{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,9]],"date-time":"2026-10-09T05:46:04Z","timestamp":1791524764928,"version":"4.3.4"},"reference-count":68,"publisher":"Cambridge University Press (CUP)","issue":"1","license":[{"start":{"date-parts":[[2012,7,6]],"date-time":"2012-07-06T00:00:00Z","timestamp":1341532800000},"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":[[2013,2]]},"abstract":"<jats:p>The paper presents a cointegration model in continuous time, where the linear combinations of the integrated processes are modeled by a multivariate Ornstein\u2013Uhlenbeck process. The integrated processes are defined as vector-valued L\u00e9vy processes with an additional noise term. Hence, if we observe the process at discrete time points, we obtain a multiple regression model. As an estimator for the regression parameter we use the least squares estimator. We show that it is a consistent estimator and derive its asymptotic behavior. The limit distribution is a ratio of functionals of Brownian motions and stable L\u00e9vy processes, whose characteristic triplets have an explicit analytic representation. In particular, we present the Wald and the<jats:italic>t<\/jats:italic>-ratio statistic and simulate asymptotic confidence intervals. For the proofs we derive some central limit theorems for multivariate Ornstein\u2013Uhlenbeck processes.<\/jats:p>","DOI":"10.1017\/s0266466612000217","type":"journal-article","created":{"date-parts":[[2012,7,6]],"date-time":"2012-07-06T11:54:06Z","timestamp":1341575646000},"page":"28-67","source":"Crossref","is-referenced-by-count":7,"title":["TIME SERIES REGRESSION ON INTEGRATED CONTINUOUS-TIME PROCESSES WITH HEAVY AND LIGHT TAILS"],"prefix":"10.1017","volume":"29","author":[{"given":"Vicky","family":"Fasen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"56","published-online":{"date-parts":[[2012,7,6]]},"reference":[{"key":"S0266466612000217_ref10","doi-asserted-by":"crossref","first-page":"506","DOI":"10.1017\/S0266466600005983","article-title":"Weak convergence of matrix stochastic integral with stable processes","volume":"13","author":"Caner","year":"1997","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref47","doi-asserted-by":"crossref","first-page":"1145","DOI":"10.1016\/S0895-7177(01)00123-6","article-title":"Statistical inference in regression with heavy-tailed integrated variables","volume":"34","author":"Mittnik","year":"2001","journal-title":"Mathematical and Computer Modelling"},{"key":"S0266466612000217_ref22","doi-asserted-by":"crossref","first-page":"251","DOI":"10.2307\/1913236","article-title":"Co-integration and error correction: Representation, estimation and testing","volume":"55","author":"Engle","year":"1987","journal-title":"Econometrica"},{"key":"S0266466612000217_ref66","volume-title":"Probability","author":"Shiryaev","year":"1995"},{"key":"S0266466612000217_ref21","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1080\/14697680500149370","article-title":"Pairs trading","volume":"5","author":"Elliott","year":"2005","journal-title":"Quantitative Finance"},{"key":"S0266466612000217_ref56","volume-title":"Handbook of Heavy Tailed Distributions in Finance","author":"Rachev","year":"2003"},{"key":"S0266466612000217_ref6","doi-asserted-by":"crossref","DOI":"10.2202\/1558-3708.1355","article-title":"Analytic approximations for the price dynamics of spark spread options","volume":"10","author":"Benth","year":"2006","journal-title":"Studies in Nonlinear Dynamics Econometrics"},{"key":"S0266466612000217_ref63","volume-title":"L\u00e9vy Processes and Infinitely Divisible Distributions","author":"Sato","year":"1999"},{"key":"S0266466612000217_ref49","doi-asserted-by":"crossref","first-page":"775","DOI":"10.1214\/aoap\/1028903450","article-title":"Cointegrated processes with infinite variance innovations","volume":"8","author":"Paulauskas","year":"1998","journal-title":"Annals of Applied Probability"},{"key":"S0266466612000217_ref30","doi-asserted-by":"crossref","first-page":"888","DOI":"10.1017\/S0266466608080365","article-title":"Regression asymptotics using martingale convergence methods","volume":"24","author":"Ibragimov","year":"2008","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref15","volume-title":"An Introduction to the Theory of Point Processes, vol. I, Elementary Theory and Methods","author":"Daley","year":"2003"},{"key":"S0266466612000217_ref28","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/0304-4076(81)90079-8","article-title":"Some properties of time series data and their use in econometric model specification","volume":"16","author":"Granger","year":"1981","journal-title":"Journal