High quantile estimation for some Stochastic Volatility models

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3 High quantile estimation for some Stochastic Volatility models Ling Luo Thesis submitted to the Faculty of Graduate and Postdoctoral Studies in partial fulfillment of the requirements for the degree of Master of Science in Mathematics Department of Mathematics and Statistics Faculty of Science University of Ottawa c Ling Luo, Ottawa, Canada, 20 The M.Sc. program is a joint program with Carleton University, administered by the Ottawa- Carleton Institute of Mathematics and Statistics

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5 Abstract In this thesis we consider estimation of the tail index for heavy tailed stochastic volatility models with long memory. We prove a central limit theorem for a Hill estimator. In particular, it is shown that neither the rate of convergence nor the asymptotic variance is affected by long memory. The theoretical findings are verified by simulation studies. iii

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7 Acknowledgements I would like to thank my supervisors, Rafal Kulik and Mahmoud Zarepour, for their friendly guidance, inspiring discussions and unending patience throughout my research project. Specially, I would also like to thank my husband, Hao, for his understanding and support and my lovely son, Travis, for bringing happiness into my life. v

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9 Contents List of Figures List of Tables x xii Introduction Time Series Models 7. White noise ARMA models ARFIMA model Gaussian long memory sequences Definition of long memory Gaussian sequences and Hermite polynomials Stochastic Volatility models Long Memory in Stochastic Volatility Limit theorems under long memory High quantiles estimation for long memory stochastic volatility models Tail empirical process with deterministic levels Weak convergence of the martingale part M n Weak convergence of long memory part R n vii

10 viii CONTENTS 2.2 Tail empirical process with random levels Asymptotic normality of the Hill estimator Numerical experiments Estimators for the tail index α Peaks Over Threshold (POT) Maximum Likelihood (ML) estimator Hill estimator Pickands estimator Simulations: POT method for different choices of threshold u Simulations: Hill and Pickands estimators Hill plots Appendix A - D[0, ) space 55 Appendix B - R codes 56 Bibliography 6

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12 List of Figures USD versus Swiss Franc Exchange Rates Log-differenced USD vs. Swiss Franc Exchange Rates White noise and its ACF AR() and its ACF MA() and its ACF SV model and its ACF Statistics for POT simulation. The graphs illustrate empirical distribution of ˆα,..., ˆα M based on M = 000 Monte Carlo runs Hill plot for iid Pareto Hill plots with different dependence parameters x

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14 List of Tables 3. Hill estimator simulation; standard deviation β = Hill estimator simulation; standard deviation β = Pickands estimator simulation; standard deviation β = Pickands estimator simulation; standard deviation β = xii

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16 Introduction Motivation Typical financial data have the following behavior: they are uncorrelated, they have strong dependence in squares, they are heavy-tailed. Notice that the autocovariance function may not be defined if the variance is infinite. To illustrate better, we consider the daily exchange rates between US Dollar and Swiss Franc for the period Figure shows that the time series {P i, i =,..., n} seems to be nonstationary, it has a decreasing trend. Consider the log-differences, i.e. S i = log(p i /P i ), i = 2,..., n. The time series plot of S i, i = 2,..., n, seems to be stationary. There is no correlation among different lags, hence the log-difference series {S i, i = 2,..., n} is uncorrelated. Figure 2, lower panel depicts that significant autocorrelations exist in Si 2, i = 2,..., n. Therefore the log-difference series is not independent. The QQ-plot shows that S i has heavier tails than the normal distribution.

17 2 Introduction Time series plot for Pi Pi Time Figure : Exchange rates The natural question is about modeling the time series so that it can capture the above features. Furthermore, we would like to estimate relevant parameters, in particular the so-called tail index in such models. Structure of the thesis In Chapter we give a brief overview of the standard time series, long memory and stochastic volatility (SV) models. We will illustrate that only SV models can capture the above mentioned properties. In particular, a pure SV process and EGARCH (Exponential Generalized Autoregressive Conditionally Heteroscedastic) process can have both dependence and heavy-tailed features. These processes are defined in Section.5. and appropriate tail and dependence properties are discussed. In particular we introduce a notion of the tail index. Furthermore, in Section.5.2 we collect some limit theorems for long memory models. We use tools such as Hermite polynomials

