1. SAMSI RISK WS 2007 1 Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University www.stat.columbia.edu/~rdavis Thomas Mikosch University of Copenhagen

2. SAMSI RISK WS 2007 2 Outline F Financial time series modeling General comments Characteristics of financial time series Examples (exchange rate, Merck, Amazon) Multiplicative models for log-returns (GARCH, SV) Regular variation Multivariate case F Applications of regular variation Stochastic recurrence equations (GARCH) Stochastic volatility Extremes and extremal index Limit behavior of sample correlations F Wrap-up

3. SAMSI RISK WS 2007 3 Financial Time Series Modeling One possible goal: Develop models that capture essential features of financial data. Strategy: Formulate families of models that at least exhibit these key characteristics. (e.g., GARCH and SV) Linkage with goal: Do fitted models actually capture the desired characteristics of the real data? Answer wrt to GARCH and SV models: Yes and no. Answer may depend on the features. Stǎricǎs paper: Is GARCH(1,1) Model as Good a Model as the Nobel Accolades Would Imply? Stǎricǎs paper discusses inadequacy of GARCH(1,1) model as a data generating process for the data.

4. SAMSI RISK WS 2007 4 Financial Time Series Modeling (cont) Goal of this talk: compare and contrast some of the features of GARCH and SV models. Regular-variation of finite dimensional distributions Extreme value behavior Sample ACF behavior

5. SAMSI RISK WS 2007 5 Characteristics of financial time series Define X t = ln (P t ) - ln (P t-1 ) (log returns) heavy tailed P(|X 1 | > x) ~ RV(- ), uncorrelated near 0 for all lags h > 0 |X t | and X t 2 have slowly decaying autocorrelations converge to 0 slowly as h increases. process exhibits volatility clustering.

6. SAMSI RISK WS 2007 6 Example: Pound-Dollar Exchange Rates (Oct 1, 1981 – Jun 28, 1985; Koopman website)

7. SAMSI RISK WS 2007 7 Example: Pound-Dollar Exchange Rates Hills estimate of alpha (Hill Horror plots-Resnick)

8. SAMSI RISK WS 2007 8 Stǎricǎ Plots for Pound-Dollar Exchange Rates 15 realizations from GARCH model fitted to exchange rates + real exchange rate data. Which one is the real data?

9. SAMSI RISK WS 2007 9 Stǎricǎ Plots for Pound-Dollar Exchange Rates ACF of the squares from the 15 realizations from the GARCH model on previous slide.

10. SAMSI RISK WS 2007 10 Stǎricǎ Plots for Pound-Dollar Exchange Rates 15 realizations from SV model fitted to exchange rates + real data. Which one is the real data?

11. SAMSI RISK WS 2007 11 Example: Pound-Dollar Exchange Rates ACF of the squares from the 15 realizations from the SV model on previous slide.

12. SAMSI RISK WS 2007 12 Example: Merck log(returns) (Jan 2, 2003 – April 28, 2006; 837 observations)

13. SAMSI RISK WS 2007 13 Example: Merck log-returns Hills estimate of alpha (Hill Horror plots-Resnick)

14. SAMSI RISK WS 2007 14 Example: Amazon (May 16, 1997 – June 16, 2004)

15. SAMSI RISK WS 2007 15 Example: Amazon-returns (May 16, 1997 – June 16, 2004)

16. SAMSI RISK WS 2007 16 Example: Amazon-returns Hills estimate of alpha (Hill Horror plots-Resnick)

17. SAMSI RISK WS 2007 17 Stǎricǎ Plots for the Amazon Data 15 realizations from GARCH model fitted to Amazon + exchange rate data. Which one is the real data?

18. SAMSI RISK WS 2007 18 Stǎricǎ Plots for Amazon ACF of the squares from the 15 realizations from the GARCH model on previous slide.

19. SAMSI RISK WS 2007 19 Multiplicative models for log(returns) Basic model X t = ln (P t ) - ln (P t-1 ) (log returns) = t Z t, where { Z t } is IID with mean 0, variance 1 (if exists). (e.g. N(0,1) or a t-distribution with df.) { t } is the volatility process t and Z t are independent. Properties: EX t = 0, Cov(X t, X t+h ) = 0, h>0 (uncorrelated if Var(X t ) < ) conditional heteroscedastic (condition on t ).

