Testing extreme value conditions  an overview and recent approaches

1. Testing extreme value conditionsan overview and recent approaches Isabel Fraga AlvesCEAUL & DEIO University Lisbon, PortugalCláudia NevesUIMA & DM University Aveiro, Portugal

2. Contents • Introduction • Preliminaries and notation • Testing extremes • Parametric Approaches • Annual Maxima (AM) • Peaks Over Threshold (POT) • Largest Observations (LO) • Semi-Parametric Approaches • Testing EV Conditions • PORT approach  Three Tests • A case study  S&P500 data

3. Introduction • In analysis of extreme large (or small) values it is of relevant importance the model assumptions on the right (or left) tail of the underlying distribution function (d.f.) F to the sample data. • We focus on the problem of extreme large values. By an obvious transformation, the problem of extreme small values is analogous. • Statistical inference about rare events can clearly be deduced only from those observations which are extreme in some sense: • classical Gumbel method of block of annual maxima (AM) • peaks-over-threshold (POT) methods • peaks-over-random-threshold (PORT) methods. • Statistical inference is clearly improved if one make an a priori statistical choice about the more appropriate tail decay for the underlying df: light tails with finite right endpoint orpolynomial exponential • This is supported by Extreme Value Theory (EVT).

4. Theory and Extreme Values Analysis • Extreme Values Analysis Models for Extreme Values, not central values; modelling the tail of the underlying distribution • Problem:How to make inference beyond the sample data? • One Answer: use techniques based on EVT in such a way that it is possible to make statistical inference about rare events, using only a limited amount of data! • Notation:

5. Basic Theory – distribution of the Maximum • Gnedenko (1943) • Then [GEV- Generalized Extreme Value] von Mises-Jenkinson Representation

6. Extreme Value Distributions (maxima) • The GEV(g) incorporates the 3 types:[Fisher-Tippett] • Fréchet: limit for heavy tailed distributions • Weibull: limit for short tailed distributions with • Gumbel: limit for exponential tailed distributions

7. Parametric aprochesFitting GEV(g) to Anual Maxima (AM) – GUMBEL METHOD • Inclusion of location l and scale dparametersin GEV(g) df tail index (shape) g Block 1 Block 2 Block 3 Block 4 Block 5

8. Testing problem in GEV(g) The shape parameter g determines the weight of the tail Choice between Gumbel, Weibull or Fréchet or • Van Montfort (1970) • Bardsley (1977) • Otten and Van Montfort (1978) • Tiago de Oliveira (1981) • Gomes (1982) • Tiago de Oliveira (1984) • Tiago de Oliveira and Gomes (1984) • Hosking (1984) • Marohn (1994) • Wang, Cooke, and Li (1996) • Marohn (2000)

9. Heavy Tail • Pareto: bounded support • Beta: • Exponential: Exponential tail Generalized Pareto distribution GP(g) • GP(g) df includes the models:

10. Excesses over high thresholds – POT ( Peaks Over Thresholds ) • Balkema-de Haan’74+Pickands’75 u

11. Testing problem in GP(g) The shape parameter g determines the weight of the tail Choice between Exponential, Beta or Pareto or • Fitting GPdf to data • Castillo and Hadi (1997) • Goodness-of-fit tests for GPdf model • Choulakian and Stephens (2001) • Goodness-of-fit problem heavy tailed Pareto-type dfs • Beirlant, de Wet and Goegebeur (2006) • Fitting GPdf to data • Castillo and Hadi (1997) • Goodness-of-fit tests for GPdf model • Choulakian and Stephens (2001) • Goodness-of-fit problem heavy tailed Pareto-type dfs • Beirlant, de Wet and Goegebeur (2006) • Van Montfort and Witter (1985) • Gomes and Van Montfort (1986) • Brilhante (2004) • Marohn (2000) AM & POT

12. LO (Larger Observations) k largest observations of the sample: are modeledbyjoint pdf GEV(g) - extremal process

13. Testing problem in GEV(g) GEV(g)-extremal process The shape parameter g determines the weight of the tail Choice between Gumbel, Weibull or Fréchet or • Gomes and Alpuim (1986) • Gomes (1989) LO & AM • Goodness-of-fit tests • Gomes (1987)

14. Semi-Parametric Approach– Upper Order Statistics upper intermediate o.s.

15. Peaks Over Random Threshold - PORT Excesses Over Random Threshold

16. Testing Problem: Max-Domains of Attraction The shape parameter g determines the weight of the tail Choice between Domains of Attraction or • Galambos (1982) • Castillo, Galambos and Sarabia (1989) • Hasofer and Wang (1992) • Falk (1995) • Fraga Alves and Gomes (1996) • Fraga Alves (1999) • Marohn (1998a,b) • Segers and Teugels (2000) • PORT approach • Neves, Picek and Fraga Alves(2006) • Neves and Fraga Alves (2006)