18 Introduction 3 to establish our results. In Chapter 2, we establish asymptotic theory for high quantile estimation of EGARCH and SV processes. The core problem resolved in this paper is to study the asymptotic behavior of a tail empirical process with deterministic levels and obtain the corresponding behavior of a tail empirical process with random levels. We conclude in Theorem 2.. that the long memory influences the behaviour of the tail empirical process with deterministic levels, however, this is not the case for random levels; see Section 2.2. We use the latter result to obtain asymptotic normality of the Hill estimator, the most popular estimator of the tail index. In Chapter 3, we study the numerical performance of three tail index estimators: Peak over Threshold (POT) estimator, Hill estimator and Pickands estimator. Note that the Hill estimator can be thought of as the POT estimator based on random levels. Therefore, the theoretical results in Chapter 2 suggest that the Hill estimator should be robust against long memory, contrary to the standard POT estimator, which is based on deterministic levels. We illustrate that under the SV model with different dependence and index, the dependence does not influence the performance of the Hill estimator. Contribution Convergence of the tail empirical processes was studied in Rootzen (2009) in a weakly dependent case and in Kulik and Soulier (20) for pure long memory stochastic volatility models. Our main contribution is the extension of the latter result to so-called EGARCH models. This extension requires a careful application of the martingale central limit theorem; see Section 2.. Furthermore, we conduct simulations which confirm our theoretical findings. Similar work has been done in McElroy and

19 4 Introduction Jach (20) or Mikosch and Rezapour (20); in both cases for a pure stochastic volatility model.

20 Introduction 5 Time series plot for Si Si Time Series Si Series Si^2 Normal Q Q Plot ACF ACF Sample Quantiles Lag Lag Theoretical Quantiles Figure 2: Graph of log-differences: time series, autocovariance function, autocovariance function of squares, QQ-plot

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22 Chapter Time Series Models Let {X i, < i < } be a stationary sequence with a finite variance. We will use the following notation: γ X (h) = Cov(X i, X i+h ) = Cov(X 0, X h ), ρ X (h) = Cor(X 0, X h ) = γ X(h) Var(X).. White noise Let {Z i, < i < } be a sequence of independent standard normal random variables. Of course, these random variables, as well as their squares are uncorrelated. To illustrate this, we simulate 000 observations from a i.i.d. N(0, ) process. Figure. shows 000 simulated values, the sequence seems to be stationary. Furthermore, the figure shows the corresponding autocorrelation function, the values ρ Z (h) are close to 0 for all h > 0. It indicates that the series {Z i, < i < } is uncorrelated. Also, the squares are uncorrelated. We will write {Z i, < i < } IID(0, ). 7

23 8. Time Series Models Time series plot for White Noise Zi Time Series Zi Series Zi^2 ACF ACF Lag Lag.2 ARMA models Figure.: Simulated white noise and its ACF Definition.2. We say that {X i, < i < } is an ARMA(p,q) process if {X i, < i < } is stationary and if for every i, X i ϕ X i ϕ p X i p = Z i + θ Z i + + θ q Z i q, (.2.) where {Z i, < i < } White Noise(0, σ 2 ) and the polynomials ϕ p (z) = ϕ z ϕ p z p

24 .2. ARMA models 9 and have no common factors. We may write θ q (z) = + θ z + + θ q z q ϕ p (B)X i = θ q (B)Z i, (.2.2) where B is the backward shift operator (B j X i = X i j, j = 0, ±,...). AR(), 0.5 AR Time Series AR Series AR^2 ACF ACF Lag Lag Figure.2: Simulated AR() and its ACF We simulate 000 observations from an AR()=ARMA(,0) process. Figure.2 shows that the AR() series is stationary. The ACF decays exponentially to zero.

25 0. Time Series Models There is correlation in the process for different lags h. Theoretically, we will show that the covariance function is not equal to 0. Example.2.2 Consider AR() model: X i = ϕx i + Z i, where ϕ <, and Z i, i =,..., n, is a white noise (i.e, the sequence of uncorrelated random variables, with mean zero and variance σz 2 ). Then X i = ϕx i + Z i = ϕ j Z i j j=0 which is a linear process with ACF γ X (h) = j=0 ϕ j ϕ j+h σ 2 Z = σ 2 Zϕ h ϕ 2 = Cϕh, h > 0. This means that the AR() process is correlated. We simulate 000 observations from an MA()=ARMA(0,) process. Figure.3 shows that the MA() series is stationary. The ACF is almost zero after lag of the process and it shows that the series is correlated. Again, the squares of the time series are correlated. Example.2.3 Consider MA() process X i = Z i + θz i, where {Z i, < i < } IID(0, σ 2 ). We have γ X (h) = E(X i X i h ) 0 = E[(Z i + θz i )(Z i h + θz i h )] = E(Z i Z i h + θz i Z i h + θz i Z i h + θ 2 Z i Z i h ). When h = 0, γ X (0) = E(Z 2 i + θz i Z i + θz i Z i + θ 2 Z 2 i ) = σ 2 + θ 2 σ 2 = σ 2 ( + θ 2 )

