Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-31T16:18:48.326Z Has data issue: false hasContentIssue false
  • English
  • Français

An Asymptotic Expansion in the GARCH(l, 1) Model

Published online by Cambridge University Press:  11 February 2009

Oliver Linton
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop order T−1 asymptotic expansions for the quasi-maximum likelihood estimator (QMLE) and a two-step approximate QMLE in the GARCH(l,l) model. We calculate the approximate mean and skewness and, hence, the Edgeworth-B distribution function. We suggest several methods of bias reduction based on these approximations.

Type
Research Article
Information
Econometric Theory , Volume 13 , Issue 4 , February 1997 , pp. 558 - 581
Copyright
Copyright © Cambridge University Press 1997

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

Akahira, M. & Takeuchi, K. (1981) Asymptotic Efficiency of Statistical Estimators: Concepts and Higher Order Asymptotic Efficiency. Berlin: Springer-Verlag.10.1007/978-1-4612-5927-5CrossRefGoogle Scholar
Anderson, T.W. (1974) An asymptotic expansion of the distribution of the limited information maximum likelihood estimate of a coefficient in a simultaneous equation system. Journal of the American Statistical Association 69, 565–573.10.1080/01621459.1974.10482994CrossRefGoogle Scholar
Anderson, T.W. (1977) Asymptotic expansions of the distributions of estimates in simultaneous equations for alternative parameter sequences. Econometrica 45, 509–518.10.2307/1911225CrossRefGoogle Scholar
Anderson, T.W. & Sawa, T. (1973) Distributions of estimates of coefficients of a single equation in a simultaneous system and their asymptotic expansion. Econometrica 41, 683–714.10.2307/1914090CrossRefGoogle Scholar
Anderson, T.W. & Sawa, T. (1979) Evaluation of the distribution function of the two stage least squares estimate. Econometrica 47, 163–182.10.2307/1912353CrossRefGoogle Scholar
Andrews, D.W.K. (1993) Exactly median unbiased estimation of first order autoregressive/unit root models. Econometrica 61, 139–166.10.2307/2951781CrossRefGoogle Scholar
Baillie, R.T. & Bollerslev, T. (1992) Prediction in dynamic models with time-dependent conditional variances. Journal of Econometrics 52, 91–113.10.1016/0304-4076(92)90066-ZCrossRefGoogle Scholar
Barndorff-Nielsen, O.E. & Cox, D.R. (1989) Asymptotic Techniques for Use in Statistics. London: Chapman and Hall.10.1007/978-1-4899-3424-6CrossRefGoogle Scholar
Bartlett. M.S. (1953) Approximate confidence intervals II. Biometrika 40, 306–317.Google Scholar
Bollerslev, T., Engle, R.F., & Nelson, D.B. (1994) ARCH models. In McFadden, D.F. & Engle, R.F. HI (eds.). The Handbook of Econometrics, vol. IV, pp. 2959–3038. Amsterdam: North-Holland.10.1016/S1573-4412(05)80018-2CrossRefGoogle Scholar
Bollerslev, T. & Wooldridge, J.M. (1992) Quasi maximum likelihood estimation and inference in dynamic models with time varying covariates. Econometric Reviews 11, 143–172.10.1080/07474939208800229CrossRefGoogle Scholar
Cooley, T.F. & Parke, W.R. (1990) Asymptotic likelihood-based prediction functions. Econometrica 58, 1215–1234.10.2307/2938307CrossRefGoogle Scholar
Cox, D.R. & Hinkley, D.V. (1974) Theoretical Statistics. London: Chapman and Hall.10.1007/978-1-4899-2887-0CrossRefGoogle Scholar
Engle, R.F., Hendry, D.F., & Trumble, D. (1985) Small sample properties of ARCH estimators and tests. Canadian Journal of Economics 18, 66–93.10.2307/135114CrossRefGoogle Scholar
Firth, D. (1993) Bias reduction of maximum likelihood estimates. Biometrika 80, 27–38.10.1093/biomet/80.1.27CrossRefGoogle Scholar
Ghosh, J.K., Sinja, B.K., & Wieand, H.S. (1980) Second order efficiency of the MLE with respect to any bowl shaped loss function. Annals of Statistics 8, 506–521.10.1214/aos/1176345005CrossRefGoogle Scholar
Hajivassiliou, V.A. & Ruud, FA. (1994) Classical estimation methods for LDV models using simulation. In McFadden, D.F. & Engle, R.F. III (eds.), The Handbook of Econometrics, vol. IV, pp. 2384–2441. Amsterdam: North-Holland.Google Scholar
Haldane, J.B.S. & Smith, S.M. (1956) The sampling distribution of a maximum likelihood estimate. Biometrika 63, 96–103.10.1093/biomet/43.1-2.96CrossRefGoogle Scholar
