{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T10:10:51Z","timestamp":1760523051057},"reference-count":18,"publisher":"Cambridge University Press (CUP)","issue":"5","license":[{"start":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T00:00:00Z","timestamp":1361404800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Econom. Theory"],"published-print":{"date-parts":[[2013,10]]},"abstract":"<jats:p>Relevant sample quantities such as the sample autocorrelation function and extremes contain useful information about autoregressive time series with heteroskedastic errors. As these quantities usually depend on the tail index of the underlying heteroskedastic time series, estimating the tail index becomes an important task. Since the tail index of such a model is determined by a moment equation, one can estimate the underlying tail index by solving the sample moment equation with the unknown parameters being replaced by their quasi-maximum likelihood estimates. To construct a confidence interval for the tail index, one needs to estimate the complicated asymptotic variance of the tail index estimator, however. In this paper the asymptotic normality of the tail index estimator is first derived, and a profile empirical likelihood method to construct a confidence interval for the tail index is then proposed. A simulation study shows that the proposed empirical likelihood method works better than the bootstrap method in terms of coverage accuracy, especially when the process is nearly nonstationary.<\/jats:p>","DOI":"10.1017\/s0266466612000801","type":"journal-article","created":{"date-parts":[[2013,2,21]],"date-time":"2013-02-21T09:20:02Z","timestamp":1361438402000},"page":"920-940","source":"Crossref","is-referenced-by-count":10,"title":["TAIL INDEX OF AN AR(1) MODEL WITH ARCH(1) ERRORS"],"prefix":"10.1017","volume":"29","author":[{"given":"Ngai Hang","family":"Chan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Deyuan","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang","family":"Peng","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rongmao","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"56","published-online":{"date-parts":[[2013,2,21]]},"reference":[{"key":"S0266466612000801_ref13","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1111\/j.1467-9868.2004.00432.x","article-title":"Estimation and testing stationarity for double autoregressive models","volume":"66","author":"Ling","year":"2004","journal-title":"Journal of the Royal Statistical Society B"},{"key":"S0266466612000801_ref2","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1017\/S0266466603194030","article-title":"Estimation of the maximal moment exponent of a GARCH (1,1) sequence","volume":"19","author":"Berkes","year":"2003","journal-title":"Econometric Theory"},{"key":"S0266466612000801_ref17","doi-asserted-by":"crossref","first-page":"300","DOI":"10.1214\/aos\/1176325370","article-title":"Empirical likelihood and general estimating equations","volume":"22","author":"Qin","year":"1994","journal-title":"Annals of Statistics"},{"key":"S0266466612000801_ref8","volume-title":"Financial Modeling with Jump Processes","author":"Cont","year":"2004"},{"key":"S0266466612000801_ref18","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1111\/j.1467-9892.1984.tb00382.x","article-title":"ARMA models with ARCH errors","volume":"3","author":"Weiss","year":"1984","journal-title":"Journal of Time Series Analysis"},{"key":"S0266466612000801_ref16","doi-asserted-by":"crossref","DOI":"10.1201\/9781420036152","volume-title":"Empirical Likelihood","author":"Owen","year":"2001"},{"key":"S0266466612000801_ref7","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s11749-009-0159-5","article-title":"A review on empirical likelihood methods for regression","volume":"18","author":"Chen","year":"2009","journal-title":"Test"},{"key":"S0266466612000801_ref3","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1016\/S0304-4149(99)00073-3","article-title":"Extremal behavior of the autoregressive process with ARCH(1) errors","volume":"85","author":"Borkovec","year":"2000","journal-title":"Stochastic Processes and Their Applications"},{"key":"S0266466612000801_ref5","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1214\/aoap\/1015345401","article-title":"The tail of the stationary distribution of an autoregressive process with ARCH(1) errors","volume":"11","author":"Borkovec","year":"2001","journal-title":"Annals of Applied Probability"},{"key":"S0266466612000801_ref4","doi-asserted-by":"crossref","first-page":"847","DOI":"10.2307\/3318623","article-title":"Asymptotic behavior of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH errors","volume":"7","author":"Borkovec","year":"2001","journal-title":"Bernoulli"},{"key":"S0266466612000801_ref6","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1093\/biomet\/92.2.477","article-title":"Weighted least absolute deviations estimation for an AR(1) process with ARCH(1) errors","volume":"92","author":"Chan","year":"2005","journal-title":"Biometrika"},{"key":"S0266466612000801_ref15","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1214\/aos\/1176347494","article-title":"Empirical likelihood confidence regions","volume":"18","author":"Owen","year":"1990","journal-title":"Annals of Statistics"},{"key":"S0266466612000801_ref11","volume-title":"Martingale Limit Theory and Its Application.","author":"Hall","year":"1980"},{"key":"S0266466612000801_ref12","first-page":"48","article-title":"Empirical likelihood method for an AR(1) process with ARCH(1) errors","volume":"1","author":"Kl\u00fcppelberg","year":"2006","journal-title":"International Journal of Statistics and Management Systems"},{"key":"S0266466612000801_ref10","unstructured":"Guo, B. & Phillips, P.C.B. (1998) Efficient Estimation of Second Moment Parameters in ARCH Models. Unpublished manuscript."},{"key":"S0266466612000801_ref1","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1017\/S0266466600006587","article-title":"The Bahadur-Kiefer representation of Lp regression estimators","volume":"12","author":"Arcones","year":"1996","journal-title":"Econometric Theory"},{"key":"S0266466612000801_ref14","first-page":"1427","article-title":"and C. 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