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CHARACTERISTIC FUNCTION–BASED TESTING FOR MULTIFACTOR CONTINUOUS-TIME MARKOV MODELS VIA NONPARAMETRIC REGRESSION

  • Bin Chen (a1) and Yongmiao Hong (a2)

Abstract

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.

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Corresponding author

*Address correspondence to Bin Chen, Department of Economics, University of Rochester, Rochester, NY 14627, USA; e-mail: bchen8@mail.rochester.edu
Yongmiao Hong, Departments of Economics and Statistical Science, Cornell University, Ithaca, NY 14850, USA; e-mail: yh20@cornell.edu.

References

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

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