Cointegrated processes with infinite variance innovations. (English) Zbl 0941.62092

Recently, stable non-Gaussian processes are used to model some important economics variables (in finance or macroeconomics). The authors apply this approach to cointegration theory and obtain a proof of the main asymptotic result under the assumption that innovations are independent, identically distributed and are in the domain of attraction of an \((\alpha_1, \ldots ,\alpha_r)\)-stable law. The paper generalizes some of the results of J.Y. Park and {it P.C.B. Phillips} [Econometric Theory 4, No. 3, 468-497 (1988)].


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60F17 Functional limit theorems; invariance principles
62P20 Applications of statistics to economics
Full Text: DOI


[1] Akgiray, V. and Booth, G. G. (1988). The stable law model of stock returns. J. Bus. Econ. Statist. 6 51-57.
[2] 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.
[3] Billingsley, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201
[4] Caner, M. (1995). Tests for cointegration with infinite variance errors. · Zbl 1041.62515
[5] Chan, N. H. and Tran, L. T. (1989). On the first order autoregressive process with infinite variance. Econometric Theory 5 354-362. JSTOR:
[6] Chan, N. H. and Wei, C. Z. (1988). Limiting distribution of least squares estimates of unstable autoregressive processes. Ann. Statist. 16 367-401. · Zbl 0666.62019
[7] Davis, R. and Resnick, S. (1985). Limit theorems for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179-195. · Zbl 0562.60026
[8] Elliott, R. J. (1982). Stochastic Calculus and Applications. Springer, New York. · Zbl 0503.60062
[9] Engle, R. F. and Granger, C. W. J. (1987). Cointegration and error correction: representation, estimation and testing. Econometrica 55 251-276. JSTOR: · Zbl 0613.62140
[10] Engle, R. F. and Yoo, S. B. (1987). Forecasting and testing in cointegrated sy stems. J. Econometrics 35 143-159. · Zbl 0649.62108
[11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Characterization and Convergence. Wiley, New York. · Zbl 0592.60049
[12] Fama, E. (1965). The behavior of stock market prices. J. Business 38 34-105. · Zbl 0129.11903
[13] Gikhman, I. I. and Skorokhod, A. V. (1969). Introduction to the Theory of Random Processes. W. B. Saunders, Philadelphia. · Zbl 0132.37902
[14] Granger, C. W. J. (1981). Some properties of time series data and their use in econometric model specification. J. Econometrics 16 121-130.
[15] Jacod, J. and Shiry aev, A. N. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin.
[16] 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. · Zbl 0638.60049
[17] 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. JSTOR: · Zbl 04505239
[18] Johansen, S. (1988). Statistical analysis of cointegration vectors. J. Econom. Dy nam. Control 12 231-254. · Zbl 0647.62102
[19] Johansen, S. (1991). Estimation and hy pothesis testing of cointegration vectors in Gaussian vector autoregressive models. Econometrica 59 1551-1580. JSTOR: · Zbl 0755.62087
[20] Koedijk, K. and Kool, C. M. J. (1992). Tail estimates of East European exchange rates. J. Bus. Econom. Statist. 10 83-96.
[21] 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. · Zbl 0742.60053
[22] Memin, J. and Slominski, L. (1991). Condition UT et stabilité en loi des solutions d’equations differentielles stochastiques. Séminaire de Probabilités XXV. Lecture Notes in Math. 1485 162-177. Springer, Berlin. · Zbl 0746.60063
[23] Mittnik, S. and Rachev, S. T. (1993). Modeling asset returns with alternative stable distributions. Econometric Rev. 12 261-330. · Zbl 0801.62096
[24] 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.
[25] Park, J. Y. and Phillips, P. C. B. (1988). Statistical inference in regressions with integrated processes I. Econometric Theory 4 468-497. JSTOR:
[26] Phillips, P. C. B. (1990). Time series regression with a unit root and infinite-variance errors. Econometric Theory 6 44-62. JSTOR:
[27] Phillips, P. C. B. and Durlauf, S. N. (1986). Multiple time series regression with integrated processes. Rev. Econom. Stud. 53 473-495. JSTOR: · Zbl 0599.62103
[28] Protter, P. (1990). Stochastic Integration and Differential Equations. Springer, Berlin. · Zbl 0694.60047
[29] 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. · Zbl 0888.90034
[30] Resnick, S. (1987). Extreme Values, Regular Variation and Point Processes. Springer, New York. · Zbl 0633.60001
[31] Resnick, S. and Greenwood, P. (1979). Bivariate stable characterization and domains of attraction. J. Multivariate Anal. 9 206-221. · Zbl 0409.62038
[32] Samorodnitsky, G. and Taqqu, M. S. (1993). Stable Non-Gaussian Random Processes: Models with Infinite Variance. Chapman and Hall, London. · Zbl 0925.60027
[33] Sharpe, M. (1969). Operator-stable probability distributions on vector groups. Trans. Amer. Math. Soc. 136 51-65. · Zbl 0192.53603
[34] Skorokhod, A. V. (1957). Limit theorems for stochastic processes with independent increments. Theory Probab. Appl. 2 138-171. · Zbl 0097.13001
[35] Stock, J. H. and Watson, W. W. (1988). Testing for common trends. J. Amer. Statist. Assoc. 83 1097-1107. JSTOR: · Zbl 0673.62099
[36] Stricker, C. (1985). Lois de semimartingales et criteres de compacite. Séminaire de Probabilités XIX. Lecture Notes in Math. 1123 209-217. Springer, Berlin. · Zbl 0558.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.