of Econometrics"},{"key":"S0266466612000217_ref3","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1214\/aop\/1176989938","article-title":"Weak convergence of sums of moving average processes in the \u03b1-stable domain of attraction","volume":"20","author":"Avram","year":"1991","journal-title":"Annals of Probability"},{"key":"S0266466612000217_ref43","doi-asserted-by":"crossref","first-page":"97","DOI":"10.3150\/bj\/1077544605","article-title":"On multidimensional Ornstein-Uhlenbeck processes driven by a general L\u00e9vy process","volume":"10","author":"Masuda","year":"2004","journal-title":"Bernoulli"},{"key":"S0266466612000217_ref67","first-page":"403","article-title":"Gaussian diffusions and autoregressive processes: Weak convergence and statistical inference","volume":"21","author":"Stockmarr","year":"1994","journal-title":"Scandinavian Journal Statistics"},{"key":"S0266466612000217_ref17","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1214\/aop\/1176993074","article-title":"Limit theory for moving averages of random variables with regularly varying tail probabilities","volume":"13","author":"Davis","year":"1985","journal-title":"Annals of Probability"},{"key":"S0266466612000217_ref23","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1086\/294743","article-title":"The behavior of stock market prices","volume":"38","author":"Fama","year":"1965","journal-title":"Journal of of Business"},{"key":"S0266466612000217_ref55","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1554-7","volume-title":"Subsampling","author":"Politis","year":"1999"},{"key":"S0266466612000217_ref68","doi-asserted-by":"crossref","DOI":"10.1002\/9781118673362","volume-title":"Modeling and Forecasting Electricity Loads and Prices: A Statistical Approach","author":"Weron","year":"2006"},{"key":"S0266466612000217_ref62","volume-title":"Stable Non-Gaussian Random Processes","author":"Samorodnitsky","year":"1994"},{"key":"S0266466612000217_ref34","volume-title":"Likelihood-Based Inference on Cointegration in the Vector Autoregressive Model","author":"Johansen","year":"1996"},{"key":"S0266466612000217_ref16","first-page":"26.2","article-title":"Extremal properties of a class of multivariate moving averages","volume":"51","author":"Davis","year":"1985","journal-title":"Bulletin of the International Statistical Institute"},{"key":"S0266466612000217_ref2","volume-title":"A Practical Guide to Heavy Tails","author":"Adler","year":"1998"},{"key":"S0266466612000217_ref24","unstructured":"Fasen, V. (2011) Limit theory for high frequency sampled MCARMA models. Manuscript, ETH, Z\u00fcrich."},{"key":"S0266466612000217_ref11","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1007\/978-3-540-71297-8_30","volume-title":"Handbook of Financial Time Series","author":"Chan","year":"2009"},{"key":"S0266466612000217_ref36","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1017\/CBO9780511662980.010","article-title":"Characterizing the weak convergence of stochastic integrals","volume":"167","author":"Kurtz","year":"1991","journal-title":"Stochastic Analysis"},{"key":"S0266466612000217_ref52","doi-asserted-by":"crossref","first-page":"967","DOI":"10.2307\/2938169","article-title":"Error correction and long-run equilibrium in continuous time","volume":"59","author":"Phillips","year":"1991","journal-title":"Econometrica"},{"key":"S0266466612000217_ref26","unstructured":"Ferstl, C. (2009) Cointegration in discrete and continuous time, Diploma Thesis, Technische Universit\u00e4t M\u00fcnchen."},{"key":"S0266466612000217_ref4","doi-asserted-by":"crossref","first-page":"6497","DOI":"10.1007\/s10958-006-0366-5","article-title":"Student\u2019s t-test for Gaussian scale mixtures","volume":"139","author":"Bakirov","year":"2006","journal-title":"Journal of Mathematical Science"},{"key":"S0266466612000217_ref44","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1080\/07350015.1997.10524689","article-title":"Measuring tail thickness to estimate the stable index \u03b1: A critique","volume":"15","author":"McCulloch","year":"1997","journal-title":"Journal of Business and Economic Statistics"},{"key":"S0266466612000217_ref59","volume-title":"Adventures in Stochastic Processes","author":"Resnick","year":"1992"},{"key":"S0266466612000217_ref27","doi-asserted-by":"crossref","first-page":"797","DOI":"10.1093\/rfs\/hhj020","article-title":"Pairs trading: Performance of a relative-value arbitrage rule","volume":"19","author":"Gatev","year":"2006","journal-title":"Review of Financial Studies"},{"key":"S0266466612000217_ref57","volume-title":"Stable Paretian Models in Finance","author":"Rachev","year":"2000"},{"key":"S0266466612000217_ref29","doi-asserted-by":"crossref","first-page":"838","DOI":"10.3150\/08-BEJ125","article-title":"Tail probabilities for infinite series of regularly varying random vectors","volume":"14","author":"Hult","year":"2008","journal-title":"Bernoulli"},{"key":"S0266466612000217_ref37","unstructured":"Lindskog, F. (2004) Multivariate extremes and regular variation for stochastic processes, Ph.D. Dissertation, ETH Z\u00fcrich."