IDEAS home Printed from https://ideas.repec.org/p/uww/wpaper/14-01.html
   My bibliography  Save this paper

Temporal Aggregation of Random Walk Processes and Implications for Asset Prices

Author

Listed:
  • Yamin Ahmad

    (Department of Economics, University of Wisconsin - Whitewater)

  • Ivan Paya

    (Department of Economics, Lancaster University Management School)

Abstract

This paper examines the impact of time averaging and interval sampling data assuming that the data generating process for a given series follows a random walk with uncorrelated increments. We provide expressions for the corresponding variances, and covariances, for both the levels and differences of the aggregated series, demonstrating how the degree of temporal aggregation impacts these particular properties. Moreover, we analytically derive any differences that arise between the aggregated series and its disaggregated counterpart, and show that they can be decomposed into a distortionary and small sample effect. We also provide exact expressions for the variance and sharpe ratios, and correlation coefficients for any level of aggregation. We discuss our results in the context of asset prices, which have utilized these extensively.

Suggested Citation

  • Yamin Ahmad & Ivan Paya, 2014. "Temporal Aggregation of Random Walk Processes and Implications for Asset Prices," Working Papers 14-01, UW-Whitewater, Department of Economics.
  • Handle: RePEc:uww:wpaper:14-01
    as

    Download full text from publisher

    File URL: http://www.uww.edu/documents/colleges/cobe/economics/wpapers/14-01_Ahmad_Paya.pdf
    File Function: Full text
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Chow, K. Victor & Denning, Karen C., 1993. "A simple multiple variance ratio test," Journal of Econometrics, Elsevier, vol. 58(3), pages 385-401, August.
    2. Andrew W. Lo, A. Craig MacKinlay, 1988. "Stock Market Prices do not Follow Random Walks: Evidence from a Simple Specification Test," Review of Financial Studies, Society for Financial Studies, vol. 1(1), pages 41-66.
    3. Andrea Silvestrini & David Veredas, 2008. "Temporal Aggregation Of Univariate And Multivariate Time Series Models: A Survey," Journal of Economic Surveys, Wiley Blackwell, vol. 22(3), pages 458-497, July.
    4. Huizinga, John, 1987. "An empirical investigation of the long-run behavior of real exchange rates," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 27(1), pages 149-214, January.
    5. Ahmad, Yamin & Craighead, William D., 2011. "Temporal aggregation and purchasing power parity persistence," Journal of International Money and Finance, Elsevier, vol. 30(5), pages 817-830, September.
    6. Christiano, Lawrence J & Eichenbaum, Martin & Marshall, David, 1991. "The Permanent Income Hypothesis Revisited," Econometrica, Econometric Society, vol. 59(2), pages 397-423, March.
    7. Taylor, Alan M, 2001. "Potential Pitfalls for the Purchasing-Power-Parity Puzzle? Sampling and Specification Biases in Mean-Reversion Tests of the Law of One Price," Econometrica, Econometric Society, vol. 69(2), pages 473-498, March.
    8. Ivan Paya & David A. Peel, 2006. "Temporal aggregation of an ESTAR process: some implications for purchasing power parity adjustment," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 21(5), pages 655-668, July.
    9. John Y. Campbell & N. Gregory Mankiw, 1987. "Are Output Fluctuations Transitory?," The Quarterly Journal of Economics, Oxford University Press, vol. 102(4), pages 857-880.
    10. Liu, Christina Y & He, Jia, 1991. "A Variance-Ratio Test of Random Walks in Foreign Exchange Rates," Journal of Finance, American Finance Association, vol. 46(2), pages 773-785, June.
