Identification and Estimation of Structural VARMA Models Using Higher Order Dynamics

Abstract

We use information from higher order moments to achieve identification of non-Gaussian structural vector autoregressive moving average (SVARMA) models, possibly nonfundamental or noncausal, through a frequency domain criterion based on higher order spectral densities. This allows us to identify the location of the roots of the determinantal lag matrix polynomials and to identify the rotation of the model errors leading to the structural shocks up to sign and permutation. We describe sufficient conditions for global and local parameter identification that rely on simple rank assumptions on the linear dynamics and on finite order serial and component independence conditions for the non-Gaussian structural innovations. We generalize previous univariate analysis to develop asymptotically normal and efficient estimates exploiting second and higher order cumulant dynamics given a particular structural shocks ordering without assumptions on causality or invertibility. Finite sample properties of estimates are explored with real and simulated data.

Keywords:

• Causality
• Higher-order spectra
• Independence of components
• Invertibility
• Minimum distance
• Rank condition

Supplementary Materials

The Supplemental Materials contain an Appendix with proofs of results, additional discussion and further numerical results, and Matlab code for the Monte Carlo simulations and empirical section.

Acknowledgments

Thanks to an Associate Editor and two referees, and to J. C. Escanciano, F. J. Hidalgo, I. N. Lobato, G. Sucarrat and seminar participants at Nuffield College, LSE, QMUL, BI Norwegian BS and at Workshop on Time Series Econometrics 2018, Waseda International Symposium 2018, CFE-CM Statistics Conference 2019, EC² Conference on Identification in Macroeconomics 2019 for helpful discussions and comments.

Notes

1 Notice that in Theorem 1 of Lippi and Reichlin (1994) it is fixed that ${\mathbf{K}}_0={\mathbf{I}}_d,$ but ${\mathbf{K}}_0$ may need to be different from identity.

Additional information

Funding

Financial support from the Ministerio Economía y Competitividad (Spain) grants ECO2017-86009-P and PID2020-114664GB-I00 is gratefully acknowledged.