Skip to main content Accessibility help
×
Home
Hostname: page-component-55b6f6c457-xdj6x Total loading time: 0.368 Render date: 2021-09-24T08:17:42.500Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true, "newUsageEvents": true }

ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES

Published online by Cambridge University Press:  20 August 2013

Karim M. Abadir
Affiliation:
Imperial College London
Walter Distaso
Affiliation:
Imperial College London
Liudas Giraitis*
Affiliation:
Queen Mary, University of London
Hira L. Koul
Affiliation:
Michigan State University
*
*Address correspondence to Liudas Giraitis, School of Ecnomics and Finance, Queen Mary, University of London, Mile End Rd., London E14NS, United Kingdom; e-mail: l.giraitis@qmul.ac.uk

Abstract

We establish asymptotic normality of weighted sums of linear processes with general triangular array weights and when the innovations in the linear process are martingale differences. The results are obtained under minimal conditions on the weights and innovations. We also obtain weak convergence of weighted partial sum processes. The results are applicable to linear processes that have short or long memory or exhibit seasonal long memory behavior. In particular, they are applicable to GARCH and ARCH(∞) models and to their squares. They are also useful in deriving asymptotic normality of kernel-type estimators of a nonparametric regression function with short or long memory moving average errors.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2013 

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

Abadir, K.M. (1993) The limiting distribution of the autocorrelation coefficient under a unit root. Annals of Statistics 21, 10581070.CrossRefGoogle Scholar
Abadir, K.M. (1995) The limiting distribution of the t ratio under a unit root. Econometric Theory 11, 775793.CrossRefGoogle Scholar
Anderson, T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. Annals of Mathematical Statistics 30, 676687.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bollerslev, T. (1988) On the correlation structure for the generalized autoregressive conditional heteroskedastic process. Journal of Time Series Analysis., 9, 121131.CrossRefGoogle Scholar
Borovskikh, Y.V. & Korolyuk, V.S. (1997) Martingale Approximation. VSP.Google Scholar
Breidt, F.J., Crato, N. and de Lima, P. (1998) On the detection and estimation of long memory in stochastic volatility. Journal of Econometrics 83, 325348.CrossRefGoogle Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed.Springer Series in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Davydov, Y.A. (1970) The invariance principle for stationary processes. Theory of Probability and its Applications 15, 487498.CrossRefGoogle Scholar
Dharmadhikari, S.W., Fabian, V. & Jogdeo, K. (1968) Bounds on the moments of martingales. Annals of Mathematical Statistics 39, 17191723.CrossRefGoogle Scholar
Dickey, D.A., & Fuller, W.A. (1979) Distribution of estimators of autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427431.Google Scholar
Dickey, D.A. & Fuller, W.A. (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 10571072.CrossRefGoogle Scholar
Dolado, J.J., Gonzalo, J., & Mayoral, L. (2002) A fractional Dickey-Fuller test for unit roots. Econometrica 70, 19632006.CrossRefGoogle Scholar
Engle, R.F. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 9871008.CrossRefGoogle Scholar
Giraitis, L., Kokoszka, P., Leipus, R. (2000) Stationary ARCH models: Dependence structure and central limit theorems. Econometric Theory 16, 322.CrossRefGoogle Scholar
Giraitis, L.R.Leipus, & Surgailis, D. (2007) Recent advances in ARCH modelling. In Teyssiere, G. & Kirman, A.P. (eds.), Long Memory in Economics, pp. 338Springer.CrossRefGoogle Scholar
Gordin, M.I. (1969) The central limit theorem for stationary processes. Soviet Mathematics Doklady 10, 11741176.Google Scholar
Granger, C.W.J. & Joyeux, R. (1980) An introduction to long-memory time series models and fractional differencing. Journal of Time Series Analysis 1, 1529.CrossRefGoogle Scholar
Gray, H.L., Zhang, N.-F. & Woodward, W.A. (1989) On generalized fractional processes. Journal of Time Series Analysis 10, 233257.CrossRefGoogle Scholar
Hájek, J. & Sidák, Z. (1967) Theory of Rank Tests. Academic Press.Google Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Applications. Academic Press.Google Scholar
Harvey, A.C. (1998) Long memory in stochastic volatility. In Knight, J., & Satchell, S. (eds.), Forecasting Volatility in the Financial Markets, pp. 307320, Butterworth & Heineman.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165176CrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. and Shin, Y. (1992). Testing the null hypothesis of stationary against the alternative of a unit root: how sure are we that economic time series have a unit root? Journal of Econometrics 54, 159178.CrossRefGoogle Scholar
Lamperti, J.W. (1962) Semi-stable stochastic processes. Transactions of the American Mathematical Society 104, 6278.CrossRefGoogle Scholar
Marinucci, D. & Robinson, P.M. (1999) Alternative forms of fractional Brownian motion. Journal of Statistical Planning and Inference 80, 111122.CrossRefGoogle Scholar
Merlevède, F., Peligrad, M., & Utev, S. (2006) Recent advances in invariance principles for stationary sequences. Probability Surveys 3, 136.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347370.CrossRefGoogle Scholar
Peligrad, M. & Utev, S. (1997) Central limit theorem for stationary linear processes. Annals of Probability 25, 443456.Google Scholar
Peligrad, M. & Utev, S. (2005) A new maximal inequality and invariance principle for stationary sequences. Annals of Probability 33, 798815.CrossRefGoogle Scholar
Peligrad, M. & Utev, S. (2006) Central limit theorem for stationary linear processes. Annals of Probability 34, 16081622.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277301.CrossRefGoogle Scholar
Phillips, P.C.B. & Magdalinos, T. (2007) Limit theory for moderate deviations from a unit root. Journal of Econometrics 136, 115130.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 9711001.CrossRefGoogle Scholar
Pollard, D. (1984) Convergence of Stochastic Processes. Springer Series in Statistics. Springer-Verlag.CrossRefGoogle Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 6784.CrossRefGoogle Scholar
Robinson, P.M. (1997) Large-sample inference for non-parametric regression with dependent errors. Annals of Statistics 25, 20542083.CrossRefGoogle Scholar
Robinson, P.M. (2001) The memory of stochastic volatility models. Journal of Econometrics 101, 195218.CrossRefGoogle Scholar
Silverman, B.W. (1986) Density Estimation. Chapman & Hall.CrossRefGoogle Scholar
Stout, W. (1974) Almost Sure Convergence. Academic Press.Google Scholar
Surgailis, D. & Viano, M.-C. (2002) Long memory properties and covariance structure of the EGARCH model. ESAIM: Probability & Statistics 6, 311329.CrossRefGoogle Scholar
Taqqu, M.S. (1975) Weak convergence to fractional Brownian motion and to the Rosenblatt process. Zeitschritt für Wahrscheinlichkeitstheorie und verwandte Gebiete 31, 287302.CrossRefGoogle Scholar
Wu, W.B. & Woodroofe, M. (2004) Martingale approximations for sums of stationary processes. Annals of Probability 32, 16741690.Google Scholar
17
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ASYMPTOTIC NORMALITY FOR WEIGHTED SUMS OF LINEAR PROCESSES
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *