# The role of initial values in nonstationary fractional time series models

## Author Info

• Søren Johansen

() (University of Copenhagen and CREATES)

• Morten Ørregaard Nielsen

() (Queen?s University and CREATES)

## Abstract

We consider the nonstationary fractional model $\Delta^{d}X_{t}=\varepsilon _{t}$ with $\varepsilon_{t}$ i.i.d.$(0,\sigma^{2})$ and $d>1/2$. We derive an analytical expression for the main term of the asymptotic bias of the maximum likelihood estimator of $d$ conditional on initial values, and we discuss the role of the initial values for the bias. The results are partially extended to other fractional models, and three different applications of the theoretical results are given.

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File URL: ftp://ftp.econ.au.dk/creates/rp/12/rp12_47.pdf

## Bibliographic Info

Paper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2012-47.

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Length: 29
Date of revision:
Handle: RePEc:aah:create:2012-47

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Web page: http://www.econ.au.dk/afn/

## Related research

Keywords: Asymptotic expansion; bias; conditional inference; fractional integration; initial values; likelihood inference.;

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Find related papers by JEL classification:
• C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models

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## References

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1. Søren Johansen & Morten Ørregaard Nielsen, 2012. "Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model," Econometrica, Econometric Society, vol. 80(6), pages 2667-2732, November.
2. Dolado, Juan José & Gonzalo, Jesús & Mayoral, Laura, . "A fractional Dickey-Fuller test for unit roots," Open Access publications from Universidad Carlos III de Madrid info:hdl:10016/770, Universidad Carlos III de Madrid.
3. Morten Ã˜rregaard Nielsen, 2011. "Asymptotics for the conditional-sum-of-squares estimator in fractional time series models," Working Papers 1259, Queen's University, Department of Economics.
4. Tschernig, Rolf & Weber, Enzo & Weigand, Roland, 2010. "Long-run Identification in a Fractionally Integrated System," University of Regensburg Working Papers in Business, Economics and Management Information Systems 447, University of Regensburg, Department of Economics.
5. Donald W.K. Andrews & Offer Lieberman, 2002. "Higher-order Improvements of the Parametric Bootstrap for Long-memory Gaussian Processes," Cowles Foundation Discussion Papers 1378, Cowles Foundation for Research in Economics, Yale University.
6. Søren Johansen & Morten Ørregaard Nielsen, 2010. "Likelihood inference for a nonstationary fractional autoregressive model," Working Papers 1172, Queen's University, Department of Economics.
7. David Byers & James Davidson & David Peel, 1997. "Modelling Political Popularity: an Analysis of Long-range Dependence in Opinion Poll Series," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 160(3), pages 471-490.
8. Eduardo Rossi & Paolo Santucci de Magistris, 2013. "A No‐Arbitrage Fractional Cointegration Model for Futures and Spot Daily Ranges," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 33(1), pages 77-102, 01.
9. Johansen, SØren, 2008. "A Representation Theory For A Class Of Vector Autoregressive Models For Fractional Processes," Econometric Theory, Cambridge University Press, vol. 24(03), pages 651-676, June.
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