[go: up one dir, main page]
IDEAS home Printed from https://ideas.repec.org/p/chf/rpseri/rp1733.html
   My bibliography  Save this paper

Super-Exponential RE Bubble Model with Efficient Crashes

Author

Listed:
  • Jerome L Kreuser

    (ETH Zurich)

  • Didier Sornette

    (ETH Zürich and Swiss Finance Institute)

Abstract
We propose a dynamic Rational Expectations (RE) bubble model of prices with the intention to exploit it for and evaluate it on optimal investment strategies. Our bubble model is defined as a geometric Brownian motion combined with separate crash (and rally) discrete jump distributions associated with positive (and negative) bubbles. We assume that crashes tend to efficiently bring back excess bubble prices close to a “normal” or fundamental value (“efficient crashes”). Then, the RE condition implies that the excess risk premium of the risky asset exposed to crashes is an increasing function of the amplitude of the expected crash, which itself grows with the bubble mispricing: hence, the larger the bubble price, the larger its subsequent growth rate. This positive feedback of price on return is the archetype of super-exponential price dynamics, which has been previously proposed as a general definition of bubbles. Our bubble model also allows for a sequence of small jumps or long-term corrections. We use the RE condition to estimate the realtime crash probability dynamically through an accelerating probability function depending on the increasing expected return. After showing how to estimate the model parameters, we examine the optimal investment problem in the context of the bubble model by obtaining an analytic expression for maximizing the expected log of wealth (Kelly criterion) for the risky asset and a risk-free asset. We also obtain a closed-form approximation for the optimal investment. We demonstrate, on seven historical crashes, the promising outperformance of the method compared to a 60/40 portfolio, the classic Kelly allocation, and the risky asset, and how it mitigates jumps, both positive and negative.

Suggested Citation

  • Jerome L Kreuser & Didier Sornette, 2017. "Super-Exponential RE Bubble Model with Efficient Crashes," Swiss Finance Institute Research Paper Series 17-33, Swiss Finance Institute.
    • Handle: RePEc:chf:rpseri:rp1733
    as

