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Semi-nonparametric estimation and misspecification testing of diffusion models. (English) Zbl 1441.62784

Summary: Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [D. Kristensen, J. Econom. 156, No. 2, 239–259 (2010; Zbl 1431.62342)] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.

MSC:

62P20 Applications of statistics to economics
62G10 Nonparametric hypothesis testing
62M05 Markov processes: estimation; hidden Markov models
60J60 Diffusion processes
62G05 Nonparametric estimation
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference

Citations:

Zbl 1431.62342
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References:

[1] Ai, C., A semiparametric maximum likelihood estimator, Econometrica, 65, 933-963 (1997) · Zbl 0902.62031
[2] Aït-Sahalia, Y., Nonparametric pricing of interest rate derivative securities, Econometrica, 64, 527-560 (1996) · Zbl 0844.62094
[3] Aït-Sahalia, Y., Testing continuous-time models of the spot interest rate, Review of Financial Studies, 9, 385-426 (1996)
[4] Aït-Sahalia, Y., Maximum likelihood estimation of discretely sampled diffusions: a closed-form approximation approach, Econometrica, 70, 223-262 (2002) · Zbl 1104.62323
[5] Aït-Sahalia, Y.; Fan, J.; Peng, H., Nonparametric transition-based tests for jump diffusions, Journal of the American Statistical Association, 104, 1102-1116 (2009) · Zbl 1388.62124
[6] Andrews, D. W.K., Asymptotics for semiparametric econometric models via stochastic equicontinuity, Econometrica, 62, 43-72 (1994) · Zbl 0798.62104
[7] Andrews, D. W.K., Nonparametric kernel estimation for semiparametric models, Econometric Theory, 11, 560-596 (1995)
[8] Andrews, D. W.K., Higher-order improvements of the parametric bootstrap for Markov processes, (Andrews, D. W.K.; Stock, J. H., Identification and Inference for Econometric Models: A Festschrift in Honor of T.J. Rothenberg (2005), Cambridge University Press: Cambridge University Press Cambridge) · Zbl 1154.62412
[9] Bandi, F. M.; Phillips, P. C.B., Fully nonparametric estimation of scalar diffusion models, Econometrica, 71, 241-283 (2003) · Zbl 1136.62365
[10] Bandi, F. M.; Phillips, P. C.B., A simple approach to the parametric estimation of potentially nonstationary diffusions, Journal of Econometrics, 137, 354-395 (2005) · Zbl 1360.62443
[11] Bhardwaj, G.; Corradi, V.; Swanson, N. R., A simulation-based specification test for diffusion processes, Journal of Business & Economic Statistics, 26, 176-193 (2008)
[12] Bibby, B. M.; Jacobsen, M.; Sørensen, M., Estimating functions for discretely sampled diffusion-type models, (Aït-Sahalia, Y.; Hansen, L. P., Handbook of Financial Econometrics, vol. 1 (2009), Elsevier: Elsevier Amsterdam)
[13] Bickel, P. J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, Annals of Statistics, 1, 1071-1095 (1973) · Zbl 0275.62033
[14] Björk, T., Arbitrage Theory in Continuous Time (2004), Oxford University Press: Oxford University Press Oxford · Zbl 1140.91038
[15] Chan, K. C.; Karolyi, G. A.; Longstaff, F. A.; Sanders, A. B., An empirical comparison of alternative models of the short-term interest rate, Journal of Finance, 47, 1209-1227 (1992)
[16] Chen, S. X.; Gao, J.; Tang, C. Y., A test for model specification of diffusion processes, Annals of Statistics, 36, 167-198 (2009)
[17] Chen, X.; Hansen, L. P.; Scheinkman, J., Nonlinear principal components and long run implications of multivariate diffusions, Annals of Statistics, 37, 4279-4312 (2009) · Zbl 1191.62107
[18] Chen, X.; Hansen, L. P.; Carrasco, M., Nonlinearity and temporal dependence, Journal of Econometrics, 155, 155-169 (2010) · Zbl 1431.62600
[19] Corradi, V.; Swanson, N. R., Bootstrap specification tests for diffusion processes, Journal of Econometrics, 124, 117-148 (2005) · Zbl 1337.62193
[20] Corradi, V.; White, H., Specification tests for the variance of a diffusion, Journal of Time Series Analysis, 20, 253-270 (1999) · Zbl 0932.62095
[21] Doukhan, P.; Massart, P.; Rio, E., The central limit theorem for strongly mixing processes, Annales de l’Institut Henri Poincaré, Section B, 30, 63-82 (1994) · Zbl 0790.60037
[22] Doukhan, P.; Massart, P.; Rio, E., Invariance principles for absolutely regular empirical processes, Annales de l’Institut Henri Poincaré, Section B, 31, 393-427 (1995) · Zbl 0817.60028
[23] Escanciano, J. C., On the lack of power of omnibus specification tests, Econometric Theory, 25, 162-194 (2009) · Zbl 1231.62079
[24] Eubank, R. L.; LaRiccia, V. N., Asymptotic comparison of Cramér-Von Mises and nonparametric function estimation techniques for testing goodness-of-fit, Annals of Statistics, 20, 2071-2086 (1992) · Zbl 0769.62033
[25] Fan, Y., Testing the goodness-of-fit of a parametric density function by kernel method, Econometric Theory, 10, 316-356 (1994)
[26] Fan, Y., Bootstrapping a consistent nonparametric goodness-of-fit test, Econometric Reviews, 14, 367-382 (1995) · Zbl 0832.62038
[27] Fan, J.; Zhang, C.; Zhang, J., Generalized likelihood ratio statistics and Wilks phenomenon, Annals of Statistics, 29, 153-193 (2001) · Zbl 1029.62042