20. SAMSI RISK WS 2007 20 Multiplicative models for log(returns)-cont X t = t Z t (observation eqn in state-space formulation) Two classes of models for volatility: (i) GARCH(p,q) process (General AutoRegressive Conditional Heteroscedastic-observation-driven specification) Special case: ARCH(1): (stochastic recurrence eqn)

21. SAMSI RISK WS 2007 21 Multiplicative models for log(returns)-cont GARCH(2,1): Then follows the SRE given by Questions: Existence of a unique stationary solution to the SRE? Regular variation of the joint distributions?

22. SAMSI RISK WS 2007 22 Multiplicative models for log(returns)-cont X t = t Z t (observation eqn in state-space formulation) (ii) stochastic volatility process (parameter-driven specification) Question: Joint distributions of process regularly varying if distr of Z 1 is regularly varying?

23. SAMSI RISK WS 2007 23 Two models for log(returns)-cont GARCH(1,1): Stochastic Volatility: Main question: What intrinsic features in the data (if any) can be used to discriminate between these two models?

24. SAMSI RISK WS 2007 24 Two models for log(returns)-cont X t = t Z t (observation eqn in state-space formulation) (i) GARCH(1,1) (General AutoRegressive Conditional Heteroscedastic – observation-driven specification): (ii) Stochastic Volatility (parameter-driven specification): Main question: What intrinsic features in the data (if any) can be used to discriminate between these two models?

25. SAMSI RISK WS 2007 25 Regular variation univariate case Def: The random variable X is regularly varying with index if P(|X|> t x)/P(|X|>t) x and P(X> t)/P(|X|>t) p, or, equivalently, if P(X> t x)/P(|X|>t) px and P(X t) qx where 0 p 1 and p+q=1. Equivalence: X is RV( if and only if P(X t ) /P(|X|>t) v ( ) R ( v vague convergence of measures on R\{0}). In this case, dx) = ( p x I(x>0) + q (-x) - I(x<0) ) dx Note: tA) = t - A) for every t and A bounded away from 0.

26. SAMSI RISK WS 2007 26 Another formulation (polar coordinates): Define the 1 valued rv P( ) = p, P( ) = 1 p = q. Then X is RV( if and only if or S ( v vague convergence of measures on S 0 = {-1,1}). Regular variation univariate case (cont)

27. SAMSI RISK WS 2007 27 Equivalence: R is a measure on R m which satisfies for x > 0 and A bounded away from 0, (xA) = x (A). Multivariate regular variation of X=(X 1,..., X m ): There exists a random vector S m-1 such that P(|X|> t x, X/|X| )/P(|X|>t) v x P( ) S ( v vague convergence on S m-1, unit sphere in R m ). P( ) is called the spectral measure is the index of X. Regular variation multivariate case

28. SAMSI RISK WS 2007 28 Examples: 1. If X 1 > 0 and X 2 > 0 are iid RV(, then X= (X 1, X 2 ) is multivariate regularly varying with index and spectral distribution P( =(0,1) ) = P( =(1,0) ) =.5 (mass on axes). Interpretation: Unlikely that X 1 and X 2 are very large at the same time. Figure: plot of (X t1,X t2 ) for realization of 10,000. Regular variation multivariate case (cont)

29. SAMSI RISK WS 2007 29 2. If X 1 = X 2 > 0, then X= (X 1, X 2 ) is multivariate regularly varying with index and spectral distribution P( = (1/ 2, 1/ 2) ) = 1. 3. AR(1): X t =.9 X t-1 + Z t, {Z t }~IID symmetric stable (1.8) sqrt(1.81), W.P..9898, W.P..0102 Figure: scatter plot of (X t, X t+1 ) for realization of 10,000 Distr of

30. SAMSI RISK WS 2007 30 Domain of attraction for sums of iid random vectors (Rvaceva, 1962). That is, when does the partial sum converge for some constants a n ? Spectral measure of multivariate stable vectors. Domain of attraction for componentwise maxima of iid random vectors (Resnick, 1987). Limit behavior of Weak convergence of point processes with iid points. Solution to stochastic recurrence equations, Y t = A t Y t-1 + B t Weak convergence of sample autocovariances. Applications of multivariate regular variation

31. SAMSI RISK WS 2007 31 Use vague convergence with A c ={y: c T y > 1}, i.e., where t - L(t) = P(|X| > t). Linear combinations: X ~RV( ) all linear combinations of X are regularly varying i.e., there exist and slowly varying fcn L(.), s.t. P(c T X> t)/(t - L(t)) w(c), exists for all real-valued c, where w(tc) = t w(c). AcAc Applications of multivariate regular variation (cont)