},{"key":"S0266466612000217_ref45","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1017\/S026646660218501X","article-title":"Robust inference for the mean in the presence of serial correlation and heavy tailed distributions","volume":"18","author":"McElroy","year":"2002","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref54","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1214\/aos\/1176348666","article-title":"Asymptotics for linear processes","volume":"20","author":"Phillips","year":"1992","journal-title":"Annals of Statistics"},{"key":"S0266466612000217_ref14","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1016\/S0304-4076(98)00025-6","article-title":"Discrete and continuous time cointegration","volume":"88","author":"Comte","year":"1999","journal-title":"Journal of Econometrics"},{"key":"S0266466612000217_ref1","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1016\/S0378-3758(02)00419-6","article-title":"Generalized least-squares estimators for the thickness of heavy tails","volume":"119","author":"Aban","year":"2004","journal-title":"Journal of Statistical Planning and Inference"},{"key":"S0266466612000217_ref58","doi-asserted-by":"crossref","DOI":"10.1007\/978-0-387-75953-1","volume-title":"Extreme Values, Regular Variation, and Point Processes","author":"Resnick","year":"1987"},{"key":"S0266466612000217_ref33","doi-asserted-by":"crossref","first-page":"171","DOI":"10.2298\/PIM0694171J","article-title":"Regularly varying functions","volume":"80","author":"Jessen","year":"2006","journal-title":"Publications de l\u2019Institut Math\u00e9matique"},{"key":"S0266466612000217_ref20","doi-asserted-by":"crossref","first-page":"247","DOI":"10.1007\/978-3-642-18412-3_9","volume-title":"Advanced Mathematical Methods for Finance","author":"Ekstr\u00f6m","year":"2011"},{"key":"S0266466612000217_ref35","volume-title":"Foundations of Modern Probability","author":"Kallenberg","year":"1997"},{"key":"S0266466612000217_ref46","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1111\/1467-9892.00187","article-title":"Moving averages of random vectors with regularly varying tails","volume":"21","author":"Meerschaert","year":"2000","journal-title":"Journal of Time Series Analysis"},{"key":"S0266466612000217_ref48","doi-asserted-by":"crossref","first-page":"907","DOI":"10.1017\/S0266466610000538","article-title":"weak convergence to a matrix stochastic integral with stable processes","volume":"27","author":"Paulauskas","year":"2011","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref60","volume-title":"Heavy-Tail Phenomena: Probabilistic and Statistical Modeling","author":"Resnick","year":"2007"},{"key":"S0266466612000217_ref9","doi-asserted-by":"crossref","first-page":"23","DOI":"10.2307\/2331251","article-title":"Arbitrage, cointegration, and testing the unbiasedness hypothesis in financial markets","volume":"10","author":"Brenner","year":"1995","journal-title":"Journal of Financial Quantitative Analysis"},{"key":"S0266466612000217_ref25","article-title":"Statistical estimation of multivariate Ornstein-Uhlenbeck processes and applications to co-integration","author":"Fasen","year":"2012","journal-title":"Journal of of Econometrics"},{"key":"S0266466612000217_ref38","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/0927-5398(94)90004-3","article-title":"Testing the covariance stationarity of heavy tailed time series","volume":"1","author":"Loretan","year":"1994","journal-title":"Journal of Empirical Finance"},{"key":"S0266466612000217_ref51","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1017\/S0266466600004904","article-title":"Time series regression with a unit root and infinite-variance errors","volume":"6","author":"Phillips","year":"1990","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref41","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1086\/294632","article-title":"The variation of certain speculative prices","volume":"36","author":"Mandelbrot","year":"1963","journal-title":"Journal of Business"},{"key":"S0266466612000217_ref31","volume-title":"Limit Theorems for Stochastic