IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v235y2023i2p1934-1954.html
   My bibliography  Save this article

Uniform and Lp convergences for nonparametric continuous time regressions with semiparametric applications

Author

Listed:
  • Bu, Ruijun
  • Kim, Jihyun
  • Wang, Bin

Abstract

We obtain uniform and Lp convergence rates of kernel type nonparametric estimators for the instantaneous conditional mean and variance functions of continuous time regressions, where the regressor is assumed to be a general recurrent diffusion. Our asymptotics are developed under a general set-up, with a shrinking sampling interval and an increasing time span, and without the stationarity assumption. Based on our convergence results, we develop a semiparametric inferential procedure for continuous time predictive regressions. In particular, a robust semiparametric likelihood ratio test for linear predictability is proposed, with its limit distribution established. We also apply our convergence results to obtain the asymptotics of a semiparametric maximum likelihood estimator of the drift of recurrent diffusions. In our simulation study, we examine the finite sample performance of our robust test against several existing tests in the literature. An empirical illustration is presented to test the predictability of the excess returns of two major stock indices using the commonly used dividend–price ratio and earnings–price ratio as the predictor.

Suggested Citation

  • Bu, Ruijun & Kim, Jihyun & Wang, Bin, 2023. "Uniform and Lp convergences for nonparametric continuous time regressions with semiparametric applications," Journal of Econometrics, Elsevier, vol. 235(2), pages 1934-1954.
  • Handle: RePEc:eee:econom:v:235:y:2023:i:2:p:1934-1954
    DOI: 10.1016/j.jeconom.2023.02.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304407623000726
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.jeconom.2023.02.006?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Jiang, George J. & Knight, John L., 1997. "A Nonparametric Approach to the Estimation of Diffusion Processes, With an Application to a Short-Term Interest Rate Model," Econometric Theory, Cambridge University Press, vol. 13(5), pages 615-645, October.
    2. Kim, Jihyun & Park, Joon Y., 2017. "Asymptotics for recurrent diffusions with application to high frequency regression," Journal of Econometrics, Elsevier, vol. 196(1), pages 37-54.
    3. Chen, Xiaohong & Liao, Zhipeng, 2014. "Sieve M inference on irregular parameters," Journal of Econometrics, Elsevier, vol. 182(1), pages 70-86.
    4. Dominguez, Kathryn M., 1998. "Central bank intervention and exchange rate volatility1," Journal of International Money and Finance, Elsevier, vol. 17(1), pages 161-190, February.
    5. Andrews, Donald W K, 1994. "Asymptotics for Semiparametric Econometric Models via Stochastic Equicontinuity," Econometrica, Econometric Society, vol. 62(1), pages 43-72, January.
    6. Kasparis, Ioannis & Andreou, Elena & Phillips, Peter C.B., 2015. "Nonparametric predictive regression," Journal of Econometrics, Elsevier, vol. 185(2), pages 468-494.
    7. Kim, Jihyun & Meddahi, Nour, 2020. "Volatility Regressions with Fat Tails," TSE Working Papers 20-1097, Toulouse School of Economics (TSE).
    8. Fan J. & Zhang C., 2003. "A Reexamination of Diffusion Estimators With Applications to Financial Model Validation," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 118-134, January.
    9. Li, Degui & Phillips, Peter C. B. & Gao, Jiti, 2016. "Uniform Consistency Of Nonstationary Kernel-Weighted Sample Covariances For Nonparametric Regression," Econometric Theory, Cambridge University Press, vol. 32(3), pages 655-685, June.
    10. Hansen, Lars Peter & Scheinkman, Jose Alexandre, 1995. "Back to the Future: Generating Moment Implications for Continuous-Time Markov Processes," Econometrica, Econometric Society, vol. 63(4), pages 767-804, July.
    11. Kristensen, Dennis, 2009. "Uniform Convergence Rates Of Kernel Estimators With Heterogeneous Dependent Data," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1433-1445, October.
    12. Giampiero Gallo & Barbara Pacini, 2000. "The effects of trading activity on market volatility," The European Journal of Finance, Taylor & Francis Journals, vol. 6(2), pages 163-175.
