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Efficient Likelihood Inference In Nonstationary Univariate Models

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  • Nielsen, Morten rregaard

Abstract

Recent literature shows that embedding fractionally integrated time series models with spectral poles at the long-run and/or seasonal frequencies in autoregressive frameworks leads to estimators and test statistics with non-standard limiting distributions that must be simulated on a case-by-case basis. However, we show that by embedding the models in a general I(d) framework the resulting estimators and tests regain all the desirable properties from standard statistical analysis. We derive the time domain maximum likelihood estimator and show that it is consistent, asymptotically normal, and under Gaussianity asymptotically efficient in the sense that it has asymptotic variance equal to the inverse of the Fisher information matrix. The three likelihood based test statistics (Wald, likelihood ratio, and Lagrange multiplier) are asymptotically equivalent and have the usual asymptotic chi-squared distribution and under the additional assumption of Gaussianity they are locally most powerful. In the special case where the dynamics of the model is characterized by a scalar parameter, we show that, in addition, the two-sided tests achieve the Gaussian power envelope of all invariant and unbiased tests, i.e. they are uniformly most powerful invariant unbiased. The finite sample properties of the tests are evaluated by Monte Carlo experiments. In contrast to what might be expected from the literature, the likelihood ratio test is found to outperform the Lagrange multiplier and Wald tests.

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Bibliographic Info

Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 20 (2004)
Issue (Month): 01 (February)
Pages: 116-146

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Handle: RePEc:cup:etheor:v:20:y:2004:i:01:p:116-146_20

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References

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  1. Herman J. Bierens, 2000. "Complex Unit Roots and Business Cycles: Are They Real?," Econometric Society World Congress 2000 Contributed Papers 0197, Econometric Society.
  2. Graham Elliott & Thomas J. Rothenberg & James H. Stock, 1992. "Efficient Tests for an Autoregressive Unit Root," NBER Technical Working Papers 0130, National Bureau of Economic Research, Inc.
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Cited by:
  1. Haldrup, Niels & Nielsen, Morten Orregaard, 2006. "A regime switching long memory model for electricity prices," Journal of Econometrics, Elsevier, Elsevier, vol. 135(1-2), pages 349-376.
  2. Luis F. Martins & Paulo M.M. Rodrigues, 2010. "Testing for Persistence Change in Fractionally Integrated Models: An Application to World Inflation Rates," Working Papers w201030, Banco de Portugal, Economics and Research Department.
  3. Morten Oerregaard Nielsen, . "Optimal Residual Based Tests for Fractional Cointegration and Exchange Rate Dynamics," Economics Working Papers, School of Economics and Management, University of Aarhus 2002-7, School of Economics and Management, University of Aarhus.
  4. Haldrup; Niels & Morten Oerregaard Nielsen, 2005. "Directional Congestion and Regime Switching in a Long Memory Model for Electricity Prices," Economics Working Papers, School of Economics and Management, University of Aarhus 2005-18, School of Economics and Management, University of Aarhus.
  5. Uwe Hassler & Paulo M.M. Rodrigues & Antonio Rubia, 2012. "Quantile regression for long memory testing: A case of realized volatility," Working Papers w201207, Banco de Portugal, Economics and Research Department.
  6. Giuseppe Cavaliere & Morten Ørregaard Nielsen & A.M. Robert Taylor, 2014. "Bootstrap Score Tests for Fractional Integration in Heteroskedastic ARFIMA Models, with an Application to Price Dynamics in Commodity Spot and Futures Markets," CREATES Research Papers 2014-22, School of Economics and Management, University of Aarhus.
  7. Paulo M.M. Rodrigues & Antonio Rubia & João Valle e Azevedo, 2009. "Finite Sample Performance of Frequency and Time Domain Tests for Seasonal Fractional Integration," Working Papers w200902, Banco de Portugal, Economics and Research Department.
  8. Søren Johansen & Morten Ørregaard Nielsen, 2007. "Likelihood inference for a nonstationary fractional autoregressive model," CREATES Research Papers 2007-33, School of Economics and Management, University of Aarhus.
  9. Morten Ørregaard Nielsen & Per Frederiksen, 2005. "Finite Sample Comparison of Parametric, Semiparametric, and Wavelet Estimators of Fractional Integration," Working Papers, Queen's University, Department of Economics 1189, Queen's University, Department of Economics.
  10. Shao, Xiaofeng & Wu, Wei Biao, 2007. "Local asymptotic powers of nonparametric and semiparametric tests for fractional integration," Stochastic Processes and their Applications, Elsevier, Elsevier, vol. 117(2), pages 251-261, February.
  11. Giuseppe Cavaliere & Morten Ørregaard Nielsen & A. M. Robert Taylor, 2014. "Quasi-Maximum Likelihood Estimation of Heteroskedastic Fractional Time Series Models," Working Papers, Queen's University, Department of Economics 1324, Queen's University, Department of Economics.
  12. Fleming, Jeff & Kirby, Chris, 2011. "Long memory in volatility and trading volume," Journal of Banking & Finance, Elsevier, vol. 35(7), pages 1714-1726, July.
  13. Per Frederiksen & Frank S. Nielsen, 2008. "Estimation of Dynamic Models with Nonparametric Simulated Maximum Likelihood," CREATES Research Papers 2008-59, School of Economics and Management, University of Aarhus.

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