The occurrence of extreme observations in a time series depends on the heaviness of the tails of its distribution. The paper proposes a dynamic conditional score model (DCS) for modelling dynamic shape parameters that govern the tail index. The model is based on the Generalised t family of conditional distributions, allowing for the presence of asymmetric tails and therefore the possibility of specifying different dynamics for the left and right tail indices. The paper examines through simulations both the convergence properties of the model and the implications of the link functions used. In addition the paper introduces and studies the size and power properties of a new Lagrange Multiplier (LM) test based on fitted scores to detect the presence of dynamics in the tail index parameter. The paper also shows that the novel LM test is more effective than existing tests based on fitted scores. The model is fitted to Equity Indices and Credit Default Swaps returns. It is found that the tail index for equities has dynamics driven mainly by either the upper or lower tail depending if leverage is taken or not into account. In the case of Credit Default Swaps the test identifies very persistent dynamics for both the tails. Finally the implications of dynamic tail indices for the estimated conditional distribution are assessed in terms of conditional distribution forecasting showing that the novel model predicts more accurately expected shortfalls and value-at-risk than existing models.