Simulation study shows that tail‑scale parameter estimates are biased in small samples but become reliable with larger data sets. The appendix reports contour plots of the log‑likelihood for the parameter w, which display multiple local maxima when the sample size is small. Maximum‑likelihood estimators of w are systematically biased downward, inflating the value of log(w). This effect persists regardless of simultaneous estimation of the volatility parameter σ, whose estimates cluster around the true value a = 0.01. As the sample size T increases, the multimodal pattern disappears and the w estimator becomes asymptotically normal, illustrated in Figure D.2. Kernel‑density plots (Figure D.3) compare –log(w) under known and estimated scale parameters, using 100 Monte‑Carlo replicates and scatter plots with 50 000 observations. The results, based on up to 50 000 observations, are presented in ECB Working Paper Series No 3166.
© European Central Bank, 2025.
Summary derived from the ECB website (https://www.ecb.europa.eu ).
https://www.ecb.europa.eu/pub/pdf/scpwps/ecb.wp3166~4e485ab256.en.pdf
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