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Infinite-Variance Error Structure in Finance and Economics

Year 2018, , 14 - 23, 16.04.2018
https://doi.org/10.33818/ier.306676

Abstract

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.

References

  • 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.
Year 2018, , 14 - 23, 16.04.2018
https://doi.org/10.33818/ier.306676

Abstract

References

  • 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.
There are 51 citations in total.

Details

Journal Section Articles
Authors

Fatma Özgü Serttaş

Publication Date April 16, 2018
Submission Date April 17, 2017
Published in Issue Year 2018

Cite

APA Serttaş, F. Ö. (2018). Infinite-Variance Error Structure in Finance and Economics. International Econometric Review, 10(1), 14-23. https://doi.org/10.33818/ier.306676
AMA Serttaş FÖ. Infinite-Variance Error Structure in Finance and Economics. IER. April 2018;10(1):14-23. doi:10.33818/ier.306676
Chicago Serttaş, Fatma Özgü. “Infinite-Variance Error Structure in Finance and Economics”. International Econometric Review 10, no. 1 (April 2018): 14-23. https://doi.org/10.33818/ier.306676.
EndNote Serttaş FÖ (April 1, 2018) Infinite-Variance Error Structure in Finance and Economics. International Econometric Review 10 1 14–23.
IEEE F. Ö. Serttaş, “Infinite-Variance Error Structure in Finance and Economics”, IER, vol. 10, no. 1, pp. 14–23, 2018, doi: 10.33818/ier.306676.
ISNAD Serttaş, Fatma Özgü. “Infinite-Variance Error Structure in Finance and Economics”. International Econometric Review 10/1 (April 2018), 14-23. https://doi.org/10.33818/ier.306676.
JAMA Serttaş FÖ. Infinite-Variance Error Structure in Finance and Economics. IER. 2018;10:14–23.
MLA Serttaş, Fatma Özgü. “Infinite-Variance Error Structure in Finance and Economics”. International Econometric Review, vol. 10, no. 1, 2018, pp. 14-23, doi:10.33818/ier.306676.
Vancouver Serttaş FÖ. Infinite-Variance Error Structure in Finance and Economics. IER. 2018;10(1):14-23.