"Frontmatter". In: Analysis of Financial Time Series

EXTREME VALUE THEORY 275{ [] }r n(i) − β 1/kn nF ∗ [r n(i) ]=1 − exp − 1 + k nα n(7.20)Consequently, using Eqs. (7.19) and (7.20) and approximating expectation by anobserved value, we have{ [] }ig + 1 = 1 − exp r n(i) − β 1/kn n− 1 + k n .α nTherefore,{ [] }r n(i) − β 1/kn nexp − 1 + k n = 1 − iα n g + 1 = g + 1 − ig + 1 , i = 1,...,g.Taking natural logarithm twice, the prior equation gives[ ( g + 1 − iln − lng + 1)]= 1 []r n(i) − β nln 1 + k n , i = 1,...,g.k n α nIn practice, letting e i be the deviation between the previous two quantities and assumingthat the series {e t } is not serially correlated, we have a regression setup[ ( g + 1 − iln − lng + 1)]= 1 []r n(i) − β nln 1 + k n + e i , i = 1,...,g. (7.21)k n α nThe least squares estimates of k n ,β n ,andα n can be obtained by minimizing the sumof squares of e i .When k n = 0, the regression setup reduces to[ ( g + 1 − iln − lng + 1)]= 1 α nr n(i) − β nα n+ e i ,i = 1,...,g.The least squares estimates are consistent, but less efficient than the likelihood estimates.We use the likelihood estimates in this chapter.7.5.2.2 The Nonparametric ApproachThe shape parameter k can be estimated using some nonparametric methods. Wemention two such methods here. These two methods are proposed by Hill (1975)and Pickands (1975) and are referred to as the Hill estimator and Pickands estimator,respectively. Both estimators apply directly to the returns {r t }t=1 T . Thus, there is noneed to consider subsamples. Denote the order statistics of the sample asr (1) ≤ r (2) ≤···≤r (T ) .Let q be a positive integer. The two estimators of k are defined as