Let $r_{s, t}$ and $r_{f, t}$ be the return rates of the spot and futures of a commodity at time $t$. The hedging ratio based on variance minimization is calculated by finding the minimum of the variance of the combined returns:
$$\beta_t = \rho_{sf, t} \frac{\sigma_{s, t}}{\sigma_{f, t}},$$
where $\rho_{sf, t}$ is the time-varying correlation and $\sigma_{s, t}$ and $\sigma_{f, t}$ are the corresponding time-varying volatility of the spot and future returns respectively. From the several papers that I went through, the volatility measures are calculated using different GARCH-type models and the filtered standardized residuals are used to estimate various Copula models. My question is how the estimated static Copula models are then transformed to the time-varying correlation that is later used to calculate the hedge ratio? I would really be thankful for any kind of guidance.