"Frontmatter". In: Analysis of Financial Time Series

276 VALUEATRISKk p (q) =− 1 [ ]ln(2) ln −r(q) + r (2q)−r (2q) + r (4q)k h (q) = −1q(7.22)q∑ {ln[−r(i) ]−ln[−r (q+1) ] } , (7.23)i=1where the argument (q) is used to emphasize that the estimators depend on q. Thechoice of q differs between Hill and Pickands estimators. It has been investigatedby several researchers, but there is no general consensus on the best choice available.Dekkers and De Haan (1989) show that k p (q) is consistent if q increases at aproperly chosen pace with the sample size T . In addition, √ q[k p (q)−k] is asymptoticallynormal with mean zero and variance k 2 (2 −2k+1 + 1)/[2(2 −k − 1) ln(2)] 2 .TheHill estimator is applicable to the Fréchet distribution only, but it is more efficientthan the Pickands estimator when applicable. Goldie and Smith (1987) show that√ q[kh (q)−k] is asymptotically normal with mean zero and variance k 2 . In practice,one may plot the Hill estimator k h (q) against q and find a proper q such that theestimate appears to be stable.(a) Monthly maximum log returnsmax2 4 6 8 10 121970 1980 1990 2000year(b) Monthly minimum log returnsmin-25 -20 -15 -10 -5 01970 1980 1990 2000Figure 7.3. Maximum and minimum daily log returns of IBM stock when the subperiod is 21trading days. The data span is from July 3, 1962 to December 31, 1998: (a) positive returns,and (b) negative returns.year