Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-24T20:35:45.170Z Has data issue: false hasContentIssue false
  • English
  • Français

HIGHER ORDER ASYMPTOTIC THEORY WHEN A PARAMETER IS ON A BOUNDARY WITH AN APPLICATION TO GARCH MODELS

Published online by Cambridge University Press:  06 September 2007

Emma M. Iglesias  and
Oliver B. Linton
Emma M. Iglesias
Affiliation:
Michigan State University
Oliver B. Linton
Affiliation:
London School of Economics
Get access Rights & Permissions [Opens in a new window]

Abstract

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 6 , December 2007 , pp. 1136 - 1161
Copyright
© 2007 Cambridge University Press

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.)

References

REFERENCES

Andrews, D.W.K. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 1341–1383.Google Scholar
Andrews, D.W.K. (2000) Inconsistency of the bootstrap when a parameter is on the boundary of the parameter space. Econometrica 68, 399–405.Google Scholar
Barndorff-Nielsen, O.E. & D.R. Cox (1989) Asymptotic Techniques for Use in Statistics. Chapman and Hall.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Bollerslev, T. & J.M. Wooldridge (1992) Quasi maximum likelihood estimation and inference in dynamic models with time varying covariances. Econometric Reviews 11, 143–172.Google Scholar
Chong, C.W., M.I. Ahmad, & M.Y. Abdullah (1999) Performance of GARCH models in forecasting stock market volatility. Journal of Forecasting 18, 333–343.Google Scholar
Comte, F. & O. Lieberman (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.Google Scholar
Cox, D.R. & P. Hall (2002) Estimation in a simple random effects model with nonnormal distributions. Biometrika 89, 831–840.Google Scholar
Doornik, J.A. & M. Ooms (2003) Multimodality in the GARCH regression model. Economics Working papers 2003-W20, Nuffield.
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Fiorentini, G., E. Sentana, & G. Calzolari (2003) Maximum likelihood estimation and inference in multivariate conditionally heteroscedastic dynamic regression models with Student t innovations. Journal of Business & Economic Statistics 21, 532–546.Google Scholar
Giot, P. & S. Laurent (2003) Value at risk for long and short trading positions. Journal of Applied Econometrics 18, 641–664.Google Scholar
Götze, F. & C. Hipp (1983) Asymptotic expansions for sums of weakly dependent random vectors. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 64, 311–239.Google Scholar
Hansen, P.R. & A. Lunde (2005) A forecast comparison of volatility models: Does anything beat a GARCH(1,1)? Journal of Applied Econometrics 20, 873–889.Google Scholar
Iglesias, E.M. & G.D.A. Phillips (2005) Small Sample Estimation Bias in GARCH Models with Any Number of Exogenous Variables in the Mean Equation. Working paper, Michigan State University.
Jeantheau, T. (1998) Strong consistency of estimators of multivariate ARCH models. Econometric Theory 14, 70–86.Google Scholar
Jensen, S.T. & A. Rahbek (2004a) Asymptotic normality of the QMLE of ARCH in the nonstationary case. Econometrica 72, 641–646.Google Scholar
Jensen, S.T. & A. Rahbek (2004b) Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203–1226.Google Scholar
Kim, T.-H., D. Stone, & H. White (2000) Asymptotic and Bayesian Confidence Intervals for Sharpe Style Weights. Discussion paper 2000–27, University of California at San Diego.
Lee, S.-W. & B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.Google Scholar
Leeb, H. & B. Pötscher (2003) Can one estimate the unconditional distribution of post-model-selection estimators? Cowles Foundation Discussion Paper 1444, Yale University.
Ling, S. & M. McAleer (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 278–308.Google Scholar
Linton, O. (1997) An asymptotic expansion in the GARCH(1,1) model. Econometric Theory 13, 558–581.Google Scholar
Lumsdaine, R.L. (1995) Finite sample properties of the maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models: A Monte Carlo investigation. Journal of Business & Economic Statistics 13, 1–10.Google Scholar
Lumsdaine, R.L. (1996) Asymptotic properties of the maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models. Econometrica 64, 575–596.Google Scholar
Newey, W.K. & D.G. Steigerwald (1997) Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models. Econometrica 65, 587–599.Google Scholar
Nocedal, J. & S.J. Wright (1999) Numerical Optimization. Springer-Verlag.
Otten, R. & D. Bams (2001) Statistical tests for return-based style analysis. Paper presented at the EFMA meetings, Lugano.
Phillips, P.C.B. (1977a) Approximations to some finite sample distributions associated with a first-order stochastic difference equation. Econometrica 45, 463–485.Google Scholar
Phillips, P.C.B. (1977b) A general theorem in the theory of asymptotic expansions as approximations to the finite sample distributions of econometric estimators. Econometrica 45, 1517–1534.Google Scholar
Phillips, P.C.B. (1978) Edgeworth and saddle point approximations in the first-order noncircular autoregression. Biometrika 65, 91–98.Google Scholar
Rothenberg, T.J. (1984) Approximating the distributions of econometric estimators and test statistics. In Z. Griliches & M.D. Intriligator (eds.), Handbook of Econometrics, vol. 2, 881–935. North-Holland.
Sargan, J.D. (1974) On the validity of Nagar's expansion for the moments of econometric estimators. Econometrica 42, 169–176.Google Scholar
Sargan, J.D. (1976) Econometric estimators and the Edgeworth expansion. Econometrica 44, 421–448; erratum, 45, 272.Google Scholar
Srinivasan, T.N. (1970) Approximations to finite sample moments of estimators whose exact sampling distributions are unknown. Econometrica 38, 533–541.Google Scholar
Taniguchi, M. & Y. Kakizawa (2000) Asymptotic Theory of Statistical Inference for Time Series. Springer-Verlag.
Tse, Y.K. (1998) The conditional heteroscedasticity of the yen-dollar exchange rate. Journal of Applied Econometrics 13, 49–55.Google Scholar
Van Dijk, D., P.H. Franses, & A. Lucas (1999) Testing for ARCH in the presence of additive outliers. Journal of Applied Econometrics 14, 539–562.Google Scholar
Weiss, A.A. (1984) ARMA models with ARCH errors. Journal of Time Series Analysis 5, 129–143.Google Scholar
Weiss, A.A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.Google Scholar