    11. Campbell, John Y. & Mankiw, N. Gregory, 1989. "International evidence on the persistence of economic fluctuations," Journal of Monetary Economics, Elsevier, vol. 23(2), pages 319-333, March.
    12. Lo, Andrew W. & MacKinlay, A. Craig, 1989. "The size and power of the variance ratio test in finite samples : A Monte Carlo investigation," Journal of Econometrics, Elsevier, vol. 40(2), pages 203-238, February.
    13. Rossana, Robert J & Seater, John J, 1995. "Temporal Aggregation and Economic Time Series," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(4), pages 441-451, October.
    14. Haug, Alfred A., 1991. "The random walk hypothesis of consumption and time aggregation," Journal of Macroeconomics, Elsevier, vol. 13(4), pages 691-700.
    15. Grilli, Vittorio & Kaminsky, Graciela, 1991. "Nominal exchange rate regimes and the real exchange rate : Evidence from the United States and Great Britain, 1885-1986," Journal of Monetary Economics, Elsevier, vol. 27(2), pages 191-212, April.
    16. Cochrane, John H, 1988. "How Big Is the Random Walk in GNP?," Journal of Political Economy, University of Chicago Press, vol. 96(5), pages 893-920, October.
    17. Choi, Chi-Young & Mark, Nelson C. & Sul, Donggyu, 2006. "Unbiased Estimation of the Half-Life to PPP Convergence in Panel Data," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 38(4), pages 921-938, June.
    18. Fama, Eugene F, 1970. "Efficient Capital Markets: A Review of Theory and Empirical Work," Journal of Finance, American Finance Association, vol. 25(2), pages 383-417, May.
    19. Yilmaz, Kamil, 2003. "Martingale Property of Exchange Rates and Central Bank Interventions," Journal of Business & Economic Statistics, American Statistical Association, vol. 21(3), pages 383-395, July.
    20. Belaire-Franch, Jorge & Opong, Kwaku K., 2005. "Some evidence of random walk behavior of Euro exchange rates using ranks and signs," Journal of Banking & Finance, Elsevier, vol. 29(7), pages 1631-1643, July.
    21. Brewer, K. R. W., 1973. "Some consequences of temporal aggregation and systematic sampling for ARMA and ARMAX models," Journal of Econometrics, Elsevier, vol. 1(2), pages 133-154, June.
    22. Campos, Julia & Ericsson, Neil R. & Hendry, David F., 1990. "An analogue model of phase-averaging procedures," Journal of Econometrics, Elsevier, vol. 43(3), pages 275-292, March.
    23. Fong, Wai Mun & Koh, Seng Kee & Ouliaris, Sam, 1997. "Joint Variance-Ratio Tests of the Martingale Hypothesis for Exchange Rates," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 51-59, January.
    24. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    25. Cecchetti, Stephen G & Lam, Pok-sang, 1994. "Variance-Ratio Tests: Small-Sample Properties with an Application to International Output Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 177-186, April.
    26. Rossana, Robert J & Seater, John J, 1992. "Aggregation, Unit Roots and the Time Series Structure on Manufacturing Real Wages," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(1), pages 159-179, February.
    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. Ahmad, Yamin & Craighead, William D., 2011. "Temporal aggregation and purchasing power parity persistence," Journal of International Money and Finance, Elsevier, vol. 30(5), pages 817-830, September.
    2. Jesús Otero & Theodore Panagiotidis & Georgios Papapanagiotou, 2021. "Testing for exuberance in house prices using data sampled at different frequencies," Working Paper series 21-13, Rimini Centre for Economic Analysis.