26 .2. ARMA models MA(), 0.5 MA Time Series MA Series MA^2 ACF ACF Lag Lag Figure.3: Simulated MA() and its ACF When h =, γ X () = E(Z i Z i + θz i Z i 2 + θz 2 i + θ 2 Z i Z i 2 ) = θe(z 2 i ) = θσ 2 When h =, γ X ( ) = E(Z i Z i+ + θz 2 i + θz i Z i+ + θ 2 Z i Z i ) = θe(z 2 i ) = θσ 2

27 2. Time Series Models When h = 2, γ X (2) = E(Z i Z i 2 + θz i Z t 3 + θz i Z i 2 + θ 2 Z i Z t 3 ) = 0. Similarly, γ X (h) = 0 if h > 2. Thus we have, γ X (h) = σ 2 ( + θ 2 ) if h = 0; σ 2 θ if h = ±; 0 if h >. In summary, ARMA models can not capture the following properties of financial data: the original time series is uncorrelated, but their squares are correlated..3 ARFIMA model Consider a process {X i, < i < } defined as ϕ p (B)( B) d X i = θ q (B)Z i, {Z i, < i < } IID(0, σz), 2 (.3.) where d is a rational number and B is the backward shift operator; ϕ p (B) = ϕ B ϕ 2 B 2... ϕ p B p θ q (B) = + θ B + θ 2 B θ q B q are respectively the autoregressive and the moving average characteristic polynomials, satisfying ϕ(z) 0 and θ(z) 0 for all z such that z. For simplicity, to find the range of d for the process (.3.), we consider the case, where ϕ p (B) = and θ q (B) =, ( B) d X i = Z i (.3.2)

28 .3. ARFIMA model 3 If the process (.3.2) is stationary, then we may write it as (see Brockwell and Davis 2002) X i = ( B) d Z i. (.3.3) The operator ( B) d in equation (.3.3) is defined by the binomial expansion ( B) d = = ( ) d ( ) d ( B) j = ( ) j B j j j j=0 j=0 ψ j B j, j=0 where ψ j = ( ) d j ( ) j = Γ(j+d). Therefore X Γ(j+)Γ(d) i in equation (.3.3) is X i = ( B) d Z i = ψ j B j Z i = j=0 ψ j Z i j. (.3.4) j=0 Lemma.3. We have Proof: Using Stirling s formula ψ j Γ(d)j d. Γ(x) 2πe x x x+/2 as x, we have ψ j = = Now, we prove that equivalently, Γ(j + d) Γ(j + )Γ(d) e d+ (j + d) j+d 2. Γ(d) (j + ) j+ 2 e d+ (j + d) j+d 2 Γ(d) (j + ) j+ 2 2πe (j+d) (j + d) j+d /2 2πe (j+) (j + ) j+ /2 Γ(d) lim e d+ (j + d) j+d j j d (j + ) j+ 2 Γ(d)j d, 2 =.

29 4. Time Series Models ( where Since we have j j+d LHS = e d+ j d (j + d) j+d 2 lim j (j + ) j+ 2 ( ) j+/2 j + d = e d+ lim j d (j + d) d j j + ( ) d ( ) j+/2 j j + d = e d+ lim j j + d j + ) d, ( = e d+ lim j ( j j + d ) j j+d j e d and ) d ( j + d ( j j ) j+/2 ( ) j+/2 j, j + j+) j e as j, so ( ) d ( ) j+/2 j j + d e d+ lim = e d+ e d = = RHS. j j + d j + ψj 2 j=0 ψ j Γ(d)j d (Γ(d)) 2 j=0 j 2( d), which is convergent if 2( d) >, i.e., d < 0.5, therefore {ψ j } is square summable if and only if d < 0.5 and in this paper we are interested in d > 0. The process in (.3.) with 0 < d < 0.5 is denoted the ARFIMA(p,d,q) (AutoRegressive Fractionally Integrated Moving Average) model and the process X i in (.3.2) with 0 < d < 0.5 is called the fractionally integrated noise. The linear representation (.3.4) with the coefficients ψ j = Γ(d)j d following behaviour of ACF (Beran 994, p. 63): yields the Lemma.3.2 Consider the stationary ARFIMA(0,d,0) process with d (0, /2). Then ρ X (k) Γ( d) k 2d, k. Γ(d)