    Download full text from publisher

    File URL: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3064668
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Refet S. Gürkaynak, 2008. "Econometric Tests Of Asset Price Bubbles: Taking Stock," Journal of Economic Surveys, Wiley Blackwell, vol. 22(1), pages 166-186, February.
    2. Matthias LEISS & Heinrich H. NAX & Didier SORNETTE, 2014. "Super-Exponential Growth Expectations and the Global Financial Crisis," Swiss Finance Institute Research Paper Series 14-52, Swiss Finance Institute, revised Sep 2015.
    3. Peter C. B. Phillips & Shu-Ping Shi & Jun Yu, 2011. "Testing for Multiple Bubbles," Working Papers CoFie-03-2011, Singapore Management University, Sim Kee Boon Institute for Financial Economics.
    4. Didier Sornette & Ryan Woodard, & Wanfeng Yan & Wei-Xing Zhou, "undated". "Clarifications to Questions and Criticisms on the Johansen-Ledoit-Sornette bubble Model," Working Papers ETH-RC-11-004, ETH Zurich, Chair of Systems Design.
    5. Edwin J. Elton & Martin J. Gruber, 1974. "Note--On the Maximization of the Geometric Mean with Lognormal Return Distribution," Management Science, INFORMS, vol. 21(4), pages 483-488, December.
    6. Peter C. B. Phillips & Yangru Wu & Jun Yu, 2011. "EXPLOSIVE BEHAVIOR IN THE 1990s NASDAQ: WHEN DID EXUBERANCE ESCALATE ASSET VALUES?," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(1), pages 201-226, February.
    7. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold, 2007. "Roughing It Up: Including Jump Components in the Measurement, Modeling, and Forecasting of Return Volatility," The Review of Economics and Statistics, MIT Press, vol. 89(4), pages 701-720, November.
    8. D. Sornette, 2003. "Critical Market Crashes," Papers cond-mat/0301543, arXiv.org.
    9. Leiss, Matthias & Nax, Heinrich H. & Sornette, Didier, 2015. "Super-exponential growth expectations and the global financial crisis," LSE Research Online Documents on Economics 65434, London School of Economics and Political Science, LSE Library.
    10. repec:zbw:bofism:2012_047 is not listed on IDEAS
    11. Didier Sornette & Ryan Woodard & Maxim Fedorovsky & Stefan Reimann & Hilary Woodard & Wei-Xing Zhou, 2010. "The Financial Bubble Experiment: Advanced Diagnostics and Forecasts of Bubble Terminations Volume II-Master Document," Papers 1005.5675, arXiv.org, revised Nov 2010.
    12. George A. Akerlof, 2009. "How Human Psychology Drives the Economy and Why It Matters," American Journal of Agricultural Economics, Agricultural and Applied Economics Association, vol. 91(5), pages 1175-1175.
    13. Yacine Aït-Sahalia & Jean Jacod, 2012. "Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data," Journal of Economic Literature, American Economic Association, vol. 50(4), pages 1007-1050, December.
    14. Lux, Thomas & Sornette, Didier, 2002. "On Rational Bubbles and Fat Tails," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 34(3), pages 589-610, August.
    15. Peter C. B. Phillips & Jun Yu, 2011. "Dating the timeline of financial bubbles during the subprime crisis," Quantitative Economics, Econometric Society, vol. 2(3), pages 455-491, November.
    16. Xin Huang & George Tauchen, 2005. "The Relative Contribution of Jumps to Total Price Variance," Journal of Financial Econometrics, Oxford University Press, vol. 3(4), pages 456-499.
    17. Grzegorz Andruszkiewicz & Mark H. A. Davis & S'ebastien Lleo, 2014. "Risk-sensitive investment in a finite-factor model," Papers 1407.5278, arXiv.org, revised Jan 2016.
    18. Hüsler, A. & Sornette, D. & Hommes, C.H., 2013. "Super-exponential bubbles in lab experiments: Evidence for anchoring over-optimistic expectations on price," Journal of Economic Behavior & Organization, Elsevier, vol. 92(C), pages 304-316.
    19. Mark H A Davis & Sébastien Lleo, 2014. "Risk-Sensitive Investment Management," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 9026, August.
    20. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007, January-A.
    21. Sornette, Didier & Woodard, Ryan & Yan, Wanfeng & Zhou, Wei-Xing, 2013. "Clarifications to questions and criticisms on the Johansen–Ledoit–Sornette financial bubble model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4417-4428.
    22. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
    23. Bali, Turan G. & Zhou, Hao, 2016. "Risk, Uncertainty, and Expected Returns," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 51(3), pages 707-735, June.
    24. Evans, George W, 1991. "Pitfalls in Testing for Explosive Bubbles in Asset Prices," American Economic Review, American Economic Association, vol. 81(4), pages 922-930, September.
