In this paper, we study the hedging effectiveness of crude oil futures on the basis of the lower partial moments (LPMs). An improved kernel density estimation method is proposed to estimate the optimal hedge ratio. We investigate crude oil price hedging by contributing to the literature in the following twofold: First, unlike the existing studies which focus on univariate kernel density method, we use bivariate kernel density to calculate the estimated LPMs, wherein the two bandwidths of the bivariate kernel density are not limited to the same, which is our main innovation point. According to ...