Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T21:36:37.358Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Estimation in ARCH Models

Published online by Cambridge University Press:  11 February 2009

Oliver Linton
Oliver Linton
Affiliation:
Nuffield College
Get access Rights & Permissions [Opens in a new window]

Abstract

We construct efficient estimators of the identifiable parameters in a regression model when the errors follow a stationary parametric ARCH(P) process. We do not assume a functional form for the conditional density of the errors, but do require that it be symmetric about zero. The estimators of the mean parameters are adaptive in the sense of Bickel [2]. The ARCH parameters are not jointly identifiable with the error density. We consider a reparameterization of the variance process and show that the identifiable parameters of this process are adaptively estimable.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 4 , August 1993 , pp. 539 - 569
Copyright
Copyright © Cambridge University Press 1993

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

1.Abramson, I.S.On bandwidth variation in kernel estimates – a square root law. The Annals of Statistics 10 (1980): 1217–1223.Google Scholar
2.Bickel, P.J.On adaptive estimation. Annals of Statistics 10 (1982): 647–671.CrossRefGoogle Scholar
3.Bickel, P.J. & Ritov, Y.. Achieving information bounds in non and semiparametric models. Annals of Statistics 18 (1990): 925–938.Google Scholar
4.Bickel, P.J., Klaassen, C.A.J., Ritov, Y. & Wellner, J.A.. Efficient and Adaptive Inference in Semiparametric Models, Johns Hopkins University Press, 1991 (forthcoming monograph).Google Scholar
5.Bollerslev, T.Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31 (1986): 307–327.CrossRefGoogle Scholar
6.Bollerslev, T. & Wooldridge, J.M.. Quasi-maximum likelihood estimation of dynamic models with time varying covariances. Econometric Reviews (1991) (forthcoming).Google Scholar
7.Bollerslev, T., Chou, R. & Kroner, K.. ARCH modelling in finance: A review of the theory and empirical evidence. Journal of Econometrics 52 (1992): 5–59.CrossRefGoogle Scholar
8.Bougerol, P. & Picard, N.. Stationarity of GARCH processes and some non-negative time series. Journal of Econometrics 52 (1992): 115–128.CrossRefGoogle Scholar
9.Cramer, H.Mathematical Methods in Statistics. Princeton, NJ: Princeton University Press, 1946.Google Scholar
10.Engle, R.F.Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50 (1982): 987–1008.CrossRefGoogle Scholar
11.Engle, R.F.Estimates of the variance of U.S. inflation based upon the ARCH model. Journal of Money, Credit, and Banking 15 (1983): 286–301.CrossRefGoogle Scholar
12.Engle, R.F. & Bollerslev, T.. Modelling the persistence of conditional variances. Econometric Reviews 5 (1986): 1–50.CrossRefGoogle Scholar
13.Engle, R.F. & Gonzalez-Rivera, G.. Semiparametric ARCH models. Journal of Business and Economic Statistics 9 (1991): 345–360.Google Scholar
14.Fabian, V. & Hannan, J.. On estimation and adaptive estimation for locally asymptotically normal families. Zeitschrift für Wahrscheinlichskeitstheorie und verwandte Gebiete 59 (1982): 459–478.CrossRefGoogle Scholar
15.Hajek, J.Asymptotically most powerful rank order tests. Annals of Mathematical Statistics 33 (1962): 1124–1147.CrossRefGoogle Scholar
16.Hajek, J.Local asymptotic minimax and admissibility in estimation. Proceedings of the Sixth Berkeley Symposium in Mathematical Statistics, 1972.Google Scholar
17.Gallant, A.R., Hsieh, D. & Tauchen, G.. On fitting a recalcitrant series: The dollar/pound exchange rate, 1974–1983. In Barnett, W., Powell, J., and Tauchen, G. (eds.) Nonparametric and Semiparametric Methods in Econometrics and Statistics. Cambridge University Press, 1991.Google Scholar
18.Geweke, J.Comment on: modelling the persistence of conditional variances. Econometric Reviews 5 (1986): 57–61.CrossRefGoogle Scholar
19.Hall, P. & Heyde, C.C.. Martingale Limit Theory. New York: Academic Press, 1980.Google Scholar