32. SAMSI RISK WS 2007 32 Converse? X ~RV( ) all linear combinations of X are regularly varying? There exist and slowly varying fcn L(.), s.t. (LC) P(c T X> t)/(t - L(t)) w(c), exists for all real-valued c. Theorem (Basrak, Davis, Mikosch, `02). Let X be a random vector. 1.If X satisfies (LC) with non-integer, then X is RV( ). 2.If X > 0 satisfies (LC) for non-negative c and is non-integer, then X is RV( ). 3.If X > 0 satisfies (LC) with an odd integer, then X is RV( ). Applications of multivariate regular variation (cont)

33. SAMSI RISK WS 2007 33 1.If X satisfies (LC) with non-integer, then X is RV( ). 2.If X > 0 satisfies (LC) for non-negative c and is non-integer, then X is RV( ). 3.If X > 0 satisfies (LC) with an odd integer, then X is RV( ). Applications of multivariate regular variation (cont) There exist and slowly varying fcn L(.), s.t. (LC) P(c T X> t)/(t - L(t)) w(c), exists for all real-valued c. Remarks: 1 cannot be extended to integer (Hult and Lindskog `05) 2 cannot be extended to integer (Hult and Lindskog `05) 3 can be extended to even integers (Lindskog et al. `07, under review).

34. SAMSI RISK WS 2007 34 1. Kesten (1973). Under general conditions, (LC) holds with L(t)=1 for stochastic recurrence equations of the form Y t = A t Y t-1 + B t, (A t, B t ) ~ IID, A t d d random matrices, B t random d-vectors. It follows that the distributions of Y t, and in fact all of the finite diml distrs of Y t are regularly varying (no longer need to be non-even). 2. GARCH processes. Since squares of a GARCH process can be embedded in a SRE, the finite dimensional distributions of a GARCH are regularly varying. Applications of theorem

35. SAMSI RISK WS 2007 35 Example of ARCH(1): X t =( 0 1 X 2 t-1 ) 1/2 Z t, {Z t }~IID. found by solving E| Z 2 | = 1..312.5771.001.57 8.004.002.001.00 Distr of P( ) = E { ||(B,Z)|| I(arg((B,Z)) ) } / E||(B,Z)|| where P(B = 1) = P(B = -1) =.5 Examples

36. SAMSI RISK WS 2007 36 Figures: plots of (X t, X t+1 ) and estimated distribution of for realization of 10,000. Example of ARCH(1): 0 1 =1, =2, X t =( 0 1 X 2 t-1 ) 1/2 Z t, {Z t }~IID Examples (cont)

37. SAMSI RISK WS 2007 37 Results: i) IID sequences {Z t } are time-reversible. ii) Linear time series (with a couple obvious exceptions) are time- reversible iff Gaussian. (Breidt and Davis `91) Application: If plot of time series does not look time- reversible, then it cannot be modeled as IID or a Gaussian process. Use the flip and compare inspection test! Reversibility. A stationary sequence of random variables {X t } is time- reversible if (X 1,...,X n ) = d (X n,...,X 1 ) for all n > 1. Excursion to time-reversibility

38. SAMSI RISK WS 2007 38 Reversibility. Does the following series look time-reversible?

39. SAMSI RISK WS 2007 39 Is this process time-reversible? Figures: plots of (X t, X t+1 ) and (X t+1, X t ) imply non-reversibility. Example of ARCH(1): 0 1 =1, =2, X t =( 0 1 X 2 t-1 ) 1/2 Z t, {Z t }~IID Examples (cont)

40. SAMSI RISK WS 2007 40 Example: SV model X t = t Z t Suppose Z t ~ RV( ) and Then Z n =(Z 1,…,Z n ) is regulary varying with index and so is X n = (X 1,…,X n ) = diag( 1,…, n ) Z n with spectral distribution concentrated on ( 1,0), (0, 1). Figure: plot of (X t,X t+1 ) for realization of 10,000. Examples (cont)

41. SAMSI RISK WS 2007 41 Example: SV model X t = t Z t Examples (cont) SV processes are time-reversible if log-volatility is Gaussian. Asymptotically time-reversible if log-volatility is nonGaussian

