Infinite-Variance Error Structure in Finance and Economics

Fatma Özgü Serttaş [1]

7 6

Many macroeconomic and financial data exhibit large outliers and high volatility so that their returns are usually modeled to follow an infinite-variance stable process. Extreme behaviors in such data tend to exist especially for emerging markets due to frequent existence of high economic turmoil. A relatively new area of research studies that model the financial returns as infinite-variance stable errors exists for emerging markets as well as for industrialized countries. This study aims to briefly introduce the reader the concept of infinite-variance stable distributions, discuss some existing studies on unit root and co-integration tests that assume infinite-variance stable error structure, and then to point out the potential lines of research while showing the significance of this relatively new concept.

Infinite-variance errors, Stable distributions, Financial returns, Unit root tests
  • 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.
  • Bagshaw, M. L. and Humpage, O. F. (1986). Intervention, exchange rate volatility, and the stable Paretian distribution. Federal Reserve Bank of Cleveland Working Paper 8608.
  • Basterfield, D., Bundt, T., and Murphy, G. (2003). Statistical properties of African FX rates: An application of the stable Paretian hypothesis. In Proceedings of the IEEE 2003 International Conference on Computational Intelligence for Financial Engineering (CIFEr), pages 223–229, Hong Kong.
  • Bidarkota, P. and McCulloch, J. H. (1998). Optimal univariate inflation forecasting with symmetric stable shocks. Journal of Applied Econometrics, 13:659–670.
  • Calder, M. and Davis, R. (1998). Inference for linear processes with stable noise. In Adler, R. J.,Feldman, R. E., and Taqqu, M. S., editors, A Practical Guide to Heavy Tails: Statistical Techniques and Applications, pages 159–176. Birkhäuser, Boston.
  • Caner, M. (1998). Tests for cointegration with infinite variance errors. Journal of Econometrics, 86:155–175.
  • Cavaliere, G., Georgiev I., and Taylor, A. M. R. (2016). Unit root inference for non-stationary linear processes driven by infinite variance innovations. Econometric Theory, 1–47.
  • Chan, N. H. and Tran, L. T. (1989). On the first order autoregressive process with infinite variance. Econometric Theory, 5(3):354–362.
  • Charemza, W., Burridge, P., and Hristova, D. (2005). Is inflation stationary? Applied Economics, 37:901–903.
  • Chen, P. and Hsiao, C.-Y. (2010). Subsampling the Johansen test with stable innovations. Australian & New Zealand Journal of Statistics, 52:61–73.
  • Dickey, D. and Fuller, W. (1979). Distribution of the estimates for autoregressive time series with a unit root. Journal of the American Statistical Association, 74:427–431.
  • DuMouchel, W. (1973). On the asymptotic normality of the maximum-likelihood estimate when sampling from a stable distribution. The Annals of Statistics, 1:948–957.
  • Engle, R. E. and Granger, C. W. (1987). Cointegration and error-correction: Representation, estimation, and testing. Econometrica, 55:251–276.
  • Falk, B. and Wang, C.-H. (2003). Testing long-run PPP with infinite variance returns. Journal of Applied Econometrics, 18:471–484.
  • Fama, E. (1965). The behavior of stock market prices. The Journal of Business, 38(1):34–105.
  • Fofack, H. and Nolan, J. P. (2001). Distribution of parallel exchange rates in African countries. Journal of International Money and Finance, 20:987–1001. Georgiev, I., Rodrigues, P. M. M., and Taylor, A. M. R. (2017). Unit root tests and heavy-tailed innovations. Journal of Time Series Analysis.
  • Hannsgen, G. (2008). Do the Innovations in a Monetary VAR Have Finite Variances?. Working Paper No. 546. Annandale-on-Hudson, NY: The Levy Economics Institute.
  • Hannsgen, G. (2011). Infinite-variance, Alpha-stable Shocks in Monetary SVAR: Final Working Paper Version. Working Paper No. 682. Annandale-on-Hudson, NY: The Levy Economics Institute.
  • Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. The Annals of Statistics, 3(5):1163–1174.
  • Horváth, L. and Kokoszka, P. (2003). A bootstrap approximation to a unit root test statistic for heavy-tailed observations. Statistics and Probability Letters, 62(2):163–173.
  • Ibragimov, M., and Khamidov, R. (2010). Heavy-Tailedness and Volatility in Emerging Foreign Exchange Markets: Theory and Empirics. EERC Working Paper Series 10/06e, EERC Research Network, Russia and CIS.
  • Ibragimov, M., Ibragimov, R., and Kattuman, P. (2013). Emerging markets and heavy tails, Journal of Banking & Finance, Elsevier, 37(7):2546-2559.
  • Johansen, S. (1988). Statistical analysis of co-integrating vectors. Journal of Economic Dynamics and Control, 12:231–254.
  • Johansen, S. (1991). Estimation and hypothesis testing of co-integration vectors in Gaussian vector autoregressive models. Econometrica, 59:1551–1580.
  • Kabaśinskas, A., Rachev, S. T., Sakalauskas, L., Sun, W., and Belovas, I. (2009), Alpha-stable paradigm in financial markets, in Journal of Computational Analysis and Applications, 11/4, 641-668.
  • Knight, K. and Samarakoon, M. (2009). Cointegration testing with infinite variance noise. Presented in Econometrics, Time Series Analysis and Systems Theory: A Conference in Honor of Manfred Deistler (18-20 June), Vienna, Austria.
  • Koedijk, K. G. and Kool, C. (1992). Tail estimates of East European exchange rates. Journal of Business and Economic Statistics, 10:83–96.
  • Koedijk, K. G., Schafgans, M. M. A., and Vries, C. G. D. (1990). The tail index of exchange rate returns. Journal of International Economics, 29:93–108.
  • Kurz-Kim, J.-R. and Loretan, M. (2014), On the Properties of the Coefficient of Determination in Regression Models with Infinite-Variance Variables, Journal of Econometrics, 181(1):15-24.
  • Mandelbrot, B. (1963). The variation of certain speculative prices. The Journal of Business, 36(4):394–419.
  • Mandelbrot, B. (1967). The variation of some other speculative prices. The Journal of Business, 40(4):393–413.
  • McCulloch, J. H. (1985). Interest-risk sensitive deposit insurance premia: Stable ARCH estimates. Journal of Banking and Finance, 9:137–156.
  • McCulloch, J. H. (1996). Financial applications of stable distributions. In Maddala, G. S. and Rao, C. R., editors, Handbook of Statistics: Statistical Models in Finance, Vol. 14, pages 393–425. Elsevier, Amsterdam.
  • Nolan, J. P. (2001). Maximum likelihood estimation and diagnostics for stable distributions. In Barndorff-Nielsen, O. E., Mikosch, T., and Resnick, S. I., editors, Lévy Processes: Theory and Applications, pages 379–400. Birkhäuser, Boston.
  • Patterson, K. D. and Heravi, S. M. (2003). The impact of fat-tailed distributions on some leading unit root tests. Journal of Applied Statistics, 30(6):635–667.
  • Paulauskas, V. and Rachev, S. T. (1998). Co-integrated processes with infinite-variance innovations. Annals of Applied Probability, 8:775–792.
  • Phillips, P. C. B. and Perron, P. (1988). Testing for a unit root in time series regression. Biometrika, 75:335–346.
  • Phillips, P. C. B. (1990). Time series regression with a unit root and infinite variance errors. Econometric Theory, 6(1):44–62.
  • Phillips, P. C. B. (1995). Robust non-stationary regression. Econometric Theory, 11:912–951.
  • Phillips, P. C. B. and Ouliaris, S. (1990). Asymptotic properties of residual based tests for co-integration. Econometrica, 58:165–193.
  • Rachev, S. T., Mittnik, S., and Kim, J.-R. (1998). Time series with unit roots and infinite variance disturbances. Applied Mathematics Letters, 11(5):69–74.
  • Rachev, S. T., Mittnik, S., Fabozzi, F. J., Focardi, S. M., and Jašić, T. (2007). Financial Econometrics: From Basics to Advanced Modeling Techniques, chapter 14, pages 465–494. Wiley, Hoboken.
  • Said, S.E. and Dickey D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika, 71:599–608.
  • Samarakoon, M. and Knight, K. (2009). A note on unit root tests with infinite variance noise. Econometric Reviews, 28(4):314–334.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes. Chapman and Hall, New York.
  • Serttaş, F. Ö. (2011). Essays on Infinite-Variance Stable Errors and Robust Estimation Procedures: A Monte Carlo Study with Empirical Applications. Saarbrücken, Germany: LAP Lambert Academic Publishing.
  • So, J. C. (1987). The Sub-Gaussian Distribution of Currency Futures: Stable Paretian or Nonstationary? Review of Economics and Statistics, 69:100–107.
  • Thavaneswaran, A. and Peiris, S. (1999). Estimation for regression with infinite variance errors. Mathematical and Computer Modeling, 29 (10), 177–180.
  • Westerfield, J. M. (1977). An examination of foreign exchange risk under fixed and floating rate regimes. Journal of International Economics, 7(2):181–200.
  • Wilson, H. G. (1978). Least squares versus minimum absolute deviations estimation in linear models. Decision Sciences, 9(2):322–335.
  • Zarepour, M., and Roknossadati, S. M. (2008). Multivariate Autoregression of Order One with Infinite Variance Innovations. Econometric Theory, 24(3):677–695.
Dergi Bölümü Makaleler

