## The Annals of Statistics

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

#### Abstract

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

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

Dates
Revised: October 2016
First available in Project Euclid: 3 April 2018

https://projecteuclid.org/euclid.aos/1522742429

Digital Object Identifier
doi:10.1214/17-AOS1560

Mathematical Reviews number (MathSciNet)
MR3782377

Zentralblatt MATH identifier
06870272

#### Citation

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. https://projecteuclid.org/euclid.aos/1522742429

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#### 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.