20.Hsieh, D.A. & Manski, C.F.. Monte Carlo evidence on adaptive maximum likelihood estimation of a regression. The Annals of Statistics 15 (1987): 541–551.CrossRefGoogle Scholar
21.Kreiss, J.P.On adaptive estimation in stationary ARMA processes. Annals of Statistics 15 (1987): 112–133.CrossRefGoogle Scholar
22.Kreiss, J.P.On adaptive estimation in autoregressive models when there are nuisance functions. Statistics and Decisions 5 (1987): 59–76.Google Scholar
23.Le Cam, L.Local asymptotically normal families of distributions. University of California Publications in Statistics 3 (1960): 267–284.Google Scholar
24.Le Cam, L. Likelihood functions for large numbers of independent observations. In David, F.N. (ed.), Festschrift for J. Neyman. New York: Wiley, 1966.Google Scholar
25.Le Cam, L.On the assumptions used to prove asymptotic normality of maximum likelihood estimates. The Annals of Mathematical Statistics 41 (1970): 802–828.CrossRefGoogle Scholar
26.Lind, B. & Roussas, G.. Cramer-type conditions and quadratic mean differentiability. Annals of the Institute of Statistical Mathematics 29 (1977): 189–201.CrossRefGoogle Scholar
27.Lumsdaine, R.L.Asymptotic properties of the quasi-maximum likelihood estimator in GARCH(1,1) and IGARCH(1,1) models. Manuscript. Harvard University, 1990.Google Scholar
28.Mandelbrot, B.The variation of certain speculative prices. Journal of Business 36 (1963): 394–419.CrossRefGoogle Scholar
29.Milhoj, A.The moment structure of ARCH processes. Scandinavian Journal of Statistics 12 (1985): 281–292.Google Scholar
30.Nelson, D.B.Conditional heteroscedasticity in asset returns: A new approach. Econometrica 59 (1991): 347–370.CrossRefGoogle Scholar
31.Nelson, D.B.Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6 (1990): 318–334.CrossRefGoogle Scholar
32.Nelson, D.B. & Cao, C.Q.. A note on the inequality constraints in the univariate GARCH model. Journal of Business and Economic Statistics (1991) (forthcoming).Google Scholar
33.Nemec, A.F. & Linnell, A.. Conditionally heteroscedastic autoregression. University of Washington, Department of Statistics, Technical Report #43 (1984).Google Scholar
34.Phillips, P.C.B.Optimal inference in cointegrated systems. Econometrica 59 (1991): 283–306.CrossRefGoogle Scholar
35.Roussas, G.G.Contiguity of Probability Measures: Some Applications in Statistics. New York: Cambridge University Press, 1972.CrossRefGoogle Scholar
36.Roussas, G.G.Asymptotic distribution of the log-likelihood function for stochastic processes. Zeitschrift für Wahrscheinlichskeitstheorie und verwandte Gebiete 43 (1979): 31–46.CrossRefGoogle Scholar
37.Sampson, M.A stationarity condition for the GARCH(1.1) process. Department of Economics, Concordia University (1988).Google Scholar
38.Shephard, N. A local scale model: An unobserved component alternative to integrated GARCH processes. LSE Discussion Paper EM/90/220 (1990).Google Scholar
39.Spanos, A. A parametric approach to dynamic heteroscedasticity: The students' t and related linear models. Unpublished manuscript (1990).Google Scholar
40.Steigerwald, D.Adaptive estimation in time series regression models. Working paper, Department of Economics, UC Santa Barbara (1990).Google Scholar
41.Steigerwald, D.Efficient estimation of models with conditional heteroscedasticity. Paper presented at ESWM, Louisana 1991.Google Scholar
42.Swensen, A.R. Asymptotic inference for a class of stochastic processes. Ph.D. Thesis, Department of Statistics, UC Berkeley (1980).Google Scholar
43.Swensen, A.R.The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis 16 (1985): 54–70.CrossRefGoogle Scholar
44.Weiss, A.Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory 2 (1986): 107–131.CrossRefGoogle Scholar
45.Whistler, D.Semiparametric ARCH estimation of intra-daily exchange rate volatility. Manuscript. London School of Economics (1988).Google Scholar
46.Wu, C.Asymptotic theory of nonlinear least squares estimation. The Annals of Statistics 9 (1981): 501–513.CrossRefGoogle Scholar