17. Testing EV conditions upper intermediate o.s. • Adapted Goodness-of-fit tests • (Kolmogorov-Smirnov & Cramér-von Mises type) • Dietrich, de Haan and Husler (2002) • Drees, de Haan and Li (2006)

18. PORT approach  Three Tests for Largest Observations Excesses over the Random Threshold • Define the r-Moment of Excesses

19. NPFA test statistic:Ratio between the Maximum and the Mean of Excesses Neves, Picek & FragaAlves ‘06 • The distribution does NOT depend on the location and scale • Motivation: different behaviour of the ratio between the maximum and the mean for light and heavy tails

20. Gt test statistic:Greenwood-type Statistic (Neves & FragaAlves ‘06) • The distribution does NOT depend on the location and scale • Motivation: based on the statistic Greenwood ’46

21. HW - test statistic:Hasofer and Wang Statistic (Hasofer & Wang ’92; Neves & FragaAlves ‘06) • The distribution does NOT depend on the location and scale • Motivation: based on goodness-of-fit statistic Shapiro-Wilk ’65

22. Gumbel quantile NPFA - Test at asymptotic level a under H0 + extra second order conditions on the upper tail of F + extra conditions on convergence rate of k to infinity Reject H0(light tails) in favour of H1 (bilateral) if: Reject H0(light tails) in favour of H1 (heavy tails) if: Reject H0(light tails) in favour of H1 (short tails) if:

23. e - Normal quantile Reject H0(light tails) in favour of H1 (bilateral) if: Reject H0(light tails) in favour of H1 (heavy tails) if: Reject H0(light tails) in favour of H1 (short tails) if: Gt & HW - Tests at asymptotic level a under H0 + extra second order conditions on the upper tail of F + extra conditions on convergence rate of k to infinity

24. Exact Properties of NPFA, GT & HW - Tests An extensive simulation study concerning the proposed procedures, allows us to conclude that: • The Gt-test is shown to good advantage when testing the presence of heavy-tailed distributions is in demand. • While the Gt-test barely detects small negative values of g, the HW-test is the most powerful test under study concerning alternatives in the Weibulldomain of attraction. • Since the NPFA- test based on the very simple Tn-statistic tends to be a conservative test and yet detains a reasonable power, this test proves to be a valuablecomplement to the remainder procedures.

25. Financial data: stock index log-returns • EVT offers a powerful framework to characterize financial market crashes and booms. • The exact distribution of financial returns remains an open question. • Heavy tails are consistent with a variety of financial theories. • In financial studies, the following question is relevant: • are return distributions symmetric in the tails? • Differences in the behavior of extreme positive and negative tail movements within the same market constitute a point of investigation. • The aforementioned tests can be seen as a first test for symmetry between the positive and negative tails of thelog-returns of some stock index.

26. S&P500: left and right tails of stock index log-returns • S&P500 data: n=6985 observations series of closing prices, {Si , i = 1, … , n} of S&P500 stock index taken from 4 January, 1960 up to Friday, 16 October, 1987 (the last trading day before the crash of Black Monday, October 19, 1987 ), from which we use the daily log-returns (assumed to be stationary and weakly dependent). • Study left tail of the distribution of the returns: negative log-returns, i.e., Li := - log (Si+1/ Si) , i = 1,…, n -1. • Study right tail of the distribution of the returns: positive log-returns, defined as Xi := log (Si+1/ Si)= -Li , i = 1,…, n -1.

27. S&P500: percentage log-returnsXi := log (Si+1/ Si )

28. Sample paths of the statistics T*, R* and W*,plotted against k = 5, … , 1200, applied to S&P500: negative log-returns Li := - log (Si+1/ Si) NPFA-test Gt-test HW-test

29. Sample paths of the statistics T*, R* and W*,plotted against k = 5, … , 1200, applied to S&P500: positive log-returnsXi := log (Si+1/ Si) Gt-test NPFA-test HW-test

30. S&P500: left and right tails of stock index log-returns • NPFA, HW and Gt testing procedures under the PORT approach yielded the sample paths plots presented. • This analysis suggests the consideration of the Fréchet and Gumbeldomains of attraction, respectively, for the left and righttails of the returns distribution. • This may have the following interpretation: in this stock index the crashes are much more likely than large gain values.

31. Main References • Neves, C., Picek, J. and Fraga Alves, M.I. (2006). Contribution of the maximum to the sum of excesses for testing max-domains of attraction. JSPI, 136, 4, 1281-1301. • Neves, C. and Fraga Alves, M.I. (2006). Semi-parametric Approach to Hasofer-Wang and Greenwood Statistics in Extremes. To appear in TEST.