# Connecting VIX and Stock Index ETF with VAR and Diagonal BEKK

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Department of Finance, National Chung Hsing University, Taichung City 402, Taiwan

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Department of Applied Economics, National Chung Hsing University, Hsinchu City 300, Taiwan

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Department of Finance, Asia University, Wufeng District, Taichung City 41354, Taiwan

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Econometric Institute, Erasmus School of Economics, Erasmus University Rotterdam, 3000 Rotterdam, The Netherlands

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Department of Economic Analysis and ICAE, Complutense University of Madrid, 28040 Madrid, Spain

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Institute of Advanced Sciences, Yokohama National University, Yokohama 240-8501, Japan

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Author to whom correspondence should be addressed.

Received: 17 August 2018 / Revised: 23 September 2018 / Accepted: 28 September 2018 / Published: 29 September 2018

(This article belongs to the Collection Feature Papers of JRFM)

As stock market indexes are not tradeable, the importance and trading volume of Exchange-Traded Funds (ETFs) cannot be understated. ETFs track and attempt to replicate the performance of a specific index. Numerous studies have demonstrated a strong relationship between the S&P500 Composite Index and the Volatility Index (VIX), but few empirical studies have focused on the relationship between VIX and ETF returns. The purpose of the paper is to investigate whether VIX returns affect ETF returns by using vector autoregressive (VAR) models to determine whether daily VIX returns with different moving average processes affect ETF returns. The ARCH-LM test shows conditional heteroskedasticity in the estimation of ETF returns, so that the Diagonal BEKK (named after Baba, Engle, Kraft and Kroner) model is used to accommodate multivariate conditional heteroskedasticity in the VAR estimates of ETF returns. Daily data on ETF returns that follow different stock indexes in the USA and Europe are used in the empirical analysis, which is presented for the full data set, as well as for the three sub-periods Before, During, and After the Global Financial Crisis. The estimates show that daily VIX returns have: (1) significant negative effects on European ETF returns in the short run; (2) stronger significant effects on single-market ETF returns than on European ETF returns; and (3) lower impacts on the European ETF returns than on S&P500 returns. For the European markets, the estimates of the mean equations tend to differ between the whole sample period and the sub-periods, but the estimates of the matrices A and B in the Diagonal BEKK model are quite similar for the whole sample period and at least two of the three sub-periods. For the US Markets, the estimates of the mean equations also tend to differ between the whole sample period and the sub-periods, but the estimates of the matrices A and B in the Diagonal BEKK model are very similar for the whole sample period and the three sub-periods.