    25. Mark H. A. Davis & Sébastien Lleo, 2014. "Asset and Liability Management," World Scientific Book Chapters, in: RISK-SENSITIVE INVESTMENT MANAGEMENT, chapter 4, pages 57-87, World Scientific Publishing Co. Pte. Ltd..
    26. Didier SORNETTE, 2009. "Dragon-Kings, Black Swans and the Prediction of Crises," Swiss Finance Institute Research Paper Series 09-36, Swiss Finance Institute.
    27. Evanoff, Douglas D. & Kaufman, George G. & Malliaris, A. G. (ed.), 2012. "New Perspectives on Asset Price Bubbles," OUP Catalogue, Oxford University Press, number 9780199844401.
    28. Y. Malevergne & D. Sornette, 2001. "Multi-dimensional rational bubbles and fat tails," Quantitative Finance, Taylor & Francis Journals, vol. 1(5), pages 533-541.
    29. Wanfeng Yan & Reda Rebib & Ryan Woodard & Didier Sornette, "undated". "Detection of Crashes and Rebounds in Major Equity Markets," Working Papers ETH-RC-11-001, ETH Zurich, Chair of Systems Design.
    30. Leiss, Matthias & Nax, Heinrich H. & Sornette, Didier, 2015. "Super-exponential growth expectations and the global financial crisis," Journal of Economic Dynamics and Control, Elsevier, vol. 55(C), pages 1-13.
    31. Sornette, Didier & Zhou, Wei-Xing, 2006. "Predictability of large future changes in major financial indices," International Journal of Forecasting, Elsevier, vol. 22(1), pages 153-168.
    32. D. Sornette, "undated". "Dragon-Kings, Black Swans and the Prediction of Crises," Working Papers CCSS-09-005, ETH Zurich, Chair of Systems Design.
    33. repec:zbw:bofrdp:2012_007 is not listed on IDEAS
    34. Didier SORNETTE, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based Models," Swiss Finance Institute Research Paper Series 14-25, Swiss Finance Institute.
    35. Wanfeng Yan & Ryan Woodard & Didier Sornette, 2014. "Inferring fundamental value and crash nonlinearity from bubble calibration," Quantitative Finance, Taylor & Francis Journals, vol. 14(7), pages 1273-1282, July.
    36. Te Bao & Cars Hommes & Tomasz Makarewicz, 2017. "Bubble Formation and (In)Efficient Markets in Learning‐to‐forecast and optimise Experiments," Economic Journal, Royal Economic Society, vol. 127(605), pages 581-609, October.
    37. Jiang, Zhi-Qiang & Zhou, Wei-Xing & Sornette, Didier & Woodard, Ryan & Bastiaensen, Ken & Cauwels, Peter, 2010. "Bubble diagnosis and prediction of the 2005-2007 and 2008-2009 Chinese stock market bubbles," Journal of Economic Behavior & Organization, Elsevier, vol. 74(3), pages 149-162, June.
    38. Didier Sornette & Guy Ouillon, "undated". "Dragon-kings: Mechanisms, statistical methods and empirical evidence," Working Papers ETH-RC-12-004, ETH Zurich, Chair of Systems Design.
    39. Merton, Robert C., 1980. "On estimating the expected return on the market : An exploratory investigation," Journal of Financial Economics, Elsevier, vol. 8(4), pages 323-361, December.
    40. Tauchen, George & Zhou, Hao, 2011. "Realized jumps on financial markets and predicting credit spreads," Journal of Econometrics, Elsevier, vol. 160(1), pages 102-118, January.
    41. Vogel, Harold L. & Werner, Richard A., 2015. "An analytical review of volatility metrics for bubbles and crashes," International Review of Financial Analysis, Elsevier, vol. 38(C), pages 15-28.
    42. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    43. Barndorff-Nielsen, Ole E. & Graversen, Svend Erik & Jacod, Jean & Shephard, Neil, 2006. "Limit Theorems For Bipower Variation In Financial Econometrics," Econometric Theory, Cambridge University Press, vol. 22(4), pages 677-719, August.
    44. Olivier J. Blanchard & Mark W. Watson, 1982. "Bubbles, Rational Expectations and Financial Markets," NBER Working Papers 0945, National Bureau of Economic Research, Inc.
    45. G. Demos & D. Sornette, 2017. "Birth or burst of financial bubbles: which one is easier to diagnose?," Quantitative Finance, Taylor & Francis Journals, vol. 17(5), pages 657-675, May.
    46. Harras, Georges & Sornette, Didier, 2011. "How to grow a bubble: A model of myopic adapting agents," Journal of Economic Behavior & Organization, Elsevier, vol. 80(1), pages 137-152.
    47. Francesco Audrino & Yujia Hu, 2016. "Volatility Forecasting: Downside Risk, Jumps and Leverage Effect," Econometrics, MDPI, vol. 4(1), pages 1-24, February.
    48. Evanoff, Douglas D. & Kaufman, George G. & Malliaris, A. G. (ed.), 2012. "New Perspectives on Asset Price Bubbles," OUP Catalogue, Oxford University Press, number 9780199844333.
    49. William T. Ziemba & Leonard C. MacLean, 2011. "Using the Kelly Criterion for Investing," International Series in Operations Research & Management Science, in: Marida Bertocchi & Giorgio Consigli & Michael A. H. Dempster (ed.), Stochastic Optimization Methods in Finance and Energy, edition 1, chapter 0, pages 3-20, Springer.