    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. Ahmad Yamin S & Paya Ivan, 2020. "Temporal aggregation of random walk processes and implications for economic analysis," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 24(2), pages 1-20, April.
    2. Amelie Charles & Olivier Darne, 2009. "Testing for Random Walk Behavior in Euro Exchange Rates," Economie Internationale, CEPII research center, issue 119, pages 25-45.
    3. Amélie Charles & Olivier Darné, 2009. "Variance‐Ratio Tests Of Random Walk: An Overview," Journal of Economic Surveys, Wiley Blackwell, vol. 23(3), pages 503-527, July.
    4. Yang, Yan-Hong & Shao, Ying-Hui & Shao, Hao-Lin & Stanley, H. Eugene, 2019. "Revisiting the weak-form efficiency of the EUR/CHF exchange rate market: Evidence from episodes of different Swiss franc regimes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 523(C), pages 734-746.
    5. Eduardo José Araújo Lima & Benjamin Miranda Tabak, 2008. "Exchange Rate Dynamics and the Relationship between the Random Walk Hypothesis and Official Interventions," Working Papers Series 173, Central Bank of Brazil, Research Department.
    6. Charles, Amélie & Darné, Olivier & Kim, Jae H., 2012. "Exchange-rate return predictability and the adaptive markets hypothesis: Evidence from major foreign exchange rates," Journal of International Money and Finance, Elsevier, vol. 31(6), pages 1607-1626.
    7. Benjamin Miranda Tabak, 2003. "The random walk hypothesis and the behaviour of foreign capital portfolio flows: the Brazilian stock market case," Applied Financial Economics, Taylor & Francis Journals, vol. 13(5), pages 369-378.
    8. Kim, Jae H. & Shamsuddin, Abul, 2008. "Are Asian stock markets efficient? Evidence from new multiple variance ratio tests," Journal of Empirical Finance, Elsevier, vol. 15(3), pages 518-532, June.
    9. Charles, Amélie & Darné, Olivier & Fouilloux, Jessica, 2011. "Testing the martingale difference hypothesis in CO2 emission allowances," Economic Modelling, Elsevier, vol. 28(1), pages 27-35.
    10. Shyh-wei Chen, 2009. "Random walks in asian foreign exchange markets:evidence from new multiple variance ratio tests," Economics Bulletin, AccessEcon, vol. 29(2), pages 1296-1307.
    11. Luger, Richard, 2003. "Exact non-parametric tests for a random walk with unknown drift under conditional heteroscedasticity," Journal of Econometrics, Elsevier, vol. 115(2), pages 259-276, August.
    12. Fahad Almudhaf, 2014. "Testing for random walk behaviour in CIVETS exchange rates," Applied Economics Letters, Taylor & Francis Journals, vol. 21(1), pages 60-63, January.
    13. Peter C. B. Phillips & Sainan Jin, 2014. "Testing the Martingale Hypothesis," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(4), pages 537-554, October.
    14. Patrick A. Groenendijk & André Lucas & Casper G. de Vries, 1998. "A Hybrid Joint Moment Ratio Test for Financial Time Series," Tinbergen Institute Discussion Papers 98-104/2, Tinbergen Institute.
    15. Aye, Goodness C. & Gil-Alana, Luis A. & Gupta, Rangan & Wohar, Mark E., 2017. "The efficiency of the art market: Evidence from variance ratio tests, linear and nonlinear fractional integration approaches," International Review of Economics & Finance, Elsevier, vol. 51(C), pages 283-294.
    16. Ajayi, Richard A. & Karemera, David, 1996. "A variance ratio test of random walks in exchange rates: Evidence from Pacific Basin economies," Pacific-Basin Finance Journal, Elsevier, vol. 4(1), pages 77-91, May.
    17. Lee, Chun I. & Pan, Ming-Shiun & Liu, Y. Angela, 2001. "On market efficiency of Asian foreign exchange rates: evidence from a joint variance ratio test and technical trading rules," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 11(2), pages 199-214, June.
    18. Jui-Cheng Hung & Yen-Hsien Lee & Tung-Yueh Pai, 2009. "Examining market efficiency for large- and small-capitalization of TOPIX and FTSE stock indices," Applied Financial Economics, Taylor & Francis Journals, vol. 19(9), pages 735-744.
    19. Belaire-Franch, Jorge & Opong, Kwaku K., 2005. "Some evidence of random walk behavior of Euro exchange rates using ranks and signs," Journal of Banking & Finance, Elsevier, vol. 29(7), pages 1631-1643, July.
    20. Bernard Njindan Iyke, 2019. "A Test Of The Efficiency Of The Foreign Exchange Market In Indonesia," Bulletin of Monetary Economics and Banking, Bank Indonesia, vol. 0(12th BMEB), pages 1-26, January.

    More about this item

    Keywords

    Temporal Aggregation; Random Walk; Variance Ratio; Sharpe Ratio;
    All these keywords.

    JEL classification:

    • F47 - International Economics - - Macroeconomic Aspects of International Trade and Finance - - - Forecasting and Simulation: Models and Applications
    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

    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:uww:wpaper:14-01. 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: Yamin Ahmad (email available below). General contact details of provider: https://edirc.repec.org/data/eduwwus.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.