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`Weak` trends for inference and forecasting in finite samples

  • Guillaume Chevillon

This paper studies the small sample properties of processes which exhibit both a stochastic and a deterministic trend. Whereas for estimation, inference and forecasting purposes the latter asymptotically dominates the former, it is not so when only a finite number of observations is available and large non-linearities in the parameters of the process result. To analyze this dependence, we resort to local-asymptotics and present the concept of a `weak` trend whose coefficient is of order O(T-1/2), so that the deterministic trend is O(T1/2) and the process Op(T1/2). In this framework, parameter estimates, unit-root test statistics and forecast errors are functions of `drifting` Ornstein-Uhlenbeck processes. We derive a comparison of direct and iterated multi-step estimation and forecasting of a - potentially misspecified - random walk with drift, and show that we explain well the non-linearities exhibited in finite samples. Another main benefit of direct multi-step estimation stems from some different behaviors of the `multi-step` unit-root and slope tests under the weak and strong (constant coefficient) trend frameworks which could lead to testing which framework is more relevant. A Monte Carlo analysis validates the local-asymptotics approximation to the distributions of finite sample biases and test statistics.

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File URL: http://www.economics.ox.ac.uk/materials/working_papers/paper210.pdf
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Paper provided by University of Oxford, Department of Economics in its series Economics Series Working Papers with number 210.

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Date of creation: 01 Dec 2004
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Handle: RePEc:oxf:wpaper:210
Contact details of provider: Postal: Manor Rd. Building, Oxford, OX1 3UQ
Web page: http://www.economics.ox.ac.uk/
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  1. Guillaume Chevillon & David F. Hendry, 2004. "Non-Parametric Direct Multi-step Estimation for Forecasting Economic Processes," Economics Papers 2004-W12, Economics Group, Nuffield College, University of Oxford.
  2. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  3. Francis X. Diebold & Abdelhak S. Senhadji, 1996. "Deterministic vs. Stochastic Trend in U.S. GNP, Yet Again," NBER Working Papers 5481, National Bureau of Economic Research, Inc.
  4. Banerjee, Anindya & Hendry, David F & Mizon, Grayham E, 1996. "The Econometric Analysis of Economic Policy," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 58(4), pages 573-600, November.
  5. Phillips, Peter C.B., 2005. "Challenges of trending time series econometrics," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 68(5), pages 401-416.
  6. Kemp, Gordon C.R., 1999. "The Behavior Of Forecast Errors From A Nearly Integrated Ar(1) Model As Both Sample Size And Forecast Horizon Become Large," Econometric Theory, Cambridge University Press, vol. 15(02), pages 238-256, April.
  7. Peter C. B. Phillips, 1998. "New Tools for Understanding Spurious Regressions," Econometrica, Econometric Society, vol. 66(6), pages 1299-1326, November.
  8. Douglas Staiger & James H. Stock, 1997. "Instrumental Variables Regression with Weak Instruments," Econometrica, Econometric Society, vol. 65(3), pages 557-586, May.
  9. Peter C.B. Phillips, 1986. "Regression Theory for Near-Integrated Time Series," Cowles Foundation Discussion Papers 781R, Cowles Foundation for Research in Economics, Yale University, revised Jan 1987.
  10. Sampson, Michael, 1991. "The Effect of Parameter Uncertainty on Forecast Variances and Confidence Intervals for Unit Root and Trend Stationary Time-Series Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 6(1), pages 67-76, Jan.-Marc.
  11. Clements, M.P. & Hendry, D.P., 1998. "Forecasting with Difference-Stationary and Trend-Stationary Models," The Warwick Economics Research Paper Series (TWERPS) 516, University of Warwick, Department of Economics.
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