Patton (2006) defines the upper tail dependence coefficient for a time-varying bivariate SJC copula as $$\tau^u_t=\Lambda \left(\omega_u + \beta_u \tau^u_{t-1}+\alpha_u \frac{1}{10}\sum^{10}_{i=1}|u_{t-i}-v_{t-i}| \right)$$ where $\tau^u_t$ is the upper tail dependence coefficient at time $t$, $u_t$ and $v_t$ and the univariate transformations, and $\Lambda$ is a transformation function needed to keep $\tau^u_t$ in $[0,1]$.

Although I'm working on an extension for multivariate copulas, I'd like to use a similar equation. My (simple) question though (which I guess holds for way simpler cases like an ARMA(1,1) for any time series): how many observations do I need to estimate $\omega_u$, $\alpha_u$ and $\beta_u$; what is the value of $d$ in $(\tau^u_t)_{t\in\{1, ..., d\}}$ in order to have a reliable estimate?

Patton (2006): http://www.christoffersen.com/CHRISTOP/2007/Patton_IER_2006.pdf



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