The Annals of Statistics

Gaussian pseudo-maximum likelihood estimation of fractional time series models

Javier Hualde and Peter M. Robinson

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We consider the estimation of parametric fractional time series models in which not only is the memory parameter unknown, but one may not know whether it lies in the stationary/invertible region or the nonstationary or noninvertible regions. In these circumstances, a proof of consistency (which is a prerequisite for proving asymptotic normality) can be difficult owing to nonuniform convergence of the objective function over a large admissible parameter space. In particular, this is the case for the conditional sum of squares estimate, which can be expected to be asymptotically efficient under Gaussianity. Without the latter assumption, we establish consistency and asymptotic normality for this estimate in case of a quite general univariate model. For a multivariate model, we establish asymptotic normality of a one-step estimate based on an initial √n-consistent estimate.

Article information

Ann. Statist., Volume 39, Number 6 (2011), 3152-3181.

First available in Project Euclid: 5 March 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62F12: Asymptotic properties of estimators

Fractional processes nonstationarity noninvertibility Gaussian estimation consistency asymptotic normality multiple time series


Hualde, Javier; Robinson, Peter M. Gaussian pseudo-maximum likelihood estimation of fractional time series models. Ann. Statist. 39 (2011), no. 6, 3152--3181. doi:10.1214/11-AOS931.

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Supplemental materials

  • Supplementary material: Supplement to “Gaussian pseudo-maximum likelihood estimation of fractional time series models”. The supplementary material contains a Monte Carlo experiment of finite sample performance of the proposed procedure, an empirical application to U.S. income and consumption data, and the proofs of the lemmas given in Section 5 of the present paper.