Exploring All Var Orderings for Calculating Spillovers? Yes, We Can!-a Note on Diebold and Yilmaz (2009)

Abstract

Diebold and Yilmaz (Economic Journal 2009; 119; 158-171) introduce the spillover index to measure linkages between international financial markets. As their index depends on the ordering of the variables in the underlying VAR model, they check robustness by computing the index for a small number of randomly chosen permutations, stating that it was impossible to explore the huge number of renumerations. Building on a new divide-and-conquer strategy, we provide an algorithm for swiftly calculating the spillover index's maximum and minimum over all renumerations. Using this new algorithm, we find that the true range of the spillover index can be up to three times as large as estimated by Diebold and Yilmaz. EXPLORING ALL ORDERINGS FOR SPILLOVERS 173 Building on a divide-and-conquer strategy, we provide an algorithm which allows timely calculation of the range of the spillover index even for large models. Applying it to the volatility data of Diebold and Yilmaz (2009), we show that the above-mentioned estimation of the range by randomly choosing permutations seriously underestimates the impact of the VAR ordering on the spillover index, with the estimated range sometimes comprising only one third of the true range. NOTATION AND METHODOLOGYLet us start by replicating, for the reader's convenience, the notation and methodology introduced in Diebold and Yilmaz (2009). To this end, we start with an N -dimensional VAR(p) model, Y t D (6) 4 To save space, we will restrict our attention to the maximum in equation (4), the modifications for computing the minimum being obvious. Analogously, it is also possible to compute the average of the spillover index over all variable reorderings. 5 The use of the more general formula equation (5) will become clear later. 6 The proposition's proof can be found in the Appendix. 7 The structural equivalence of the optimization problems is only true because we generalized equation (4) to equation (5). 8 The implemented algorithm uses N 1 D b N 2 c; other choices of N 1 will of course deliver the same results, but typically need more computation time.

International spillovers of policy uncertainty

“…We use the methodology for construction of spillover measures suggested by Diebold and Yilmaz (2009), with an algorithm created by Klößner and Wagner (2014) to calculate robust spillover measures.…”

“…Different orderings may thus result in significantly different spillover estimates, as shown by Klößner and Wagner (2014). We therefore apply their proposed algorithm to conveniently calculate robust spillover measures by averaging the results over all possible permutations of the system.…”

Causal ordering and inference on acyclic networks

“…Then, they suggest considering all possible permutations of the elements in Y t and conduct inferences under probability distributions on the model space, that is, distributions over all possible permutations. The idea of considering multiple permutations is also popular in the applied macroeconomics literature (Diebold and Yilmaz 2009;Klößner and Wagner 2014), particularly as a robustness check for recursive ordering suggested by theory. However, this approach can be computationally very expensive, particularly when there are a large number of observation units.…”

Bootstrap multi-step forecasts of non-Gaussian VAR models

“…Furthermore, joint multivariate forecasts are also important when forecasting future values of one variable conditional on particular values of other variables in the system; see Baumeister and Kilian (2012), Doan, Litterman, and Sims (1984), and Waggoner and Zha (1999). In order to define spillover measures, Diebold and Yilmaz (2009) also consider multivariate multi-period-ahead forecasts; see also Klobner and Wagner (2014). On the other hand, the focus of the forecasting literature is moving from point forecasts to density forecasts that incorporate the uncertainty about the future evolution of the variables of interest; see Bache, Jore, Mitchell, andVahey (2011), Clark (2011), Diebold, Hanh, and Tay (1999), and Jore, Mitchell, and Vahey (2010), among others.…”