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Published online by Cambridge University Press:  23 March 2020

Jen-Che Liao*
Fu Jen Catholic University
Wen-Jen Tsay
Academia Sinica
Address correspondence to Jen-Che Liao, Department of Economics, Fu Jen Catholic University, 510 Zhongzheng Road, Xinzhuang, New Taipei City 24205, Taiwan; e-mail:


This article proposes frequentist multiple-equation least-squares averaging approaches for multistep forecasting with vector autoregressive (VAR) models. The proposed VAR forecast averaging methods are based on the multivariate Mallows model averaging (MMMA) and multivariate leave-h-out cross-validation averaging (MCVAh) criteria (with h denoting the forecast horizon), which are valid for iterative and direct multistep forecast averaging, respectively. Under the framework of stationary VAR processes of infinite order, we provide theoretical justifications by establishing asymptotic unbiasedness and asymptotic optimality of the proposed forecast averaging approaches. Specifically, MMMA exhibits asymptotic optimality for one-step-ahead forecast averaging, whereas for direct multistep forecast averaging, the asymptotically optimal combination weights are determined separately for each forecast horizon based on the MCVAh procedure. To present our methodology, we investigate the finite-sample behavior of the proposed averaging procedures under model misspecification via simulation experiments.

© Cambridge University Press 2020

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This article was previously circulated under the title “Multivariate Least Squares Forecasting Averaging by Vector Autoregressive Models.” We thank the Editor Peter Phillips, the Co-Editor Robert Taylor, and two anonymous referees for their constructive comments on earlier versions of this article. We appreciate helpful comments and suggestions from Le-Yu Chen, Yi-Ting Chen, Graham Elliott, Bruce Hansen, Chu-An Liu, Chor-Yiu (CY) Sin, and participants of the 2016 Cross-Strait Dialogue III, the 2016 Taiwan Economics Research workshop, IAAE 2017, SETA 2019, CMES 2019, and econometrics seminars at several universities. We appreciate research support from Institute of Economics at Academia Sinica. We assume responsibility for any errors in the article.



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