Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-25T18:06:52.477Z Has data issue: false hasContentIssue false
  • English
  • Français

TRUNCATED SUM OF SQUARES ESTIMATION OF FRACTIONAL TIME SERIES MODELS WITH DETERMINISTIC TRENDS

Published online by Cambridge University Press:  01 July 2019

Javier Hualde  and
Morten Ørregaard Nielsen
Javier Hualde
Affiliation:
Universidad Pública de Navarra
Morten Ørregaard Nielsen*
Affiliation:
Queen’s University and CREATES
*
Address correspondence to Morten Ørregaard Nielsen, Department of Economics, Queen’s University, Dunning Hall 209, 94 University Avenue, Kingston, Ontario K7L 3N6, Canada; e-mail: mon@econ.queensu.ca.
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider truncated (or conditional) sum of squares estimation of a parametric model composed of a fractional time series and an additive generalized polynomial trend. Both the memory parameter, which characterizes the behavior of the stochastic component of the model, and the exponent parameter, which drives the shape of the deterministic component, are considered not only unknown real numbers but also lying in arbitrarily large (but finite) intervals. Thus, our model captures different forms of nonstationarity and noninvertibility. As in related settings, the proof of consistency (which is a prerequisite for proving asymptotic normality) is challenging due to nonuniform convergence of the objective function over a large admissible parameter space, but, in addition, our framework is substantially more involved due to the competition between stochastic and deterministic components. We establish consistency and asymptotic normality under quite general circumstances, finding that results differ crucially depending on the relative strength of the deterministic and stochastic components. Finite-sample properties are illustrated by means of a Monte Carlo experiment.

Type
MISCELLANEA
Information
Econometric Theory , Volume 36 , Issue 4 , August 2020 , pp. 751 - 772
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

We are grateful to the Co-Editor, Rob Taylor, three anonymous referees, Søren Johansen, Peter M. Robinson, and seminar participants at Universidad Carlos III, Universitat Pompeu Fabra, Universidad de Alicante, University of Nottingham, and at the 3rd CREATES Long Memory Symposium for useful comments. Javier Hualde’s research is supported by the Spanish Ministerio de Economía y Competitividad through project ECO2015-64330-P. Morten Ø. Nielsen’s research is supported by the Canada Research Chairs program and the Social Sciences and Humanities Research Council of Canada (SSHRC). Both authors are thankful to the Center for Research in Econometric Analysis of Time Series (CREATES, funded by the Danish National Research Foundation, DNRF78) for financial support.

References

REFERENCES

Abadir, K.M., Distaso, W., & Giraitis, L. (2007) Nonstationarity-extended local Whittle estimation. Journal of Econometrics 141, 1353–1384.CrossRefGoogle Scholar
Beran, J. (1995) Maximum likelihood estimation of the differencing parameter for invertible short and long memory autoregressive integrated moving average models. Journal of the Royal Statistical Society, Series B 57, 659–672.Google Scholar
Bloomfield, P. (1973) An exponential model for the spectrum of a scalar time series. Biometrika 60, 217–226.CrossRefGoogle Scholar
Box, G.E.P. & Jenkins, G.M. (1971) Time Series Analysis, Forecasting and Control. Holden-Day.Google Scholar
Dahlhaus, R. (1989) Efficient parameter estimation for self-similar processes. Annals of Statistics 17, 1749–1766.CrossRefGoogle Scholar
Fox, R. & Taqqu, M.S. (1986) Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series. Annals of Statistics 14, 517–532.CrossRefGoogle Scholar
Giraitis, L. & Surgailis, D. (1990) A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotical normality of Whittle’s estimate. Probability Theory and Related Fields 86, 87–104.CrossRefGoogle Scholar
Hannan, E.J. (1973) The asymptotic theory of linear time series models. Journal of Applied Probability 10, 130–145.CrossRefGoogle Scholar
Heyde, C.C. & Dai, W. (1996) On the robustness to small trends of estimation based on the smoothed periodogram. Journal of Time Series Analysis 17, 141–150.CrossRefGoogle Scholar
Hualde, J. & Nielsen, M.Ø. (2019) Online supplementary appendix to “Truncated sum of squares estimation of fractional time series models with deterministic trends”. Econometric Theory. Available at Cambridge Journals Online (journals.cambridge.org/ect).Google Scholar
Hualde, J. & Robinson, P.M. (2011) Gaussian pseudo-maximum likelihood estimation of fractional time series models. Annals of Statistics 39, 3152–3181.CrossRefGoogle Scholar
Iacone, F. (2010) Local Whittle estimation of the memory parameter in presence of deterministic components. Journal of Time Series Analysis 31, 37–49.CrossRefGoogle Scholar
Johansen, S. & Nielsen, M.Ø. (2010) Likelihood inference for a nonstationary fractional autoregressive model. Journal of Econometrics 158, 51–66.CrossRefGoogle Scholar
Johansen, S. & Nielsen, M.Ø. (2016) The role of initial values in conditional sum-of-squares estimation of nonstationary fractional time series models. Econometric Theory 32, 1095–1139.CrossRefGoogle Scholar
Künsch, H. (1986) Discrimination between monotonic trends and long-range dependence. Journal of Applied Probability 23, 1025–1030.CrossRefGoogle Scholar
Li, W.K. & McLeod, A.I. (1986) Fractional time series modelling. Biometrika 73, 217–221.CrossRefGoogle Scholar
Nielsen, M.Ø. (2015) Asymptotics for the conditional-sum-of-squares estimator in multivariate fractional time series models. Journal of Time Series Analysis 36, 154–188.CrossRefGoogle Scholar
Phillips, P.C.B. (2007) Regression with slowly varying regressors and nonlinear trends. Econometric Theory 23, 557–614.CrossRefGoogle Scholar
Robinson, P.M. (1994) Efficient tests of nonstationary hypotheses. Journal of the American Statistical Association 89, 1420–1437.CrossRefGoogle Scholar
Robinson, P.M. (2005) Efficiency improvements in inference on stationary and nonstationary fractional time series. Annals of Statistics 33, 1800–1842.CrossRefGoogle Scholar
Robinson, P.M. (2012) Inference on power law spatial trends. Bernoulli 18, 644–677.CrossRefGoogle Scholar
Robinson, P.M. & Iacone, F. (2005) Cointegration in fractional systems with deterministic trends. Journal of Econometrics 129, 263–298.CrossRefGoogle Scholar
Robinson, P.M. & Marinucci, D. (2000) The averaged periodogram for nonstationary vector time series. Statistical Inference for Stochastic Processes 3, 149–160.CrossRefGoogle Scholar
Velasco, C. & Robinson, P.M. (2000) Whittle pseudo-maximum likelihood estimation for nonstationary time series. Journal of the American Statistical Association 95, 1229–1243.CrossRefGoogle Scholar
White, H. & Granger, C.W.J. (2011) Consideration of trends in time series. Journal of Time Series Econometrics 3(1), Article 2.CrossRefGoogle Scholar
Wu, C.-F. (1981) Asymptotic theory of nonlinear least squares estimation. Annals of Statistics 9, 501–513.CrossRefGoogle Scholar
Supplementary material: PDF

Hualde and Nielsen supplementary material

Online Appendix

Download Hualde and Nielsen supplementary material(PDF)
PDF 471.3 KB