Processes","author":"Jacod","year":"2002"},{"key":"S0266466612000217_ref12","doi-asserted-by":"crossref","first-page":"354","DOI":"10.1017\/S0266466600012561","article-title":"On the first order autoregressive process with infinite variance","volume":"5","author":"Chan","year":"1989","journal-title":"Econometric Theory"},{"key":"S0266466612000217_ref5","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0304-4149(01)00156-9","article-title":"Regular variation of GARCH processes","volume":"99","author":"Basrak","year":"2002","journal-title":"Stochastic Processes and Their Applications"},{"key":"S0266466612000217_ref13","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/BF00344720","article-title":"Convolution tails, product tails and domains of attraction","volume":"72","author":"Cline","year":"1986","journal-title":"Probability Theory and Related Fields"},{"key":"S0266466612000217_ref7","doi-asserted-by":"crossref","DOI":"10.1142\/6811","volume-title":"Stochastic Modelling of Electricity and Related Markets","author":"Benth","year":"2008"},{"key":"S0266466612000217_ref18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-85636-8","volume-title":"Self-Normalized Processes: Limit Theory and Statistical Applications","author":"de la Pe\u00f1a","year":"2009"},{"key":"S0266466612000217_ref39","volume-title":"New Introduction to Multiple Time Series Analysis","author":"L\u00fctkepohl","year":"2007"},{"key":"S0266466612000217_ref42","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1016\/j.spa.2006.05.014","article-title":"Multivariate CARMA processes","volume":"117","author":"Marquardt","year":"2007","journal-title":"Stochastic Processes and Their Applications"},{"key":"S0266466612000217_ref61","first-page":"183","article-title":"On domains of attraction of multidimensional distributions","volume":"2","author":"Rva\u010deva","year":"1962","journal-title":"Selected Translations in Mathematical Statistics and Probability Theory"},{"key":"S0266466612000217_ref8","volume-title":"Convergence of Probability Measures","author":"Billingsley","year":"1968"},{"key":"S0266466612000217_ref53","doi-asserted-by":"crossref","first-page":"473","DOI":"10.2307\/2297602","article-title":"Multiple time series regression with integrated processes","volume":"53","author":"Phillips","year":"1986","journal-title":"Review of Economic Studies"},{"key":"S0266466612000217_ref32","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-84628-737-4","volume-title":"Mathematical Methods for Financial Markets","author":"Jeanblanc","year":"2009"},{"key":"S0266466612000217_ref40","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1013846631785","article-title":"Electricity prices and power derivatives: Evidence from the nordic power exchange","volume":"5","author":"Lucia","year":"2002","journal-title":"Review of Derivatives Research"},{"key":"S0266466612000217_ref65","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1137\/S0040585X97981032","article-title":"On the martingale measures in exponential L\u00e9vy models","volume":"49","author":"Selivanvov","year":"2005","journal-title":"Theory of Probability and Its Applications"},{"key":"S0266466612000217_ref50","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1093\/biomet\/74.3.535","article-title":"Towards a unified asymptotic theory for autoregression","volume":"74","author":"Phillips","year":"1987","journal-title":"Biometrika"},{"key":"S0266466612000217_ref19","doi-asserted-by":"crossref","first-page":"727","DOI":"10.1016\/S0165-1889(03)00042-3","article-title":"Option valuation with co-integrated assets","volume":"28","author":"Duan","year":"2004","journal-title":"Journal of Economic Dynamics and Control"},{"key":"S0266466612000217_ref64","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/0304-4149(84)90312-0","article-title":"Operator selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type","volume":"17","author":"Sato","year":"1984","journal-title":"Stochastic Processes and Their Applications"}],"container-title":["Econometric Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0266466612000217","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,4,26]],"date-time":"2024-04-26T13:53:12Z","timestamp":1714139592000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0266466612000217\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,7,6]]},"references-count":68,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,2]]}},"alternative-id":["S0266466612000217"],"URL":"https:\/\/doi.org\/10.1017\/s0266466612000217","relation":{},"ISSN":["0266-4666","1469-4360"],"issn-type":[{"value":"0266-4666","type":"print"},{"value":"1469-4360","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,7,6]]}}}