Generalized Covariance Estimator

ABSTRACT

We consider a class of semi-parametric dynamic models with iid errors, including the nonlinear mixed causal-noncausal Vector Autoregressive (VAR), Double-Autoregressive (DAR) and stochastic volatility models. To estimate the parameters characterizing the (nonlinear) serial dependence, we introduce a generic Generalized Covariance (GCov) estimator, which minimizes a residual-based multivariate portmanteau statistic. In comparison to the standard methods of moments, the GCov estimator has an interpretable objective function, circumvents the inversion of high-dimensional matrices, and achieves semi-parametric efficiency in one step. We derive the asymptotic properties of the GCov estimator and show its semi-parametric efficiency. We also prove that the associated residual-based portmanteau statistic is asymptotically chi-square distributed. The finite sample performance of the GCov estimator is illustrated in a simulation study. The estimator is then applied to a dynamic model of commodity futures.

KEYWORDS:

• Commodities
• Continuously updating GMM
• Canonical correlation
• Generalized covariance estimator
• Mixed causal-noncausal process
• Portmanteau Statistic
• Semiparametric estimator

Supplementary Materials

Supplementary Appendix contains additional theoretical results for the proofs of the Propositions, as well as a simulation study of finite sample properties of the estimator and diagnostic analysis for the empirical application.

Acknowledgments

The authors thank the Associate Editor and two anonymous referees as well as G. Imbens, D. Matteson, A. Hecq, and the participants of CMStatistics 2021 for their helpful comments.

Disclosure Statement

The authors report there are no competing interests to declare.

Notes

1 In this respect, a weaker assumption of martingale difference sequence (mds) used, for example, in Velasco (2022) would be inadequate, as it is not invariant to nonlinear transformations.

2 This implies that $g({\tilde{Y}}_t;\theta )$ is strictly stationary for any transformation g and value of parameter θ, and so are the true errors $u_t=g({\tilde{Y}}_t;{\theta }_0)$.

Additional information

Funding

The first author acknowledges financial support of the ACPR Chair “Regulation and Systemic Risk” and the ERC DYSMOIA. The second author acknowledges financial support of the Natural Sciences and Engineering Council of Canada (NSERC).