IDEAS home Printed from
   My bibliography  Save this paper

Optimal test for Markov switching


  • Marine Carrasco
  • Liang Hu


We propose a new test for the stability of parameters in a Markov switching model where regime changes are driven by an unobservable Markov chain. Testing in this context is more challenging than testing in structural change and threshold models because, besides the presence of nuisance parameters that are not identified under the null hypothesis, there is the additional difficulty due to the singularity of the information matrix under the null. In particular, a test for Markov switching does not have power against n^-1/2 alternatives, but only against n^-1/4 alternatives. Therefore we derive the behavior of the likelihood under local alternatives in n^-1/4 by using an expansion to the fourth order. We show that the densities under alternatives of order n^-1/4 are contiguous to the density under the null. We derive a class of information matrix-type tests and show that they are equivalent to the likelihood ratio test. Hence, our tests are asymptotically optimal. Besides their optimality properties, these tests are more general than the competing tests proposed by Garcia (1998) and Hansen (1992). Indeed, the underlying Markov chain driving the regime changes may have a finite or continuous state space, as long as it is exogenous. It is not restricted to linear models either. Therefore, our technique applies for instance to testing stability in random coefficient GARCH models. We use this test to investigate the presence of rational collapsing bubbles in stock markets. There is bubble if the stock price is disconnected from the market fundamental value. We regress the stock price on dividends and use the residual as proxy for the bubble size. Using US data, we find that the residuals are stationary, which could be hastily interpreted as evidence against the presence of bubbles. However, our Markov switching test strongly rejects the linearity, suggesting that at least two regimes should be used to fit the data. Estimating a two-state Markov switching model (Hamilton, 1989) reveals that one regime has a unit root, while the other is mean reverting, which is consistent with periodically collapsing bubbles.

Suggested Citation

  • Marine Carrasco & Liang Hu, 2004. "Optimal test for Markov switching," Econometric Society 2004 North American Summer Meetings 396, Econometric Society.
  • Handle: RePEc:ecm:nasm04:396

