Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T02:17:44.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationarity and Persistence in the GARCH(1,1) Model

Published online by Cambridge University Press:  11 February 2009

Daniel B. Nelson
Daniel B. Nelson
Affiliation:
University of Chicago
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper establishes necessary and sufficient conditions for the stationarity and ergodicity of the GARCH(l.l) process. As a special case, it is shown that the IGARCH(1,1) process with no drift converges almost surely to zero, while IGARCH(1,1) with a positive drift is strictly stationary and ergodic. We examine the persistence of shocks to conditional variance in the GARCH(l.l) model, and show that whether these shocks "persist" or not depends crucially on the definition of persistence. We also develop necessary and sufficient conditions for the finiteness of absolute moments of any (including fractional) order.

Type
Articles
Information
Econometric Theory , Volume 6 , Issue 3 , September 1990 , pp. 318 - 334
Copyright
Copyright © Cambridge University Press 1990

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.Abramowitz, M. & Stegun, N. (eds.). Handbook of Mathematical Functions. New York: Dover Publications Inc., 1965.Google Scholar
2.Bollerslev, T.Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31 (1986): 307–327.CrossRefGoogle Scholar
3.Davis, P.J. The gamma function and related functions. Chapter 6 and pp. 253–294 in [1].Google Scholar
4.Dudley, R.M.Real Analysis and Probability. Pacific Grove, CA: Wadsworth & Brooks/Cole, 1989.Google Scholar
5.Engle, R.F.Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 (1982): 987–1007.CrossRefGoogle Scholar
6.Engle, R.F. & Bollerslev, T.. Modeling the persistence of conditional variances. Econometric Reviews 5 (1986): 1–50.CrossRefGoogle Scholar
7.Engle, R.F. & Bollerslev, T.. Reply to comments on modeling the persistence of conditional variances. Econometric Reviews 5 (1986): 81–88.CrossRefGoogle Scholar
8.GAUSS: The GAUSS System Version 2.0. Kent, WA: Aptech Systems, Inc., 1988.Google Scholar
9.Geweke, J.Comment on modeling the persistence of conditional variances. Econometric Reviews 5 (1986): 57–62.CrossRefGoogle Scholar
10.Gradshteyn, I.S. & Ryzhik, I.M.. Table of Integrals, Series and Products. Orlando: Academic Press, 1980.Google Scholar
11.Hardy, G.H., Littlewood, J.E., & Pólya, G.. Inequalities, Second Edition. Cambridge: Cambridge University Press, 1952.Google Scholar
12.Kolmogorov, A.N. & Fomin, S.V.. Introductory Real Analysis. New York: Dover Publications, Inc.Google Scholar
13.Lebedev, N.N.Special Functions and Their Applications. New York: Dover Publications, Inc., 1972.Google Scholar
14.Nelson, D.B. ARCH models as diffusion approximations. Working Paper Series in Economics and Econometrics #89–77, Graduate School of Business, University of Chicago, 1989. Forthcoming, Journal of Econometrics.Google Scholar
15.Poterba, J.M. & Summers, L.H.. The persistence of volatility and stock market fluctuations. American Economic Review 76 (1986): 1142–1151.Google Scholar
16.Prudnikov, A.P., Brychkov, Y.A., & Marichev, O.I.. Integrals and Series: Volume 1, Elementary Functions. New York: Gordon and Breach Science Publishers, 1986.Google Scholar
17.Royden, H.L.Real Analysis, Second Edition. New York: Macmillan Publishing Co., Inc.Google Scholar
18.Sampson, M. A stationarity condition for the GARCH(l.l) process. Department of Economics, Concordia University (mimeo), 1988.Google Scholar
19.Silverman, B.W.Density Estimation for Statistics and Data Analysis. London: Chapman and Hall, 1986.Google Scholar
20.Spanier, J. & Oldham, K.B.. An Atlas of Functions. Washington: Hemisphere Publishing Corporation, 1987.Google Scholar
21.Stout, W.F.Almost Sure Convergence. London: Academic Press, Inc., 1974.Google Scholar
22.White, H.Asymptotic Theory for Econometricians. Orlando: Academic Press, Inc., 1984.Google Scholar
23.Zucker, R. Elementary transcendental functions: Logarithmic, exponential, circular and hyperbolic functions. Chapter 4 and pp. 65–226 in [1].Google Scholar