HSrdle, W. & Linton, O.B. (1994) Applied nonparametric methods. In McFadden, D.F. & Engle, R.F. III (eds.), The Handbook of Econometrics, vol. IV, pp. 2297–2339. Amsterdam: North-Holland.Google Scholar
Hendry, D.F. (1986) Discussion of Engle R.F. and T. Bollerslev, “Modelling the Persistence of Conditional Variances.” Econometric Reviews 5, 1–50.10.1080/07474938608800095Google Scholar
Lee, S.-W. & Hansen, B.E. (1994 ) Asymptotic theory for the GARCH(l.l) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.Google Scholar
Linton, O.B. & Nielsen, J.P. (1995) Kernel estimation of structured non-parametric regression models. Biometrika 82, 93–100.10.1093/biomet/82.1.93CrossRefGoogle Scholar
Lumsdaine, R.L. (1991) Consistency and Asymptotic Normality of the Quasi-Maximum Likelihood Estimator in GARCH(U) and IGARCH(1,1) Models. Manuscript, Princeton University.Google Scholar
Lumsdaine, R.L. (1995) Finite sample properties of the maximum likelihood estimator in GARCH(I.I) and IGARCH(1,1) models: A Monte Carlo investigation. Journal of Business and Economic Statistics 13, 1–10.Google Scholar
McCullagh, P. (1987) Tensor Methods in Statistics. London: Chapman and Hall.Google Scholar
Malinovskii, V.K. (1994) Asymptotic expansions in sequential estimation for the first-order random coefficient autoregressive model: A regenerative approach. Ada Applicandae Mathematicae 34, 261–281.10.1007/BF00994269CrossRefGoogle Scholar
Nagar, A.L. (1959) The bias and moment matrix of the general k-class estimators of the parameters in simultaneous equations. Econometrica 27, 573–595.Google Scholar
Nelson, D.B. (1990) Stationary and persistence in the GARCH(1,1) models. Econometric Theory 6, 318–334.10.1017/S0266466600005296CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–470.10.2307/2938260CrossRefGoogle Scholar
Pfanzagl, J. (1980) Asymptotic expansions in parametric statistical theory. In Krishnaiah, PR. (ed.). Developments in Statistics, vol. 3. New York: Academic Press.Google Scholar
Phillips, P.C.B. (1977a) Approximations to some finite sample distributions associated with a firstorder stochastic difference equation. Econometrica 45, 463–485.10.2307/1911222CrossRefGoogle Scholar
Phillips, P.C.B. (1977b) A general theorem in the theory of asymptotic expansions as approximations to the fintie sample distributions of econometric estimators. Econometrica 45, 1517–1534.10.2307/1912315CrossRefGoogle Scholar
Robinson, P.M. (1988) Root-n consistent semiparametric regression. Econometrica 56, 931–954.10.2307/1912705CrossRefGoogle Scholar
Rothenberg, T.J. (1984a) Approximate normality of generalized least squares estimates. Econometrica 52, 811–825.10.2307/1911185CrossRefGoogle Scholar
Rothenberg, T.J. (1984b) Hypothesis testing in linear models when the error covariance matrix is nonscalar. Econometrica 52, 827–842.10.2307/1911186CrossRefGoogle Scholar
Rothenberg, T. J. (1986) Approximating the distributions of econometric estimators and test statistics. In The Handbook of Econometrics, vol. II. Amsterdam: North-Holland.Google Scholar
Rothenberg, T. J. (1988) Approximate power functions for some robust tests of regression coefficients. Econometrica 56, 997–1019.10.2307/1911356CrossRefGoogle Scholar
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, 45,272.10.2307/1913972CrossRefGoogle Scholar
Shenton, L.R. & Bowman, K. (1963) Higher moments of a maximum-likelihood estimate. Journal of the Royal Statistical Society B 25, 305–317.Google Scholar
Shenton, L.R. & Bowman, K. (1969) Maximum likelihood estimate moments for the two-parameter gamma distribution. Sankha 31, 379–396.Google Scholar
Shenton, L.R. & Bowman, K. (1977) Maximum Likelihood Estimation in Small Samples. London: Griffin.Google Scholar
Taniguchi, M. (1991) Higher Order Asymptotic Theory for Time Series Analysis. Berlin: Springer-Verlag.10.1007/978-1-4612-3154-7CrossRefGoogle Scholar
Taniguchi, M. & Maekawa, K. (1990) Asymptotic expansions of the distribution of statistics related to the spectral density matrix in multivariate time series and their applications. Econometric Theory 6, 75–96.10.1017/S0266466600004928CrossRefGoogle Scholar
Weiss. A. A. (1986) Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2, 107–131.10.1017/S0266466600011397CrossRefGoogle Scholar