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Understanding the functional central limit theorems with some applications to unit root testing with structural change

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  • Juan Carlos Aquino

    (Banco Central de la Reserva del Perú)

  • Gabriel Rodríguez

    ()
    (Pontificia Universidad Católica del Perú)

Abstract

The application of different unit root statistics is by now a standard practice in empirical work. Even when it is a practical issue, these statistics have complex nonstandard distributions depending on functionals of certain stochastic processes, and their derivations represent a barrier even for many theoretical econometricians. These derivations are based on rigorous and fundamental statistical tools which are not (very) well known by standard econometricians. This paper aims to fill this gap by explaining in a simple way one of these fundamental tools: namely, the Functional Central Limit Theorem. To this end, this paper analyzes the foundations and applicability of two versions of the Functional Central Limit Theorem within the framework of a unit root with a structural break. Initial attention is focused on the probabilistic structure of the time series to be considered. Thereafter, attention is focused on the asymptotic theory for nonstationary time series proposed by Phillips (1987a), which is applied by Perron (1989) to study the effects of an (assumed) exogenous structural break on the power of the augmented Dickey-Fuller test and by Zivot and Andrews (1992) to criticize the exogeneity assumption and propose a method for estimating an endogenous breakpoint. A systematic method for dealing with efficiency issues is introduced by Perron and Rodriguez (2003), which extends the Generalized Least Squares detrending approach due to Elliot et al. (1996). An empirical application is provided.

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Bibliographic Info

Article provided by Departamento de Economía - Pontificia Universidad Católica del Perú in its journal Revista Economia.

Volume (Year): 36 (2013)
Issue (Month): 71 ()
Pages: 107-149

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Handle: RePEc:pcp:pucrev:y:2013:i:71:p:107-149

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Keywords: Unit root testing; structural break; functional central limit theorem; Ornstein-Uhlenbeck process.;

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References

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  1. Zivot, Eric & Andrews, Donald W K, 1992. "Further Evidence on the Great Crash, the Oil-Price Shock, and the Unit-Root Hypothesis," Journal of Business & Economic Statistics, American Statistical Association, American Statistical Association, vol. 10(3), pages 251-70, July.
  2. Bhargava, Alok, 1986. "On the Theory of Testing for Unit Roots in Observed Time Series," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 53(3), pages 369-84, July.
  3. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780198774037, October.
  4. Anindya Banerjee & Robin L. Lumsdaine & James H. Stock, 1990. "Recursive and Sequential Tests of the Unit Root and Trend Break Hypothesis: Theory and International Evidence," NBER Working Papers 3510, National Bureau of Economic Research, Inc.
  5. Sargan, John Denis & Bhargava, Alok, 1983. "Testing Residuals from Least Squares Regression for Being Generated by the Gaussian Random Walk," Econometrica, Econometric Society, Econometric Society, vol. 51(1), pages 153-74, January.
  6. Peter C.B. Phillips, 1986. "Regression Theory for Near-Integrated Time Series," Cowles Foundation Discussion Papers, Cowles Foundation for Research in Economics, Yale University 781R, Cowles Foundation for Research in Economics, Yale University, revised Jan 1987.
  7. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, Elsevier, vol. 10(2), pages 139-162.
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Cited by:
  1. Gabriel Rodríguez & Alfredo Vargas, 2012. "Impacto de expectativas políticas en los retornos del Índice General de la Bolsa de Valores de Lima," Revista Economía, Departamento de Economía - Pontificia Universidad Católica del Perú, Departamento de Economía - Pontificia Universidad Católica del Perú, vol. 35(70), pages 190-223.
  2. Aquino, Juan Carlos & Espino, Freddy, 2013. "Terms of Trade and Current Account Fluctuations: a Vector Autoregression Approach," Working Papers, Banco Central de Reserva del Perú 2013-008, Banco Central de Reserva del Perú.

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