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Extracting a Common Stochastic Trend:Theories with Some Applications

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This paper investigates the statistical properties of the Kalman filter for state space models including integrated time series. In particular, we derive the full asymptotics of maximum likelihood estimation for some prototypical class of such models, i.e., the models with a single latent common stochastic trend. Indeed, we establish the consistency and asymptotic mixed normality of the maximum likelihood estimator and show that the conventional method of infer- ence is valid for this class of models. The models considered explicitly in the paper comprise a special, yet useful, class of models that we may use to extract the common stochastic trend from multiple integrated time series. As we show in the paper, the models can be very useful to obtain indices that represent fluctuations of various markets or common latent factors that affect a set of economic and financial variables simultaneously. Moreover, our derivation of the asymptotics of this class makes it clear that the asymptotic Gaussianity and the validity of the conventional inference for the maximum likelihood procedure extends to a larger class of more general state space models involving integrated time series. Finally, we demonstrate the utility of the state space model by ex- tracting a common stochastic trend in three empirical analyses: interest rates, return volatility and trading volume, and Dow Jones stock prices.

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File URL: http://economics.missouri.edu/working-papers/2005/wp0507_miller.pdf
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Bibliographic Info

Paper provided by Department of Economics, University of Missouri in its series Working Papers with number 0507.

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Length: 38 pgs.
Date of creation: 15 Oct 2005
Date of revision: 18 Aug 2005
Handle: RePEc:umc:wpaper:0507

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Keywords: state space model; Kalman filter; common stochastic trend; maximum likelihood estimation; stock price index; interest rates; return volatility and trading volume.;

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  1. Park, Joon Y & Phillips, Peter C B, 2001. "Nonlinear Regressions with Integrated Time Series," Econometrica, Econometric Society, vol. 69(1), pages 117-61, January.
  2. Chang-Jin Kim & Charles R. Nelson, 1999. "State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262112388, December.
  3. Campbell, John Y & Shiller, Robert J, 1987. "Cointegration and Tests of Present Value Models," Journal of Political Economy, University of Chicago Press, vol. 95(5), pages 1062-88, October.
  4. Thomas J. Sargent, 1978. "A note on maximum likelihood estimation of the rational expectations model of the term structure," Staff Report 26, Federal Reserve Bank of Minneapolis.
  5. Hai, Weike & Mark, Nelson C & Wu, Yangru, 1997. "Understanding Spot and Forward Exchange Rate Regressions," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 12(6), pages 715-34, Nov.-Dec..
  6. Jeff Fleming & Chris Kirby & Barbara Ostdiek, 2006. "Stochastic Volatility, Trading Volume, and the Daily Flow of Information," The Journal of Business, University of Chicago Press, vol. 79(3), pages 1551-1590, May.
  7. Danny Quah, 1991. "The Relative Importance of Permanent and Transitory Components: Identi- fication and Some Theoretical Bounds," NBER Technical Working Papers 0106, National Bureau of Economic Research, Inc.
  8. Joon Y. Park & J. Isaac Miller, 2004. "Nonlinearity, Nonstationarity, and Thick Tails: How They Interact to Generate Persistency in Memory," Econometric Society 2004 North American Summer Meetings 597, Econometric Society.
  9. BAUWENS, Luc & DEPRINS, Dominique & VANDEUREN, Jean-Pierre, 1997. "Modelling interest rates with a cointegrated VAR-GARCH model," CORE Discussion Papers 1997080, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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  11. Tauchen, George E & Pitts, Mark, 1983. "The Price Variability-Volume Relationship on Speculative Markets," Econometrica, Econometric Society, vol. 51(2), pages 485-505, March.
  12. Foster, F Douglas & Viswanathan, S, 1995. "Can Speculative Trading Explain the Volume-Volatility Relation?," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(4), pages 379-96, October.
  13. Hafer, R. W. & Kutan, Ali M. & Su Zhou, 1997. "Linkage in EMS term structures: evidence from common trend and transitory components," Journal of International Money and Finance, Elsevier, vol. 16(4), pages 595-607, August.
  14. Gonzalo, Jesus & Granger, Clive W J, 1995. "Estimation of Common Long-Memory Components in Cointegrated Systems," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(1), pages 27-35, January.
  15. Engle, Robert F & Granger, Clive W J, 1987. "Co-integration and Error Correction: Representation, Estimation, and Testing," Econometrica, Econometric Society, vol. 55(2), pages 251-76, March.
  16. Modigliani, Franco & Shiller, Robert J, 1973. "Inflation, Rational Expectations and the Term Structure of Interest Rates," Economica, London School of Economics and Political Science, vol. 40(157), pages 12-43, February.
  17. Johansen, Soren, 1988. "Statistical analysis of cointegration vectors," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 231-254.
  18. Karpoff, Jonathan M., 1987. "The Relation between Price Changes and Trading Volume: A Survey," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(01), pages 109-126, March.
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Cited by:
  1. Mitra, Sinchan & Sinclair, Tara M., 2012. "Output Fluctuations In The G-7: An Unobserved Components Approach," Macroeconomic Dynamics, Cambridge University Press, vol. 16(03), pages 396-422, June.

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