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SAMSI RISK WS Heavy Tails and Financial Time Series Models Richard A. Davis Columbia University Thomas Mikosch University of Copenhagen

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SAMSI RISK WS 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

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SAMSI RISK WS 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.

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SAMSI RISK WS 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

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SAMSI RISK WS 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.

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

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

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SAMSI RISK WS 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?

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

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

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

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

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

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

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

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

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

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

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SAMSI RISK WS 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 ).

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SAMSI RISK WS 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)

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SAMSI RISK WS 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?

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SAMSI RISK WS 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?

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SAMSI RISK WS 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?

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SAMSI RISK WS 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?

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SAMSI RISK WS 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.

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SAMSI RISK WS 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)

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SAMSI RISK WS 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

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SAMSI RISK WS 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)

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SAMSI RISK WS 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) ) = 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 Figure: scatter plot of (X t, X t+1 ) for realization of 10,000 Distr of

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SAMSI RISK WS 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

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SAMSI RISK WS 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)

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SAMSI RISK WS 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)

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SAMSI RISK WS 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).

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SAMSI RISK WS 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

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SAMSI RISK WS 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 | = Distr of P( ) = E { ||(B,Z)|| I(arg((B,Z)) ) } / E||(B,Z)|| where P(B = 1) = P(B = -1) =.5 Examples

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SAMSI RISK WS 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)

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SAMSI RISK WS 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

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

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SAMSI RISK WS 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)

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SAMSI RISK WS 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)

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SAMSI RISK WS 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

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SAMSI RISK WS 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

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SAMSI RISK WS 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

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SAMSI RISK WS 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.

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SAMSI RISK WS 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

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SAMSI RISK WS (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)

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

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

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SAMSI RISK WS 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)

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SAMSI RISK WS 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)

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SAMSI RISK WS 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)

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

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

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SAMSI RISK WS Sample ACF for Squares of GARCH (1000 reps) (a) GARCH(1,1) Model, n= b) GARCH(1,1) Model, n=100000

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

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

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SAMSI RISK WS Amazon returns (GARCH model) GARCH(1,1) model fit to Amazon returns: =.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

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

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SAMSI RISK WS 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.

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