    Download full text from publisher

    File URL:
    File Function: main text
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Yingyi Qian, 1994. "Incentives and Loss of Control in an Optimal Hierarchy," Review of Economic Studies, Oxford University Press, vol. 61(3), pages 527-544.
    2. Hart, Oliver & Moore, John, 1990. "Property Rights and the Nature of the Firm," Journal of Political Economy, University of Chicago Press, vol. 98(6), pages 1119-1158, December.
    3. Williamson, Oliver E, 1979. "Transaction-Cost Economics: The Governance of Contractural Relations," Journal of Law and Economics, University of Chicago Press, vol. 22(2), pages 233-261, October.
    4. Aghion, Philippe & Tirole, Jean, 1997. "Formal and Real Authority in Organizations," Journal of Political Economy, University of Chicago Press, vol. 105(1), pages 1-29, February.
    5. Raghuram G. Rajan & Luigi Zingales, 2001. "The Firm as a Dedicated Hierarchy: A Theory of the Origins and Growth of Firms," The Quarterly Journal of Economics, Oxford University Press, vol. 116(3), pages 805-851.
    6. Michael Keren & David Levhari, 1979. "The Optimum Span of Control in a Pure Hierarchy," Management Science, INFORMS, vol. 25(11), pages 1162-1172, November.
    7. Grossman, Sanford J & Hart, Oliver D, 1986. "The Costs and Benefits of Ownership: A Theory of Vertical and Lateral Integration," Journal of Political Economy, University of Chicago Press, vol. 94(4), pages 691-719, August.
    8. Gilles, Robert P & Owen, Guillermo & van den Brink, Rene, 1992. "Games with Permission Structures: The Conjunctive Approach," International Journal of Game Theory, Springer;Game Theory Society, vol. 20(3), pages 277-293.
    9. Roth, Alvin, 2012. "The Shapley Value as a von Neumann-Morgenstern Utility," Economic Policy, Russian Presidential Academy of National Economy and Public Administration, vol. 6, pages 1-9.
    10. Gilles, R.P. & Owen, G., 1999. "Cooperative Games and Disjunctive Permission Structures," Discussion Paper 1999-20, Tilburg University, Center for Economic Research.
    11. Eric Maskin & Jean Tirole, 1999. "Unforeseen Contingencies and Incomplete Contracts," Review of Economic Studies, Oxford University Press, vol. 66(1), pages 83-114.
    12. Derks, Jean J M & Gilles, Robert P, 1995. "Hierarchical Organization Structures and Constraints on Coalition Formation," International Journal of Game Theory, Springer;Game Theory Society, vol. 24(2), pages 147-163.
    13. Gul, Faruk, 1989. "Bargaining Foundations of Shapley Value," Econometrica, Econometric Society, vol. 57(1), pages 81-95, January.
    14. Perez-Castrillo, David & Wettstein, David, 2001. "Bidding for the Surplus : A Non-cooperative Approach to the Shapley Value," Journal of Economic Theory, Elsevier, vol. 100(2), pages 274-294, October.
    15. Raghuram G. Rajan & Luigi Zingales, 1998. "Power in a Theory of the Firm," The Quarterly Journal of Economics, Oxford University Press, vol. 113(2), pages 387-432.
    16. Ichiishi,Tatsuro, 1993. "The Cooperative Nature of the Firm," Cambridge Books, Cambridge University Press, number 9780521414449, March.
    17. Oliver E. Williamson, 1967. "Hierarchical Control and Optimum Firm Size," Journal of Political Economy, University of Chicago Press, vol. 75, pages 123-123.
    18. George Baker & Robert Gibbons & Kevin J. Murphy, 2002. "Relational Contracts and the Theory of the Firm," The Quarterly Journal of Economics, Oxford University Press, vol. 117(1), pages 39-84.
    19. Wouter Dessein, 2002. "Authority and Communication in Organizations," Review of Economic Studies, Oxford University Press, vol. 69(4), pages 811-838.
    20. van den Brink, Rene & Gilles, Robert P., 1996. "Axiomatizations of the Conjunctive Permission Value for Games with Permission Structures," Games and Economic Behavior, Elsevier, vol. 12(1), pages 113-126, January.
    21. Grossman, Sanford J & Hart, Oliver D, 1983. "An Analysis of the Principal-Agent Problem," Econometrica, Econometric Society, vol. 51(1), pages 7-45, January.
    22. Klein, Benjamin & Crawford, Robert G & Alchian, Armen A, 1978. "Vertical Integration, Appropriable Rents, and the Competitive Contracting Process," Journal of Law and Economics, University of Chicago Press, vol. 21(2), pages 297-326, October.
    23. Roth, Alvin E., 1977. "Utility functions for simple games," Journal of Economic Theory, Elsevier, vol. 16(2), pages 481-489, December.
    24. Kessler, Anke S., 2000. "On Monitoring and Collusion in Hierarchies," Journal of Economic Theory, Elsevier, vol. 91(2), pages 280-291, April.
    25. Luis Garicano, 2000. "Hierarchies and the Organization of Knowledge in Production," Journal of Political Economy, University of Chicago Press, vol. 108(5), pages 874-904, October.
    26. Michael Keren & David Levhari, 1983. "The Internal Organization of the Firm and the Shape of Average Costs," Bell Journal of Economics, The RAND Corporation, vol. 14(2), pages 474-486, Autumn.
    27. Prescott, Edward Simpson & Townsend, Robert M., 2002. "Collective Organizations versus Relative Performance Contracts: Inequality, Risk Sharing, and Moral Hazard," Journal of Economic Theory, Elsevier, vol. 103(2), pages 282-310, April.
    Full references (including those not matched with items on IDEAS)


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

    Cited by:

