Obtain minimum-variance hedge ratio from a copula-GARCH model 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.

Using a static copula model implies $\rho_{s,f,t}\equiv\rho_{s,f}$. In such case fitting a copula model to obtain $\rho_{s,f}$ is an overkill, since it can be estimated very simply by the empirical correlation of the two standardized residual series from the two GARCH models. Of course, a availability of the joint distribution via a copula-GARCH model facilitates all kinds of interesting calculations, so the model may well be worth fitting, just not for estimating $\rho_{s,f}$ alone.
If you want time-varying correlation, you may be tempted to consider using BEKK-GARCH or DCC-GARCH models; this is what many authors do. However, the models seem to be seriously flawed – except for the case of diagonal BEKK-GARCH; see Caporin & McAleer (2013), McAleer (2019a), McAleer (2019b), Allen & McAleer (2018). Other alternatives are time-varying copula GARCH and GO-GARCH, among other, though I am not sure how sound they are theoretically. In any case, the latter two as well as DCC-GARCH are available in the rmgarch package in R should you decide to try them out.
References:

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*Allen, D. E., & McAleer, M. (2018). Theoretical and empirical differences between diagonal and full BEKK for risk management. Energies, 11(7), 1627.

*Caporin, M., & McAleer, M. (2013). Ten things you should know about the dynamic conditional correlation representation. Econometrics, 1(1), 115-126.

*McAleer, M. (2019). What they did not tell you about algebraic (non-) existence, mathematical (ir-) regularity, and (non-) asymptotic properties of the dynamic conditional correlation (DCC) model. Journal of Risk and Financial Management, 12(2), 61.

*McAleer, M. (2019). What they did not tell you about algebraic (non-) existence, mathematical (ir-) regularity and (non-) asymptotic properties of the full BEKK dynamic conditional covariance model. Journal of Risk and Financial Management, 12(2), 66.

