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Determination of the Number of Common Stochastic Trends Under Conditional Heteroskedasticity/Determinación del número de tendencias estocásticas comunes bajo heteroscedasticidad condicional


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    (Department of Statistical Sciences, University of Bologna. Italia.)


    (Department of Economics, University of Copenhagen. Suecia.)


    (School of Economics, University of Nottingham. Gran Bretaña.)


business cycle frequencies strongly rely on the correct detection of the number of common stochastic trends (co-integration). Standard techniques for the determination of the number of common trends, such as the well-known sequential procedure proposed in Johansen (1996), are based on the assumption that shocks are homoskedastic. This contrasts with empirical evidence which documents that many of the key macro-economic and financial variables are driven by heteroskedastic shocks. In a recent paper, Cavaliere et al., (2010, Econometric Theory, forthcoming) demonstrate that Johansen's (LR) trace statistic for co-integration rank and both its i.i.d. and wild bootstrap analogues are asymptotically valid in non-stationary systems driven by heteroskedastic (martingale difference) innovations, but that the wild bootstrap performs substantially better than the other two tests in finite samples. In this paper we analyse the behaviour of sequential procedures to determine the number of common stochastic trends present based on these tests. Numerical evidence suggests that the procedure based on the wild bootstrap tests performs best in small samples under a variety of heteroskedastic innovation processes. Tanto las descomposiciones en componentes permanentes-transitorias de las series de tiempo como el análisis de las propiedades como tales de las variables económicas en las frecuencias del ciclo económico (business cycle) dependen fuertemente de la detección correcta del número de tendencias estocásticas comunes (cointegración). Las técnicas estándar para la determinación del número de tendencias comunes, como, por ejemplo, el conocido procedimiento secuencial propuesto en Johansen (1996), se basan en la hipótesis de que los shocks son homoscedásticos. Esto contradice la evidencia empírica que demuestra que muchas de las variables financieras y macroeconómicas más importantes se mueven por shocks heteroscedásticos. En un artículo reciente, Cavaliere y otros autores (2010, Econometric Theory, de próxima aparición) demuestran que el estadístico LR de la traza para el rango de la co-integración y sus análogos (tanto los i.i.d. como los “wild” bootstrap) son válidos asintóticamente en sistemas no estacionarios dirigidos por innovaciones heteroscedásticas (diferencia de martingalas) y que, además, “wild bootstrap” funciona sustancialmente mejor que los otros dos contrastes en muestras finitas. En este artículo, basándonos en esta prueba, analizaremos el comportamiento de procedimientos secuenciales para determinar, sobre la base de esos test, el número de tendencias estocásticas comunes presentes. La evidencia numérica sugiere que el procedimiento basado en los test “wild bootstrap” funciona mejor para pequeñas muestras y bajo una variedad de procesos de innovaciones heteroscedásticas.

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Article provided by Estudios de Economía Aplicada in its journal Estudios de Economía Aplicada.

Volume (Year): 28 (2010)
Issue (Month): (Diciembre)
Pages: 519-552

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Handle: RePEc:lrk:eeaart:28_3_2

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Keywords: Co-integration; maximum eigenvalue rank tests; conditional heteroskedasticity; i.i.d. bootstrap; wild bootstrap. ; Co-integration; maximum eigenvalue rank tests; conditional heteroskedasticity; i.i.d. bootstrap; wild bootstrap..;

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  1. Gonçalves, Sílvia & KILIAN, Lutz, 2003. "Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form," Cahiers de recherche, Centre interuniversitaire de recherche en économie quantitative, CIREQ 01-2003, Centre interuniversitaire de recherche en économie quantitative, CIREQ.
  2. Mackinnon, J.G. & Haug, A.A. & Michelis, L., 1996. "Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration," G.R.E.Q.A.M., Universite Aix-Marseille III 96a09, Universite Aix-Marseille III.
  3. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2008. "Bootstrap Unit Root Tests For Time Series With Nonstationary Volatility," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 24(01), pages 43-71, February.
  4. Silvia Goncalves & Lutz Kilian, 2007. "Asymptotic and Bootstrap Inference for AR(∞) Processes with Conditional Heteroskedasticity," Econometric Reviews, Taylor & Francis Journals, Taylor & Francis Journals, vol. 26(6), pages 609-641.
  5. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, Econometric Society, vol. 64(2), pages 413-30, March.
  6. Hansen, Peter Reinhard, 2003. "Structural changes in the cointegrated vector autoregressive model," Journal of Econometrics, Elsevier, Elsevier, vol. 114(2), pages 261-295, June.
  7. Roy van der Weide, 2002. "GO-GARCH: a multivariate generalized orthogonal GARCH model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., John Wiley & Sons, Ltd., vol. 17(5), pages 549-564.
  8. Giese, Julia V., 2008. "Level, Slope, Curvature: Characterising the Yield Curve in a Cointegrated VAR Model," Economics Discussion Papers 2008-13, Kiel Institute for the World Economy.
  9. Giuseppe Cavaliere & A. M. Robert Taylor & Carsten Trenkler, 2010. "Bootstrap co-integration rank testing: the role of deterministic variables and initial values in the bootstrap recursion," Discussion Papers, University of Nottingham, Granger Centre for Time Series Econometrics 10/04, University of Nottingham, Granger Centre for Time Series Econometrics.
  10. Campbell, John Y & Shiller, Robert J, 1987. "Cointegration and Tests of Present Value Models," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 95(5), pages 1062-88, October.
  11. Giuseppe Cavaliere & A. M. Robert Taylor, 2008. "Time-Transformed Unit Root Tests for Models with Non-Stationary Volatility," Journal of Time Series Analysis, Wiley Blackwell, Wiley Blackwell, vol. 29(2), pages 300-330, 03.
  12. Nelson, Daniel B, 1991. "Conditional Heteroskedasticity in Asset Returns: A New Approach," Econometrica, Econometric Society, Econometric Society, vol. 59(2), pages 347-70, March.
  13. Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2010. "Cointegration Rank Testing Under Conditional Heteroskedasticity," Econometric Theory, Cambridge University Press, Cambridge University Press, vol. 26(06), pages 1719-1760, December.
  14. Cavaliere, Giuseppe & Taylor, A.M. Robert, 2007. "Testing for unit roots in time series models with non-stationary volatility," Journal of Econometrics, Elsevier, Elsevier, vol. 140(2), pages 919-947, October.
  15. Nielsen, Bent & Rahbek, Anders, 2000. " Similarity Issues in Cointegration Analysis," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, Department of Economics, University of Oxford, vol. 62(1), pages 5-22, February.
  16. Anders Rygh Swensen, 2006. "Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models -super-1," Econometrica, Econometric Society, Econometric Society, vol. 74(6), pages 1699-1714, November.
  17. Hall, Anthony D & Anderson, Heather M & Granger, Clive W J, 1992. "A Cointegration Analysis of Treasury Bill Yields," The Review of Economics and Statistics, MIT Press, vol. 74(1), pages 116-26, February.
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