Motivated by evidence that the tails of economic and financial time series need not be stable over time, we develop a framework for studying time-varying tails. We model the extreme value index as a smooth function of time in the locally stationary framework of Zhou and Wu (2009). We estimate it by a ratio-form kernel Hill estimator in which a single intermediate quantile thresholds the whole local window. We establish consistency and asymptotic normality, and we construct uniform confi- dence bands. We also propose a consistent test of tail stability under extremal clustering. Combined with the Weissman extrapolation method, the estimator of the extreme value index yields time- varying extreme Value-at-Risk and Expected Shortfall. In an empirical application, we find the tail of daily loss of S&P 500 is compatible with a constant shape, whereas that of volatility-filtered loss varies significantly.

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