    13. Darolles, Serge & Gourieroux, Christian, 2001. "Truncated dynamics and estimation of diffusion equations," Journal of Econometrics, Elsevier, vol. 102(1), pages 1-22, May.
    14. Jeff Fleming & Chris Kirby & Barbara Ostdiek, 2008. "The specification of GARCH models with stochastic covariates," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 28(10), pages 911-934, October.
    15. Xiaohong Chen & Oliver Linton & Ingrid Van Keilegom, 2003. "Estimation of Semiparametric Models when the Criterion Function Is Not Smooth," Econometrica, Econometric Society, vol. 71(5), pages 1591-1608, September.
    16. Bollerslev, Tim & Melvin, Michael, 1994. "Bid--ask spreads and volatility in the foreign exchange market : An empirical analysis," Journal of International Economics, Elsevier, vol. 36(3-4), pages 355-372, May.
    17. Hodrick, Robert J., 1989. "Risk, uncertainty, and exchange rates," Journal of Monetary Economics, Elsevier, vol. 23(3), pages 433-459, May.
    18. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    19. Kristensen, Dennis, 2008. "Estimation of partial differential equations with applications in finance," Journal of Econometrics, Elsevier, vol. 144(2), pages 392-408, June.
    20. Chung, Heetaik & Park, Joon Y., 2007. "Nonstationary nonlinear heteroskedasticity in regression," Journal of Econometrics, Elsevier, vol. 137(1), pages 230-259, March.
    21. Bandi, Federico M. & Renò, Roberto, 2012. "Time-varying leverage effects," Journal of Econometrics, Elsevier, vol. 169(1), pages 94-113.
    22. Renò, Roberto, 2008. "Nonparametric Estimation Of The Diffusion Coefficient Of Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 24(5), pages 1174-1206, October.
    23. Federico M. Bandi & Peter C. B. Phillips, 2003. "Fully Nonparametric Estimation of Scalar Diffusion Models," Econometrica, Econometric Society, vol. 71(1), pages 241-283, January.
    24. Andrews, Donald W.K., 1995. "Nonparametric Kernel Estimation for Semiparametric Models," Econometric Theory, Cambridge University Press, vol. 11(3), pages 560-586, June.
    25. Kristensen, Dennis, 2010. "Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models," Journal of Econometrics, Elsevier, vol. 156(2), pages 239-259, June.
    26. Bandi, Federico M. & Moloche, Guillermo, 2018. "On The Functional Estimation Of Multivariate Diffusion Processes," Econometric Theory, Cambridge University Press, vol. 34(4), pages 896-946, August.
    27. Campbell, John Y. & Yogo, Motohiro, 2006. "Efficient tests of stock return predictability," Journal of Financial Economics, Elsevier, vol. 81(1), pages 27-60, July.
    28. Brenner, Robin J. & Harjes, Richard H. & Kroner, Kenneth F., 1996. "Another Look at Models of the Short-Term Interest Rate," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 31(1), pages 85-107, March.
    29. Kristensen, Dennis, 2011. "Semi-nonparametric estimation and misspecification testing of diffusion models," Journal of Econometrics, Elsevier, vol. 164(2), pages 382-403, October.
    30. Simon van Norden & Huntley Schaller & ), 1995. "Regime Switching in Stock Market Returns," Econometrics 9502002, University Library of Munich, Germany.
    31. Niklas Wagner & Terry Marsh, 2005. "Surprise volume and heteroskedasticity in equity market returns," Quantitative Finance, Taylor & Francis Journals, vol. 5(2), pages 153-168.
    32. Gao, Jiti & Kanaya, Shin & Li, Degui & Tjøstheim, Dag, 2015. "Uniform Consistency For Nonparametric Estimators In Null Recurrent Time Series," Econometric Theory, Cambridge University Press, vol. 31(5), pages 911-952, October.
    33. Andersen, Torben G & Bollerslev, Tim, 1998. "Answering the Skeptics: Yes, Standard Volatility Models Do Provide Accurate Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 885-905, November.
    34. Aït-Sahalia, Yacine & Park, Joon Y., 2012. "Stationarity-based specification tests for diffusions when the process is nonstationary," Journal of Econometrics, Elsevier, vol. 169(2), pages 279-292.
    35. Bandi, F.M. & Renò, R., 2016. "Price and volatility co-jumps," Journal of Financial Economics, Elsevier, vol. 119(1), pages 107-146.