42. SAMSI RISK WS 2007 42 Point process application Theorem Let {X t } be an iid sequence of random vectors satisfying 1 of the 3 conditions in the theorem. Then if and only if for every c 0 where {a n } satisfies nP(| X t |> a n ) 1, and N is a Poisson process with intensity measure. {P i } are Poisson pts corresponding to the radial part, i.e., has intensity measure x (dx). { i } are iid with the spectral distribution given by the RV

43. SAMSI RISK WS 2007 43 Point process convergence Theorem (Davis & Hsing `95, Davis & Mikosch `97). Let {X t } be a stationary sequence of random m-vectors. Suppose (i) finite dimensional distributions are jointly regularly varying (let ( k,..., k ) be the vector in S (2k+1)m-1 in the definition). (ii) mixing condition A ( a n ) or strong mixing. (iii) Then (extremal index) exists. If, then

44. SAMSI RISK WS 2007 44 Point process convergence(cont) (P i ) are points of a Poisson process on (0, ) with intensity function (dy)= y 1 dy., i 1, are iid point process with distribution Q, and Q is the weak limit of Remarks: 1. GARCH and SV processes satisfy the conditions of the theorem. 2. Limit distribution for sample extremes and sample ACF follows from this theorem.

45. SAMSI RISK WS 2007 45 Setup X t = t Z t, {Z t } ~ IID (0,1) X t is RV Choose {b n } s.t. nP(X t > b n ) 1 Then Then, with M n = max{X 1,..., X n }, (i) GARCH: is extremal index ( 0 < < 1). (ii) SV model: extremal index = 1 no clustering. Extremes for GARCH and SV processes

46. SAMSI RISK WS 2007 46 (i) GARCH: (ii) SV model: Remarks about extremal index. (i) < 1 implies clustering of exceedances (ii) Numerical example. Suppose c is a threshold such that Then, if.5, (iii) is the mean cluster size of exceedances. (iv)Use to discriminate between GARCH and SV models. (v)Even for the light-tailed SV model (i.e., {Z t } ~IID N(0,1), the extremal index is 1 (see Breidt and Davis `98 ) Extremes for GARCH and SV processes (cont)

47. SAMSI RISK WS 2007 47 Extremes for GARCH and SV processes (cont) Absolute values of ARCH

48. SAMSI RISK WS 2007 48 Extremes for GARCH and SV processes (cont) Absolute values of SV process

49. SAMSI RISK WS 2007 49 Extremal Index Estimates for ARCH/SV 1 = block method 2 = 1/(mean cluster size) 3 = interval method (Ferro and Segers) 4 = interval method (Ferro and Segers)

50. SAMSI RISK WS 2007 50 Extremal Index Estimates for Amazon 1 = block method 2 = 1/(mean cluster size) 3 = interval method (Ferro and Segers) 4 = interval method (Ferro and Segers)

51. SAMSI RISK WS 2007 51 Remark: Similar results hold for the sample ACF based on |X t | and X t 2. Summary of results for ACF of GARCH(p,q) and SV models GARCH(p,q)

52. SAMSI RISK WS 2007 52 Summary of results for ACF of GARCH(p,q) and SV models (cont) SV Model

53. SAMSI RISK WS 2007 53 Sample ACF for GARCH and SV Models (1000 reps)

54. SAMSI RISK WS 2007 54 Sample ACF for Squares of GARCH (1000 reps) (a) GARCH(1,1) Model, n=10000 0.0 0.2 0.4 0.6 b) GARCH(1,1) Model, n=100000

55. SAMSI RISK WS 2007 55 Sample ACF for Squares of SV (1000 reps)

56. SAMSI RISK WS 2007 56 Example: Amazon-returns (May 16, 1997 – June 16, 2004)

57. SAMSI RISK WS 2007 57 Amazon returns (GARCH model) GARCH(1,1) model fit to Amazon returns: 0.00002493 1 =.0385, =.957, X t =( 0 1 X 2 t-1 ) 1/2 Z t, {Z t }~IID t(3.672) Simulation from GARCH(1,1) model

58. SAMSI RISK WS 2007 58 Amazon returns (SV model) Stochastic volatility model fit to Amazon returns:

59. SAMSI RISK WS 2007 59 Wrap-up Regular variation is a flexible tool for modeling both dependence and tail heaviness. Useful for establishing point process convergence of heavy-tailed time series. Extremal index < 1 for GARCH and for SV. ACF has faster convergence for SV.