    1. Agnello, Luca & Castro, Vítor & Sousa, Ricardo M., 2012. "How does fiscal policy react to wealth composition and asset prices?," Journal of Macroeconomics, Elsevier, vol. 34(3), pages 874-890.
    2. Jorge Andrés Tamayo Castaño, 2012. "Asimetrías en la demanda por trabajo en Colombia: el papel del ciclo económico," Borradores de Economia 689, Banco de la Republica de Colombia.
    3. Youngki Shin, 2009. "Misspecified Markov Switching Model," Economics Bulletin, AccessEcon, vol. 29(2), pages 957-963.
    4. Benoit Bellone, 2005. "Classical Estimation of Multivariate Markov-Switching Models using MSVARlib," Econometrics 0508017, EconWPA.
    5. Kahn, James A. & Rich, Robert W., 2007. "Tracking the new economy: Using growth theory to detect changes in trend productivity," Journal of Monetary Economics, Elsevier, vol. 54(6), pages 1670-1701, September.
    6. Balcılar, Mehmet & Demirer, Rıza & Hammoudeh, Shawkat, 2015. "Regional and global spillovers and diversification opportunities in the GCC equity sectors," Emerging Markets Review, Elsevier, vol. 24(C), pages 160-187.
    7. James D. Hamilton, 2005. "What's real about the business cycle?," Review, Federal Reserve Bank of St. Louis, issue Jul, pages 435-452.
    8. Alexander, Carol & Kaeck, Andreas, 2008. "Regime dependent determinants of credit default swap spreads," Journal of Banking & Finance, Elsevier, vol. 32(6), pages 1008-1021, June.
    9. Yilmazkuday, Hakan & Akay, Koray, 2008. "An analysis of regime shifts in the Turkish economy," Economic Modelling, Elsevier, vol. 25(5), pages 885-898, September.
    10. Brevik, Frode & d’Addona, Stefano, 2011. "Information Quality and Stock Returns Revisited," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 45(06), pages 1419-1446, January.
    11. ZHENG, Tingguo & WANG, Xia & GUO, Huiming, 2012. "Estimating forward-looking rules for China's Monetary Policy: A regime-switching perspective," China Economic Review, Elsevier, vol. 23(1), pages 47-59.
    12. Lanouar Charfeddine & Dominique Guegan, 2008. "Is it possible to discriminate between different switching regressions models? An empirical investigation," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00368358, HAL.
    13. Pierre Guérin & Massimiliano Marcellino, 2013. "Markov-Switching MIDAS Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 31(1), pages 45-56, January.
    14. Janczura, Joanna & Weron, Rafal, 2010. "Goodness-of-fit testing for regime-switching models," MPRA Paper 22871, University Library of Munich, Germany.
    15. Carol Alexander & Andreas Kaeck, 2006. "Regimes in CDS Spreads: A Markov Switching Model of iTraxx Europe Indices," ICMA Centre Discussion Papers in Finance icma-dp2006-08, Henley Business School, Reading University.
    16. Luca Agnello & Gilles Dufrénot & Ricardo M. Sousa, 2012. "Adjusting the U.S. Fiscal Policy for Asset Prices: Evidence from a TVP-MS Framework," NIPE Working Papers 20/2012, NIPE - Universidade do Minho.
    17. Spyros Andreopoulos, 2006. "The real interest rate, the real oil price, and US unemployment revisited," Bristol Economics Discussion Papers 06/592, Department of Economics, University of Bristol, UK.
    18. Silvestro Di Sanzo, 2009. "Testing for linearity in Markov switching models: a bootstrap approach," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 18(2), pages 153-168, July.
    19. Gross, Marco & Binder, Michael, 2013. "Regime-switching global vector autoregressive models," Working Paper Series 1569, European Central Bank.

    More about this item


    Asymptotics; speculative bubbles; Markov switching; optimal test; random coefficient models.;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:ecm:nasm04:396. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F. Baum). General contact details of provider: .

    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.

    We have no references for this item. You can help adding them by using 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.