    36. Chan, Nigel & Wang, Qiying, 2014. "Uniform Convergence For Nonparametric Estimators With Nonstationary Data," Econometric Theory, Cambridge University Press, vol. 30(5), pages 1110-1133, October.
    37. Bandi, Federico M. & Renò, Roberto, 2018. "Nonparametric Stochastic Volatility," Econometric Theory, Cambridge University Press, vol. 34(6), pages 1207-1255, December.
    38. Glosten, Lawrence R & Jagannathan, Ravi & Runkle, David E, 1993. "On the Relation between the Expected Value and the Volatility of the Nominal Excess Return on Stocks," Journal of Finance, American Finance Association, vol. 48(5), pages 1779-1801, December.
    39. Stambaugh, Robert F., 1999. "Predictive regressions," Journal of Financial Economics, Elsevier, vol. 54(3), pages 375-421, December.
    40. Hansen, Bruce E., 2008. "Uniform Convergence Rates For Kernel Estimation With Dependent Data," Econometric Theory, Cambridge University Press, vol. 24(3), pages 726-748, June.
    41. Lamoureux, Christopher G & Lastrapes, William D, 1990. "Heteroskedasticity in Stock Return Data: Volume versus GARCH Effects," Journal of Finance, American Finance Association, vol. 45(1), pages 221-229, March.
    42. Hoffmann, Marc, 1999. "Adaptive estimation in diffusion processes," Stochastic Processes and their Applications, Elsevier, vol. 79(1), pages 135-163, January.
    43. Hagiwara, May & Herce, Miguel A, 1999. "Endogenous Exchange Rate Volatility, Trading Volume and Interest Rate Differentials in a Model of Portfolio Selection," Review of International Economics, Wiley Blackwell, vol. 7(2), pages 202-218, May.
    44. Hansen, Lars Peter & Alexandre Scheinkman, Jose & Touzi, Nizar, 1998. "Spectral methods for identifying scalar diffusions," Journal of Econometrics, Elsevier, vol. 86(1), pages 1-32, June.
    45. Aït-Sahalia, Yacine & Park, Joon Y., 2016. "Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models," Journal of Econometrics, Elsevier, vol. 192(1), pages 119-138.
    46. Paul Berhanu Girma & Mbodja Mougoué, 2002. "An empirical examination of the relation between futures spreads volatility, volume, and open interest," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 22(11), pages 1083-1102, November.
    47. Chen, Xiaohong, 2007. "Large Sample Sieve Estimation of Semi-Nonparametric Models," Handbook of Econometrics, in: J.J. Heckman & E.E. Leamer (ed.), Handbook of Econometrics, edition 1, volume 6, chapter 76, Elsevier.
    48. Chen, Willa W. & Deo, Rohit S., 2009. "Bias Reduction And Likelihood-Based Almost Exactly Sized Hypothesis Testing In Predictive Regressions Using The Restricted Likelihood," Econometric Theory, Cambridge University Press, vol. 25(5), pages 1143-1179, October.
    49. Choi, Yongok & Jacewitz, Stefan & Park, Joon Y., 2016. "A reexamination of stock return predictability," Journal of Econometrics, Elsevier, vol. 192(1), pages 168-189.
    50. Suzanne S. Lee & Per A. Mykland, 2008. "Jumps in Financial Markets: A New Nonparametric Test and Jump Dynamics," The Review of Financial Studies, Society for Financial Studies, vol. 21(6), pages 2535-2563, November.
    51. Wang, Qiying & Xiang Rachel Wang, Ying, 2013. "Nonparametric Cointegrating Regression With Nnh Errors," Econometric Theory, Cambridge University Press, vol. 29(1), pages 1-27, February.
    52. Chunrong Ai, 1997. "A Semiparametric Maximum Likelihood Estimator," Econometrica, Econometric Society, vol. 65(4), pages 933-964, July.
    53. Reno, Roberto, 2006. "Nonparametric estimation of stochastic volatility models," Economics Letters, Elsevier, vol. 90(3), pages 390-395, March.
    54. Dzhaparidze, K. & van Zanten, J. H., 2001. "On Bernstein-type inequalities for martingales," Stochastic Processes and their Applications, Elsevier, vol. 93(1), pages 109-117, May.
    55. Elias Masry, 1996. "Multivariate Local Polynomial Regression For Time Series:Uniform Strong Consistency And Rates," Journal of Time Series Analysis, Wiley Blackwell, vol. 17(6), pages 571-599, November.
    56. F. Comte & V. Genon-Catalot & Y. Rozenholc, 2010. "Nonparametric estimation for a stochastic volatility model," Finance and Stochastics, Springer, vol. 14(1), pages 49-80, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ruijun Bu & Jihyun Kim & Bin Wang, 2020. "Uniform and Lp Convergences of Nonparametric Estimation for Diffusion Models," Working Papers 202021, University of Liverpool, Department of Economics.
