Derivation of a dynamical Generalized Pareto distribution

Question

I'm currently reading a paper for my master thesis on the tail index estimation for asset returns using the peak over threshold method. In this paper the authors introduce the cumulative distribution function of the Generalized Pareto distribution (1) $$ G_{\xi,\beta}(x) = 1 - \left(1 + \xi*\frac{x - \mu}{\beta}\right)^{-1/\xi} if \xi \neq 0 $$ I understand this but afterwards the authors introduce a rewritten version of the conditional cdf of the GDP calling it the dynamic CDF (2). $$ G_{t}^{\gamma}(r_{t}|F_{t-1}) = 1 - \left(1 + \frac{r_{t} - \gamma}{\alpha_{t}}\right)^{-\zeta_{t}} $$ Im lost inbetweeen the steps of moving from (1) to (2).I don not understand how they derive the new function and I'm not sure how they can infer the ratio for $$ \beta*\zeta_{t} = \alpha_{t} $$ I seem to be missing some steps. The cited paper does not help in providing further information Extract of paper with the rewritten function

Any help is greatly appreciated

Answer 1

This looks like a straightforward reparameterization, with some variable names changed arbitrarily - suppose that the shape parameter is a function of time, and note that (1) is already conditional on threshold exceedence, then write

$$\frac{\beta}{\xi(t)} = \alpha(t)$$ $$\frac{1}{\xi(t)} = \zeta(t)$$ $$ \mu = \gamma$$

Then you arrive at the CDF in (2), and

$$\frac{\beta}{\xi(t)} = \zeta(t)\beta = \alpha(t)$$

Incidentally, on the subject of the method itself, I’d assume allowing the shape parameter to vary over time would lead to a fairly unstable estimate compared to a time-dependent scale parameter.

It might be more commonplace in finance applications, but assuming that the tail behavior of a process can range from bounded ($\xi < 0$) to polynomial ($\xi > 0$) comes off as very strange to me.