    50. Ren, Yu & Xiong, Cong & Yuan, Yufei, 2012. "House price bubbles in China," China Economic Review, Elsevier, vol. 23(4), pages 786-800.
    51. L. Lin & D. Sornette, 2013. "Diagnostics of rational expectation financial bubbles with stochastic mean-reverting termination times," The European Journal of Finance, Taylor & Francis Journals, vol. 19(5), pages 344-365, May.
    52. Abreu, Dilip & Brunnermeier, Markus K., 2002. "Synchronization risk and delayed arbitrage," Journal of Financial Economics, Elsevier, vol. 66(2-3), pages 341-360.
    53. Anders Johansen & Didier Sornette, 2010. "Shocks, Crashes and Bubbles in Financial Markets," Brussels Economic Review, ULB -- Universite Libre de Bruxelles, vol. 53(2), pages 201-253.
    54. Anders Johansen & Olivier Ledoit & Didier Sornette, 2000. "Crashes As Critical Points," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(02), pages 219-255.
    55. Wanfeng Yan & Ryan Woodard & Didier Sornette, 2010. "Diagnosis and Prediction of Tipping Points in Financial Markets: Crashes and Rebounds," Papers 1001.0265, arXiv.org, revised Feb 2010.
    56. Brunnermeier, Markus K. & Oehmke, Martin, 2013. "Bubbles, Financial Crises, and Systemic Risk," Handbook of the Economics of Finance, in: G.M. Constantinides & M. Harris & R. M. Stulz (ed.), Handbook of the Economics of Finance, volume 2, chapter 0, pages 1221-1288, Elsevier.
    57. Peter C. B. Phillips & Shuping Shi & Jun Yu, 2015. "Testing For Multiple Bubbles: Historical Episodes Of Exuberance And Collapse In The S&P 500," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 56, pages 1043-1078, November.
    58. Camerer, Colin, 1989. "Bubbles and Fads in Asset Prices," Journal of Economic Surveys, Wiley Blackwell, vol. 3(1), pages 3-41.
    59. Blanchard, Olivier Jean, 1979. "Speculative bubbles, crashes and rational expectations," Economics Letters, Elsevier, vol. 3(4), pages 387-389.
    60. Sornette, Didier & Cauwels, Peter, 2015. "Financial Bubbles: Mechanisms and Diagnostics," Review of Behavioral Economics, now publishers, vol. 2(3), pages 279-305, October.
    61. D. Sornette & J. V. Andersen, 2001. "A Nonlinear Super-Exponential Rational Model of Speculative Financial Bubbles," Papers cond-mat/0104341, arXiv.org, revised Apr 2002.
    62. Sornette, D., 2002. "“Slimming” of power-law tails by increasing market returns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 309(3), pages 403-418.
    63. Wei Xiong, 2013. "Bubbles, Crises, and Heterogeneous Beliefs," NBER Working Papers 18905, National Bureau of Economic Research, Inc.
    64. Eric Jacquier & Cédric Okou, 2014. "Disentangling Continuous Volatility from Jumps in Long-Run Risk–Return Relationships," Journal of Financial Econometrics, Oxford University Press, vol. 12(3), pages 544-583.
    65. Ait-Sahalia, Yacine, 2004. "Disentangling diffusion from jumps," Journal of Financial Economics, Elsevier, vol. 74(3), pages 487-528, December.
    66. Didier Sornette & Ryan Woodard & Maxim Fedorovsky & Stefan Reimann & Hilary Woodard & Wei-Xing Zhou, 2009. "The Financial Bubble Experiment: advanced diagnostics and forecasts of bubble terminations," Papers 0911.0454, arXiv.org, revised May 2010.
    67. Lin, L. & Ren, R.E. & Sornette, D., 2014. "The volatility-confined LPPL model: A consistent model of ‘explosive’ financial bubbles with mean-reverting residuals," International Review of Financial Analysis, Elsevier, vol. 33(C), pages 210-225.
    68. Robert Jarrow, 2016. "Bubbles And Multiple-Factor Asset Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-19, February.
    69. Lawrance L. Evans Jr., 2003. "Why the Bubble Burst," Books, Edward Elgar Publishing, number 2808.
    70. Lux, Thomas, 1995. "Herd Behaviour, Bubbles and Crashes," Economic Journal, Royal Economic Society, vol. 105(431), pages 881-896, July.
    71. D. Sornette & J. V. Andersen, 2002. "A Nonlinear Super-Exponential Rational Model Of Speculative Financial Bubbles," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 13(02), pages 171-187.
    72. Levy, H & Markowtiz, H M, 1979. "Approximating Expected Utility by a Function of Mean and Variance," American Economic Review, American Economic Association, vol. 69(3), pages 308-317, June.
    73. L. Lin & Ren R.E. & D. Sornette, "undated". "A Consistent Model of `Explosive' Financial Bubbles With Mean-Reversing Residuals," Working Papers CCSS-09-002, ETH Zurich, Chair of Systems Design.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ivasiuc Arina, 2023. "Herding Behavior in Frontier Nordic Countries," Studia Universitatis Babeș-Bolyai Oeconomica, Sciendo, vol. 68(1), pages 21-41, April.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Zhang, Qunzhi & Sornette, Didier & Balcilar, Mehmet & Gupta, Rangan & Ozdemir, Zeynel Abidin & Yetkiner, Hakan, 2016. "LPPLS bubble indicators over two centuries of the S&P 500 index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 458(C), pages 126-139.