[28] Florens-Zmirou, D., On estimating the diffusion coefficient from discrete observations, Journal of Applied Probability, 30, 790-804 (1993) · Zbl 0796.62070
[29] Friedman, A., Stochastic Differential Equations and Applications, vol. 1 (1976), Academic Press: Academic Press New York
[30] Ghosh, B. K.; Huang, W.-M., The power and optimal kernel of the Bickel-Rosenblatt test for goodness of fit, Annals of Statistics, 19, 999-1009 (1991) · Zbl 0741.62044
[31] Gobet, E.; Hoffmann, M.; Reiß, M., Nonparametric estimation of scalar diffusions based on low frequency data, Annals of Statistics, 32, 2223-2253 (2004) · Zbl 1056.62091
[32] Gouriéroux, C.; Monfort, A.; Renault, E., Indirect inference, Journal of Applied Econometrics, 8, S85-S118 (1993)
[33] Gouriéroux, C.; Tenreiro, C., Local power properties of kernel based goodness of fit tests, Journal of Multivariate Analysis, 78, 161-190 (2001) · Zbl 1081.62529
[34] Hansen, L. P.; Scheinkman, J. A., Back to the future: generating moment implications for continuous time Markov processes, Econometrica, 63, 767-804 (1995) · Zbl 0834.60083
[35] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, Annals of Statististics, 21, 1926-1947 (1993) · Zbl 0795.62036
[36] Hong, Y.; Li, H., Nonparametric specification testing for continuous-time models with application to spot interest rates, Review of Financial Studies, 18, 37-84 (2005)
[37] Horowitz, J. L., Bootstrap methods for Markov processes, Econometrica, 71, 1049-1082 (2003) · Zbl 1154.62361
[38] Huang, L.-S., Testing goodness-of-fit based on a roughness measure, Journal of the American Statistical Association, 92, 1399-1402 (1997) · Zbl 0912.62056
[39] Karlin, S.; Taylor, H. M., A Second Course in Stochastic Processes (1981), Academic Press: Academic Press New York · Zbl 0469.60001
[40] Kristensen, D., 2007. Nonparametric estimation and misspecification testing of diffusion models. CREATES Research Papers 2007-1, University of Aarhus.; Kristensen, D., 2007. Nonparametric estimation and misspecification testing of diffusion models. CREATES Research Papers 2007-1, University of Aarhus.
[41] Kristensen, D., Estimation of partial differential equations with applications in finance, Journal of Econometrics, 144, 392-408 (2008) · Zbl 1418.62384
[42] Kristensen, D., Uniform convergence rates of kernel estimators with heterogeneous, dependent data, Econometric Theory, 25, 1433-1445 (2009) · Zbl 1286.62031
[43] Kristensen, D., Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models, Journal of Econometrics, 156, 239-259 (2010) · Zbl 1431.62342
[44] Kristensen, D., Shin, Y., 2008. Estimation of dynamic models with nonparametric simulated maximum likelihood. CREATES Research Papers 2008-58, University of Aarhus.; Kristensen, D., Shin, Y., 2008. Estimation of dynamic models with nonparametric simulated maximum likelihood. CREATES Research Papers 2008-58, University of Aarhus.
[45] Li, F., Testing the parametric specification of the diffusion function in a diffusion process, Econometric Theory, 23, 221-250 (2007) · Zbl 1237.62102
[46] Li, F.; Tkacz, G., A consistent bootstrap test for conditional density functions with time-series data, Journal of Econometrics, 133, 863-886 (2006) · Zbl 1345.62073
[47] Meyn, S. P.; Tweedie, R. L., Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Advances in Applied Probability, 25, 518-548 (1993) · Zbl 0781.60053
[48] Negri, I.; Nishiyama, Y., Goodness of fit test for ergodic diffusion processes, Annals of the Institute of Statistical Mathematics, 61, 919-928 (2009) · Zbl 1332.62301
[49] Nicolau, J., Bias reduction in nonparametric diffusion coefficient estimation, Econometric Theory, 19, 754-777 (2003) · Zbl 1441.62819
[50] Robinson, P. M., Nonparametric estimators for time series, Journal of Time Series Analysis, 4, 185-297 (1983) · Zbl 0544.62082
[51] Robinson, P. M., Root-n-consistent semiparametric regression, Econometrica, 56, 931-954 (1988) · Zbl 0647.62100
[52] Robinson, P. M., Consistent nonparametric entropy-based testing, Review of Economic Studies, 58, 437-453 (1991) · Zbl 0719.62055
[53] Rosenblatt, M., A quadratic measure of deviation of two-dimensional density estimates and a test of independence, Annals of Statistics, 3, 1-14 (1975) · Zbl 0325.62030
[54] Whang, Y.-J.; Andrews, D. W.K., Tests of specification for parametric and semiparametric models, Journal of Econometrics, 57, 277-318 (1993) · Zbl 0786.62029
[55] White, H., Maximum likelihood estimation of misspecified models, Econometrica, 50, 1-25 (1982) · Zbl 0478.62088
[56] Wong, E., The construction of a class of stationary Markoff processes, (Bellman, R., Sixteenth Symposium in Applied Mathematics-Stochastic Processes in Mathematical Physics and Engineering (1964), American Mathematical Society: American Mathematical Society Providence), 264-276
[57] Wooldridge, J. M.; White, H., Some invariance principles and central limit theorems for dependent heterogeneous processes, Econometric Theory, 4, 210-230 (1988)
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