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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
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    File URL: http://www.uww.edu/documents/colleges/cobe/economics/wpapers/14-01_Ahmad_Paya.pdf
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    References listed on IDEAS

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    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," The 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. 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.
    6. 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.
    7. 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.
    8. 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.
    9. Christiano, Lawrence J & Eichenbaum, Martin & Marshall, David, 1991. "The Permanent Income Hypothesis Revisited," Econometrica, Econometric Society, vol. 59(2), pages 397-423, March.
    10. 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.
    11. 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.
    12. John Y. Campbell & N. Gregory Mankiw, 1987. "Are Output Fluctuations Transitory?," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 102(4), pages 857-880.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. Haug, Alfred A., 1991. "The random walk hypothesis of consumption and time aggregation," Journal of Macroeconomics, Elsevier, vol. 13(4), pages 691-700.
    22. 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.
    23. 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.
    24. 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.
    25. 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.
    26. 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.
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    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.

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    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

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