    2. Li Lin & Didier Sornette, 2015. ""Speculative Influence Network" during financial bubbles: application to Chinese Stock Markets," Papers 1510.08162, arXiv.org.
    3. Damian Smug & Peter Ashwin & Didier Sornette, 2018. "Predicting financial market crashes using ghost singularities," PLOS ONE, Public Library of Science, vol. 13(3), pages 1-20, March.
    4. Li Lin & Didier Sornette, 2018. "“Speculative Influence Network” during financial bubbles: application to Chinese stock markets," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 13(2), pages 385-431, July.
    5. Jan-Christian Gerlach & Jerome Kreuser & Didier Sornette, 2020. "Awareness of crash risk improves Kelly strategies in simulated financial time series," Papers 2004.09368, arXiv.org.
    6. Riza Demirer & Guilherme Demos & Rangan Gupta & Didier Sornette, 2019. "On the predictability of stock market bubbles: evidence from LPPLS confidence multi-scale indicators," Quantitative Finance, Taylor & Francis Journals, vol. 19(5), pages 843-858, May.
    7. Lin, L. & Ren, R.E. & Sornette, D., 2014. "The volatility-confined LPPL model: A consistent model of ‘explosive’ financial bubbles with mean-reverting residuals," International Review of Financial Analysis, Elsevier, vol. 33(C), pages 210-225.
    8. Didier SORNETTE, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based Models," Swiss Finance Institute Research Paper Series 14-25, Swiss Finance Institute.
    9. Zhang, Yue-Jun & Yao, Ting, 2016. "Interpreting the movement of oil prices: Driven by fundamentals or bubbles?," Economic Modelling, Elsevier, vol. 55(C), pages 226-240.
    10. Sornette, Didier & Woodard, Ryan & Yan, Wanfeng & Zhou, Wei-Xing, 2013. "Clarifications to questions and criticisms on the Johansen–Ledoit–Sornette financial bubble model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(19), pages 4417-4428.
    11. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
    12. Leiss, Matthias & Nax, Heinrich H. & Sornette, Didier, 2015. "Super-exponential growth expectations and the global financial crisis," LSE Research Online Documents on Economics 65434, London School of Economics and Political Science, LSE Library.
    13. Wosnitza, Jan Henrik & Denz, Cornelia, 2013. "Liquidity crisis detection: An application of log-periodic power law structures to default prediction," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(17), pages 3666-3681.
    14. Wosnitza, Jan Henrik & Leker, Jens, 2014. "Can log-periodic power law structures arise from random fluctuations?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 401(C), pages 228-250.
    15. Ardila-Alvarez, Diego & Forro, Zalan & Sornette, Didier, 2021. "The acceleration effect and Gamma factor in asset pricing," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 569(C).
    16. Li Lin & Didier Sornette, 2023. "The inverse Cox-Ingersoll-Ross process for parsimonious financial price modeling," Papers 2302.11423, arXiv.org, revised Jun 2023.
    17. Leiss, Matthias & Nax, Heinrich H. & Sornette, Didier, 2015. "Super-exponential growth expectations and the global financial crisis," Journal of Economic Dynamics and Control, Elsevier, vol. 55(C), pages 1-13.
    18. Kaizoji, Taisei & Leiss, Matthias & Saichev, Alexander & Sornette, Didier, 2015. "Super-exponential endogenous bubbles in an equilibrium model of fundamentalist and chartist traders," Journal of Economic Behavior & Organization, Elsevier, vol. 112(C), pages 289-310.
    19. V. I. Yukalov & E. P. Yukalova & D. Sornette, 2015. "Dynamical system theory of periodically collapsing bubbles," Papers 1507.05311, arXiv.org.
    20. Shu, Min & Zhu, Wei, 2020. "Detection of Chinese stock market bubbles with LPPLS confidence indicator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 557(C).

    More about this item

    Keywords

    financial bubbles; efficient crashes; positive feedback; rational expectation; Kelly criterion; optimal investment;
    All these keywords.

    JEL classification:

    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods
    • G01 - Financial Economics - - General - - - Financial Crises
    • G17 - Financial Economics - - General Financial Markets - - - Financial Forecasting and Simulation

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:chf:rpseri:rp1733. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Ridima Mittal (email available below). General contact details of provider: https://edirc.repec.org/data/fameech.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.