    2. Kanaya, Shin, 2017. "Uniform Convergence Rates Of Kernel-Based Nonparametric Estimators For Continuous Time Diffusion Processes: A Damping Function Approach," Econometric Theory, Cambridge University Press, vol. 33(4), pages 874-914, August.
    3. Heejoon Han & Dennis Kristensen, 2014. "Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(3), pages 416-429, July.
    4. Kristensen, Dennis, 2010. "Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models," Journal of Econometrics, Elsevier, vol. 156(2), pages 239-259, June.
    5. Kanaya, Shin & Kristensen, Dennis, 2016. "Estimation Of Stochastic Volatility Models By Nonparametric Filtering," Econometric Theory, Cambridge University Press, vol. 32(4), pages 861-916, August.
    6. Christian Gourieroux & Hung T. Nguyen & Songsak Sriboonchitta, 2017. "Nonparametric estimation of a scalar diffusion model from discrete time data: a survey," Annals of Operations Research, Springer, vol. 256(2), pages 203-219, September.
    7. Kristensen, Dennis, 2004. "Estimation in two classes of semiparametric diffusion models," LSE Research Online Documents on Economics 24739, London School of Economics and Political Science, LSE Library.
    8. Kristensen, Dennis, 2011. "Semi-nonparametric estimation and misspecification testing of diffusion models," Journal of Econometrics, Elsevier, vol. 164(2), pages 382-403, October.
    9. Park, Joon Y. & Wang, Bin, 2021. "Nonparametric estimation of jump diffusion models," Journal of Econometrics, Elsevier, vol. 222(1), pages 688-715.
    10. Choi, Yongok & Jacewitz, Stefan & Park, Joon Y., 2016. "A reexamination of stock return predictability," Journal of Econometrics, Elsevier, vol. 192(1), pages 168-189.
    11. Zongwu Cai & Yongmiao Hong, 2013. "Some Recent Developments in Nonparametric Finance," Working Papers 2013-10-14, Wang Yanan Institute for Studies in Economics (WISE), Xiamen University.
    12. Kim, Jihyun & Park, Joon & Wang, Bin, 2020. "Estimation of Volatility Functions in Jump Diffusions Using Truncated Bipower Increments," TSE Working Papers 20-1096, Toulouse School of Economics (TSE).
    13. Bonsoo Koo & Oliver Linton, 2010. "Semiparametric Estimation of Locally Stationary Diffusion Models," STICERD - Econometrics Paper Series 551, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    14. Qi Li & Jeffrey Scott Racine, 2006. "Nonparametric Econometrics: Theory and Practice," Economics Books, Princeton University Press, edition 1, volume 1, number 8355.
    15. Chen, Bin & Song, Zhaogang, 2013. "Testing whether the underlying continuous-time process follows a diffusion: An infinitesimal operator-based approach," Journal of Econometrics, Elsevier, vol. 173(1), pages 83-107.
    16. repec:wyi:journl:002108 is not listed on IDEAS
    17. Bu, Ruijun & Hadri, Kaddour & Kristensen, Dennis, 2021. "Diffusion copulas: Identification and estimation," Journal of Econometrics, Elsevier, vol. 221(2), pages 616-643.
    18. Jianqing Fan & Yingying Fan & Jinchi Lv, 0. "Aggregation of Nonparametric Estimators for Volatility Matrix," Journal of Financial Econometrics, Oxford University Press, vol. 5(3), pages 321-357.
    19. Chen, Qiang & Zheng, Xu & Pan, Zhiyuan, 2015. "Asymptotically distribution-free tests for the volatility function of a diffusion," Journal of Econometrics, Elsevier, vol. 184(1), pages 124-144.
    20. Li, Jia & Patton, Andrew J., 2018. "Asymptotic inference about predictive accuracy using high frequency data," Journal of Econometrics, Elsevier, vol. 203(2), pages 223-240.
    21. Aït-Sahalia, Yacine & Park, Joon Y., 2016. "Bandwidth selection and asymptotic properties of local nonparametric estimators in possibly nonstationary continuous-time models," Journal of Econometrics, Elsevier, vol. 192(1), pages 119-138.

    More about this item

    Keywords

    Recurrent diffusion; Kernel estimation; Uniform convergence; Lp convergence; Non/semiparametric model; Predictability; Robust inference;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:econom:v:235:y:2023:i:2:p:1934-1954. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jeconom .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.