Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-19T16:49:33.185Z Has data issue: false hasContentIssue false
  • English
  • Français

LET’S GET LADE: ROBUST ESTIMATION OF SEMIPARAMETRIC MULTIPLICATIVE VOLATILITY MODELS

Published online by Cambridge University Press:  05 November 2014

Bonsoo Koo  and
Oliver Linton
Bonsoo Koo*
Affiliation:
Monash University
Oliver Linton
Affiliation:
University of Cambridge
*
*Address correspondence to Bonsoo Koo, Department of Econometrics and Business Statistics, Monash University, Clayton, Victoria 3800, Australia; e-mail: bonsoo.koo@monash.edu.
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate a model in which we connect slowly time varying unconditional long-run volatility with short-run conditional volatility whose representation is given as a semi-strong GARCH(1,1) process with heavy tailed errors. We focus on robust estimation of both long-run and short-run volatilities. Our estimation is semiparametric since the long-run volatility is totally unspecified whereas the short-run conditional volatility is a parametric semi-strong GARCH(1,1) process. We propose different robust estimation methods for nonstationary and strictly stationary GARCH parameters with nonparametric long-run volatility function. Our estimation is based on a two-step LAD procedure. We establish the relevant asymptotic theory of the proposed estimators. Numerical results lend support to our theoretical results.

Type
ARTICLES
Information
Econometric Theory , Volume 31 , Issue 4 , August 2015 , pp. 671 - 702
Copyright
Copyright © Cambridge University Press 2014 

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.)

Footnotes

We are grateful to Wolfgang Hardle and anonymous referees for helpful comments. Linton gratefully acknowledges financial support from the ERC.

References

REFERENCES

Ai, C. & Chen, X. (2012) The semiparametric efficiency bound for models of sequential moment restrictions containing unknown functions. Journal of Econometrics 170, 442–457.CrossRefGoogle Scholar
Andrews, D.W.K. (1994) Asymptotics for semiparametric econometric models via stochastic equicontinuity. Econometrica 62, 43–72.CrossRefGoogle Scholar
Bikbov, R. & Chernov, M. (2010) No-arbitrage macroeconomic determinants of the yield curve. Journal of Econometrics 159, 166–182.CrossRefGoogle Scholar
Black, F. (1976) Studies of stock price volatility changes. In Proceedings of the 1976 Meetings of the Business and Economic Statistics Section, pp. 171–181. American Statistical Association.Google Scholar
Bollerslev, T., Chou, R., & Kroner, K. (1992) ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52, 5–59.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.Google Scholar
Chen, X., Linton, O., & Van Keilegom, I. (2003) Estimation of semiparametric models when the criterion function is not smooth. Econometrica 71, 1591–1608.CrossRefGoogle Scholar
Dahlhaus, R. (1997) Fitting time series models to nonstationary processes. Annals of Statistics 25, 1–37.Google Scholar
Delgado, M.A. & Hidalgo, J. (2000) Nonparametric inference on structural breaks. Journal of Econometrics 96, 113–144.Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Springer.Google Scholar
Doukhan, P., Massart, P., & Rio, E. (1994) The central limit theorem for strongly mixing processes. Annales de l’Institut Henri Poincare, Series B, Probability and Statistics 30, 63–82.Google Scholar
Doukhan, P., Massart, P., & Rio, E. (1995) Invariance principles for absolutely regular processes. Annales de l’Institut Henri Poincare, Series B, Probability and Statistics 31, 393–427.Google Scholar
Drost, F.C. & Nijman, T.E. (1993) Temporal aggregation of GARCH processes. Econometrica 61, 909–927.CrossRefGoogle Scholar
Engle, R. & Rangel, J. (2008) The spline-GARCH model for low frequency volatility and its global macroeconomic causes. Review of Financial Studies 21, 1187–1222.Google Scholar
Francq, C. & Zakoïan, J. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions on the observed process. Econometric Theory 22, 815–834.CrossRefGoogle Scholar
Giurcanu, M. & Spokoiny, V. (2004) Confidence estimation of the covariance function of stationary and locally stationary processes. Statistical Decisions 22, 283–300.CrossRefGoogle Scholar
Hafner, C. & Linton, O. (2010) Efficient estimation of a multivariate multiplicative volatility model. Journal of Econometrics 159, 55–73.Google Scholar
Hall, P., Peng, L., & Yao, Q. (2002) Prediction and nonparametric estimation for time series with heavy tails. Journal of Time Series Analysis 23, 313–331.CrossRefGoogle Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.Google Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.Google Scholar
Härdle, W. & Tuan, P.-D. (1986) Some theory on M-smoothing of time series. Journal of Time Series Analysis 7, 191–204.Google Scholar
Huang, D., Wang, H., & Yao, Q. (2008) Estimating GARCH models: When to use what? Econometrics Journal 11, 27–38.CrossRefGoogle Scholar
Jensen, S. & Rahbek, A. (2004a) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.Google Scholar
Jensen, S. & Rahbek, A. (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.CrossRefGoogle Scholar
Koenker, R. (2005) Quantile Regression. Cambridge University Press.CrossRefGoogle Scholar
Kong, E., Linton, O., & Xia, Y. (2010) Uniform Bahadur representation for local polynomial estimates of M-regression and its application to the additive model. Econometric Theory 26, 1529–1564.CrossRefGoogle Scholar
Koo, B. (2012) Locally Stationary Diffusion Processes with Structural Breaks. Manuscript, Monash University.Google Scholar
Koo, B. & Linton, O. (2012) Estimation of semiparametric locally stationary diffusion models. Journal of Econometrics 170, 210–233.CrossRefGoogle Scholar
Lee, S. & Hansen, B. (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.CrossRefGoogle Scholar
Linton, O., Pan, J., & Wang, H. (2010) Estimation for a non-stationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 26, 1–28.Google Scholar
Mittnik, S. & Rachev, S.T. (2000) Stable Paretian Models in Finance. Wiley.Google Scholar
Peng, L. & Yao, Q. (2003) Least absolute deviations estimations for ARCH and GARCH models. Biometrika 90, 967–975.CrossRefGoogle Scholar
Pollard, D. (1991) Asymptotics for least absolute deviation regression estimators. Econometric Theory 7, 186–198.CrossRefGoogle Scholar
Rachev, S.T. (2003) Handbook of Heavy Tailed Distributions in Finance. North Holland.Google Scholar
Robinson, P. (1983) Nonparametric estimators for time series. Journal of Time Series Analysis 4, 185–207.CrossRefGoogle Scholar
Robinson, P. (1989) Nonparametric estimation of time-varying parameters. In Statistical Analysis and Forecasting of Economic Structural Change, pp. 253–264. Springer-Verlag.CrossRefGoogle Scholar
Rodriguez-Poo, J. & Linton, O. (2001) Nonparametric factor analysis for residual time series. Test 10, 161–182.Google Scholar
Schwert, W. (1989) Why does stock market volatility change over time? Journal of Finance 44, 1115–1153.CrossRefGoogle Scholar
van Bellegem, S. (2011) Locally stationary volatility modelling. Preprint.CrossRefGoogle Scholar
Veronesi, P. (1999) Stock market overreaction to bad news in good times: A rational expectations equilibrium model. Review of Financial Studies 12, 975–1007.CrossRefGoogle Scholar
Zhang, W., Fan, J., & Sun, Y. (2009) A semiparametric model for cluster data. Annals of Statistics 37, 2377–2408.Google ScholarPubMed
Zhou, Z. & Wu, W. (2009) Local linear quantile estimation for non-stationary time series. Annals of Statistics 37, 2696–2729.Google Scholar