Two-representation property for the Gumbel-MAG(1) process

Determine whether the copula MAG(1) process driven by a Gumbel copula, defined by V_t = K_{2|1}^{−1}(ε_t | ε_{t−1}) with K a bivariate Gumbel copula and {ε_t} i.i.d. U(0,1), admits two equivalent representations obtained by permuting the order of innovations, analogous to the known two-representation property of the Gaussian-MAG(1) process; if so, characterize the parameter transformation connecting the representations.

Background

The paper proves that a Gaussian MAG(1) process has two equivalent representations arising from permuting the i.i.d. innovations, mirroring the classical MA(1) two-representation (invertibility) phenomenon. For non-Gaussian copulas, the authors explore this possibility numerically.

For the Gumbel copula specifically, they construct an approximate ‘reciprocal’ parameter mapping based on Kendall’s tau and examine likelihood behavior but emphasize that a formal proof of a two-representation property is lacking. Establishing this would clarify identifiability and estimation behavior for Gumbel-MAG(1) models.