estimation of affine asset pricing models using the empirical characteristic function. Journal of Econometrics 102, 111141.CrossRefGoogle Scholar
Skaug, H.J. & Tjøstheim, D. (1996) Testing for serial independence using measures of distance between densities. In P.M. Robinson & M. Rosenblatt (eds.), Athena Conference on Applied Probability and Time Series. Springer Lecture Notes in Statistics 115. Springer.Google Scholar
Stone, C.J. (1977) Consistent nonparametric regression. Annals of Statistics 5, 595645.CrossRefGoogle Scholar
Su, L. & White, H. (2007) A consistent characteristic function-based test for conditional independence. Journal of Econometrics 141, 807834.CrossRefGoogle Scholar
Sundaresan, S. (2001) Continuous-time methods in finance: A review and an assessment. Journal of Finance 55, 15691622.CrossRefGoogle Scholar
Vasicek, O. (1977) An equilibrium characterization of the term structure. Journal of Financial Economics 5, 177188.CrossRefGoogle Scholar
Yu, J. (2007) Closed-form likelihood approximation and estimation of jump-diffusions with an application to the realignment risk of the chinese yuan. Journal of Econometrics 141, 12451280.CrossRefGoogle Scholar
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the or variations. ‘’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Available formats

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *