Abstract: In this paper, a new time-varying optimal copula model is proposed to precisely identify the optimal dependence structure of bivariate time series at every time point. In this model, half-rotated copulas, i.e. C_R1G(u,v;θ)=v-C_OG(1-u,v;θ) and C_R2G(u,v;θ)=u-C_OG(1-u,v;θ), are constructed to capture the asymmetric negative dependence, especially for the negative extreme dependence, i.e. lower-upper tail τ^LU(α)=Pr(X< F_X^-1(α)|Y >F_Y^-1(1-α)) and upper-lower tail dependence τ^LU(α)=Pr(X >F_X^-1(1-α)|Y< F_Y^-1(α)) for a small α, e.g. 0.05. Meanwhile, the distribution-free test for independence is introduced to verify the dependent relationship and reduce the computation time. At last, the time-varying optimal copula model is employed to analyse the dynamic dependence between energy markets, i.e. crude oil and natural gas markets, and exchange market. It is found that for Brent-USDX pair the dependence is significantly negative, the proportion of half-rotated Gumbel copula is larger than that of the original Gumbel or rotated Gumbel, the lower-upper or upper-lower tail dependence is obviously larger than the upper-upper or lower-lower tail dependence especially in the crisis period, and above empirical results for GAS-USDX pair are similar but not very remarkable. However, the dependence between Brent and GAS is positive and the upper-upper or lower-lower tail dependence is larger than lower-upper or upper-lower tail dependence. Meanwhile, the types of dependence structureacross markets vary over time and that emergencies are usually the major cause of sudden changes in the dependence structure. Resulas also show that the TVOC model captures the dynamic characteristics of the direction and intensity of the dependence as well as the dynamic characteristics of the types of dependence structure. In particular, the TVOC model canbe employed to predict the copula-dependence structure in a newway, which provides an analytical tool for market investors and risk managers to adjust their portfolio strategy, hedge the investment risk and guard against risk spillover and even a financial contagion.

Key words: tail dependence, co-movement across markets, time-varying optimal copula