The Annals of Statistics

Frequency domain minimum distance inference for possibly noninvertible and noncausal ARMA models

Carlos Velasco and Ignacio N. Lobato

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


This article introduces frequency domain minimum distance procedures for performing inference in general, possibly non causal and/or noninvertible, autoregressive moving average (ARMA) models. We use information from higher order moments to achieve identification on the location of the roots of the AR and MA polynomials for non-Gaussian time series. We propose a minimum distance estimator that optimally combines the information contained in second, third, and fourth moments. Contrary to existing estimators, the proposed one is consistent under general assumptions, and may improve on the efficiency of estimators based on only second order moments. Our procedures are also applicable for processes for which either the third or the fourth order spectral density is the zero function.

Article information

Ann. Statist., Volume 46, Number 2 (2018), 555-579.

Received: December 2015
Revised: October 2016
First available in Project Euclid: 3 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Higher-order moments higher-order spectra nonminimum phase Whittle estimate


Velasco, Carlos; Lobato, Ignacio N. Frequency domain minimum distance inference for possibly noninvertible and noncausal ARMA models. Ann. Statist. 46 (2018), no. 2, 555--579. doi:10.1214/17-AOS1560.

Export citation


  • Alekseev, V. G. (1993). Asymptotic properties of higher-order periodograms. Theory Probab. Appl. 40 409–419.
  • Alessi, L., Barigozzi, M. and Capasso, M. (2011). Non-fundamentalness in structural econometric models: A review. Int. Stat. Rev. 79 16–47.
  • Andrews, D. W. (1987). Asymptotic results for generalized Wald tests. Econometric Theory 3 348–358.
  • Andrews, B., Davis, R. A. and Breidt, F. J. (2007). Rank-based estimation for all-pass time series models. Ann. Statist. 35 844–869.
  • Anh, V. V., Leonenko, N. N. and Sakhno, L. M. (2007). Minimum contrast estimation of random processes based on information of second and third orders. J. Statist. Plann. Inference 137 1302–1331.
  • Breidt, F. J., Davis, R. A. and Trindade, A. A. (2001). Least absolute deviation estimation for all-pass time series models. Ann. Statist. 29 919–946.
  • Brillinger, D. R. (1975). Time Series: Data Analysis and Theory. Holden-Day, San Francisco, CA.
  • Brillinger, D. R. (1985). Fourier inference: Some methods for the analysis of array and nongaussian series data. Water Resour. Bull. 21 744–756.
  • Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd ed. Springer, New York.
  • Gospodinov, N. and Ng, S. (2015). Minimum distance estimation of possibly noninvertible moving average models. J. Bus. Econom. Statist. 33 403–417.
  • Hannan, E. J. (1970). Multiple Time Series. Wiley, New York.
  • Hansen, L. P. and Sargent, T. J. (1980). Formulating and estimating dynamic linear rational expectations models. J. Econom. Dynam. Control 2 7–46.
  • Hansen, L. P. and Sargent, T. J. (1991). Two difficulties in interpreting vector autoregressions. In Rational Expectations Econometrics 77–119. Westview Press, Boulder, CO.
  • Huang, J. and Pawitan, Y. (2000). Quasi-likelihood estimation of non-invertible moving average processes. Scand. J. Stat. 27 689–702.
  • Kumon, M. (1992). Identification of non-minimum phase transfer function using higher-order spectrum. Ann. Inst. Statist. Math. 44 239–260.
  • Lanne, M. and Saikkonen, P. (2011). Noncausal autoregressions for economic time series. J. Time Ser. Econom. 3 Art. 2, 32.
  • Leeper, E. M., Walker, T. B. and Yang, S.-C. S. (2013). Fiscal foresight and information flows. Econometrica 81 1115–1145.
  • Lii, K.-S. and Rosenblatt, M. (1992). An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes. J. Multivariate Anal. 43 272–299.
  • Mountford, A. and Uhlig, H. (2009). What are the effects of fiscal policy shocks? J. Appl. Econometrics 24 960–992.
  • Ramsey, J. B. and Montenegro, A. (1992). Identification and estimation of non-invertible non-Gaussian $\operatorname{MA}(q)$ processes. J. Econometrics 54 301–320.
  • Rao, C. R. and Mitra, S. K. (1972). Generalized inverse of a matrix and its applications. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 601–620. Univ. California Press, Berkeley, CA.
  • Rosenblatt, M. (1985). Stationary Sequences and Random Fields. Birkhäuser, Boston, MA.
  • Schneeweiss, H. (2014). The linear GMM model with singular covariance matrix due to the elimination of a nuisance parameter. Technical Report 165, Dept. Statistics, Univ. Munich.
  • Terdik, G. (1999). Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis: A Frequency Domain Approach. Lecture Notes in Statistics 142. Springer, New York.
  • Velasco, C. and Lobato, I. N. (2018). Supplement to “Frequency domain minimum distance inference for possibly noninvertible and noncausal ARMA models.” DOI:10.1214/17-AOS1560SUPP.
  • Whittle, P. (1953). The analysis of multiple stationary time series. J. R. Stat. Soc. Ser. B. Stat. Methodol. 15 125–139.

Supplemental materials

  • Technical Appendices to “Frequency domain minimum distance inference for possibly noninvertible and noncausal ARMA models”. This Supplementary Material contains three appendices with proofs of main results, technical lemmas and comparison with Whittle estimation.