Yazar: Fatma Özgü Serttaş
Ülke: Turkey

Bibtex @ { ier306676, journal = {International Econometric Review}, issn = {1308-8793}, address = {Ekonometrik Araştırmalar Derneği}, year = {}, volume = {10}, pages = {14 - 23}, doi = {}, title = {Infinite-Variance Error Structure in Finance and Economics}, key = {cite}, author = {Serttaş, Fatma Özgü} }
APA Serttaş, F . (). Infinite-Variance Error Structure in Finance and Economics. International Econometric Review, 10 (1), 14-23. Retrieved from
MLA Serttaş, F . "Infinite-Variance Error Structure in Finance and Economics". International Econometric Review 10 (): 14-23 <>
Chicago Serttaş, F . "Infinite-Variance Error Structure in Finance and Economics". International Econometric Review 10 (): 14-23
RIS TY - JOUR T1 - Infinite-Variance Error Structure in Finance and Economics AU - Fatma Özgü Serttaş Y1 - 2018 PY - 2018 N1 - DO - T2 - International Econometric Review JF - Journal JO - JOR SP - 14 EP - 23 VL - 10 IS - 1 SN - 1308-8793-1308-8815 M3 - UR - Y2 - 2018 ER -
EndNote %0 International Econometric Review Infinite-Variance Error Structure in Finance and Economics %A Fatma Özgü Serttaş %T Infinite-Variance Error Structure in Finance and Economics %D 2018 %J International Econometric Review %P 1308-8793-1308-8815 %V 10 %N 1 %R %U