The Annals of Applied Probability

Cointegrated processes with infinite variance innovations

Vygantas Paulauskas and Svetlozar T. Rachev

Full-text: Open access


It is widely accepted that the Gaussian assumption is too restrictive to model either financial or some important macroeconomic variables, because their distributions exhibit asymmetry and heavy tails. In this paper we develop the asymptotic theory for econometric cointegration processes under the assumption of infinite variance innovations with different distributional tail behavior. We extend some of the results of Park and Phillips which were derived under the assumption of finite variance errors.

Article information

Ann. Appl. Probab., Volume 8, Number 3 (1998), 775-792.

First available in Project Euclid: 9 August 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles 60H05: Stochastic integrals 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Cointegrated processes stable distribution Lévy processes ordinary least-squares estimators


Paulauskas, Vygantas; Rachev, Svetlozar T. Cointegrated processes with infinite variance innovations. Ann. Appl. Probab. 8 (1998), no. 3, 775--792. doi:10.1214/aoap/1028903450.

Export citation


  • Akgiray, V. and Booth, G. G. (1988). The stable law model of stock returns. J. Bus. Econ. Statist. 6 51-57.
  • Akgiray, V., Booth, G. G. and Seifert, B. (1988). Distribution properties of Latin American black market exchange rates. Journal of International Money and Finance 7 37-48.
  • Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York.
  • Caner, M. (1995). Tests for cointegration with infinite variance errors. Preprint.
  • Chan, N. H. and Tran, L. T. (1989). On the first order autoregressive process with infinite variance. Econometric Theory 5 354-362.
  • Chan, N. H. and Wei, C. Z. (1988). Limiting distribution of least squares estimates of unstable autoregressive processes. Ann. Statist. 16 367-401.
  • Davis, R. and Resnick, S. (1985). Limit theorems for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195.
  • Elliott, R. J. (1982). Stochastic Calculus and Applications. Springer, New York.
  • Engle, R. F. and Granger, C. W. J. (1987). Cointegration and error correction: representation, estimation and testing. Econometrica 55 251-276.
  • Engle, R. F. and Yoo, S. B. (1987). Forecasting and testing in cointegrated sy stems. J. Econometrics 35 143-159.
  • Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York.
  • Fama, E. (1965). The behavior of stock market prices. J. Business 38 34-105.
  • Gikhman, I. I. and Skorokhod, A. V. (1969). Introduction to the Theory of Random Processes. W. B. Saunders, Philadelphia.
  • Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. J. Econometrics 16 121-130.
  • Jacod, J. and Shiry aev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
  • Jakubowski, A., Memin, J. and Pages, G. (1989). Convergence en loi des suites d'integrales stochastiques sur l'espace D1 de Skorokhod. Probab. Theory Related Fields 81 111-137.
  • Jeganathan, P. (1991). On the asy mptotic behavior of least-squares estimators in AR time series with roots near the unit circle. Econometric Theory 7 269-306.
  • Johansen, S. (1988). Statistical analysis of cointegration vectors. J. Econom. Dy nam. Control 12 231-254.
  • Johansen, S. (1991). Estimation and hy pothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 1551-1580.
  • Koedijk, K. and Kool, C. M. J. (1992). Tail estimates of East European exchange rates. J. Bus. Econom. Statist. 10 83-96.
  • Kopp, P. E. (1984). Martingales and Stochastic Integrals. Cambridge Univ. Press. Kurtz, T. G. and Protter, P. (1991a). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070.
  • Memin, J. and Slominski, L. (1991). Condition UT et stabilit´e en loi des solutions d'equations differentielles stochastiques. S´eminaire de Probabilit´es XXV. Lecture Notes in Math. 1485 162-177. Springer, Berlin.
  • Mittnik, S. and Rachev, S. T. (1993). Modeling asset returns with alternative stable distributions. Econometric Rev. 12 261-330.
  • Nelson, C. R. and Plosser, C. I. (1982). Trends and random walks in macroeconomic time series: some evidence and implications. J. Monetary Economics 10 129.
  • Park, J. Y. and Phillips, P. C. B. (1988). Statistical inference in regressions with integrated processes I. Econometric Theory 4 468-497.
  • Phillips, P. C. B. (1990). Time series regression with a unit root and infinite-variance errors. Econometric Theory 6 44-62.
  • Phillips, P. C. B. and Durlauf, S. N. (1986). Multiple time series regression with integrated processes. Rev. Econom. Stud. 53 473-495.
  • Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin.
  • Rachev, S. T., Kim, J-R. and Mittnik, S. (1997). Econometric modeling in the presence of heavy tailed innovations: a survey of some recent advances. Comm. Statist. Stochastic Models 13 841-866.
  • Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York.
  • Resnick, S. and Greenwood, P. (1979). Bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9 206-221.
  • Samorodnitsky, G. and Taqqu, M. S. (1993). Stable Non-Gaussian Random Processes: Models with Infinite Variance. Chapman and Hall, London.
  • Sharpe, M. (1969). Operator-stable probability distributions on vector groups. Trans. Amer. Math. Soc. 136 51-65.
  • Skorokhod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2 138-171.
  • Stock, J. H. and Watson, W. W. (1988). Testing for common trends. J. Amer. Statist. Assoc. 83 1097-1107.
  • Stricker, C. (1985). Lois de semimartingales et criteres de compacite. S´eminaire de Probabilit´es XIX. Lecture Notes in Math. 1123 209-217. Springer, Berlin.