30 .4. Gaussian long memory sequences 5 In summary, the ARFIMA model itself has slowly decaying correlation. ARFIMA models cannot be used directly to model returns, but they will be used as a building block for stochastic volatility models..4 Gaussian long memory sequences.4. Definition of long memory There are different definitions of long memory. The most common form is given below (Beran 994, p. 42): Definition.4. A stationary process {X i, < i < } is said to be a short memory process if its autocorrelation function (ACF) ρ X (k) is absolutely summable: ρ X (k) <. (.4.) k=0 On the other hand, a stationary process is said to have a long memory if ACF is not absolutely summable, i.e. ρ X (k) =. k= Example.4.2 If ρ X (k) ck 2d, as k, (.4.2) where d (0, /2), then ACF is not summable. This type of behaviour of ACF is usually assumed for purpose of limit theorems. Example.4.3 Consider the AR() model of Example.2.2. If ϕ <, then γ X (h) = C ϕ h <. h=0 h=0 Thus, the AR() model has short-memory.

31 6. Time Series Models Example.4.4 Consider the ARFIMA(0,d,0) model with d (0, /2). Then γ X (h) =. h=0 Thus, the ARFIMA(0,d,0) model has long-memory..4.2 Gaussian sequences and Hermite polynomials Assume that {X i, < i < } is a stationary Gaussian process with unit variance and covariance Cov(X i, X j ) = γ X (i j) = i j 2d l 0 ( i j ), (.4.3) where d (0, /2) and l 0 is a slowly varying function at infinity. We will assume for simplicity l 0 (i) =, since if d (0, /2), Cov(X, X j ) = j=0 Thus, the sequence has long memory. A crucial tool in studying long memory Gaussian processes are Hermite polynomials. γ X (j) = j=0 j 2d = +. Definition.4.5 For any i = 0,, 2,..., the Hermite polynomials H i are defined by the formula j=0 ( ) ( ) x H i (x) = ( ) i 2 d i x 2 exp 2 dx exp. i 2 Note that {H i, < i < } is an orthogonal basis for Gaussian random variables, that is, for a standard normal random variable X with a density we have < H i (X), H j (X) >= ϕ(x) = 2π exp( x2 2 ) H i (x)h j (x)ϕ(x)dx

32 .4. Gaussian long memory sequences 7 where = Cov(H i (X), H j (X)) = E(H i (X)H j (X)) = i! if i = j, (.4.4) < H i (X), H j (X) >= 0, if i j, (.4.5) < f, g >= f(x)g(x)ϕ(x)dx. It also shows that the V ar(h i (X)) = i! and H i (X), H j (X) are uncorrelated if i j, however, H i (X), H j (X) are not independent. The first 6 Hermite polynomials are: H 0 (x) =, H (x) = x, H 2 (x) = x 2, H 3 (x) = x 3 3x, H 4 (x) = x 4 6x 2 + 3, H 5 (x) = x 5 0x 3 + 5x. Lemma.4.6 (Arcones (994)) Let a,..., a k be real numbers such that a a 2 k =. Then H q ( k j= a j x j ) = q + +q k =q q! q! q k! k j= a q j j H q j (x j ). Lemma.4.7 Let X, Y be a pair of jointly standard normal random variables with covariance γ = cov(x, Y ) and suppose that f is a measurable transformation with Hermite expansion f(x) = i=0 a i i! H i(x), where H i (x) are Hermite polynomials and a i =< f(x), H i (X) >= E(f(X)H i (X)) = f(x)h i (x)ϕ(x)dx. Then cov(h i (X), H i (Y )) = i!γ i, (.4.6) cov(h i (X), H j (Y )) = 0, for i j, (.4.7) a 2 i cov(f(x), f(y )) = i! γi. (.4.8) i=0

33 8. Time Series Models Proof of lemma.4.7: We can write Y = γx + γ 2 Z, where Z is N(0, ) and is independent of X. Using Lemma.4.6, Cov(H i (X), H i (Y )) = E(H i (X)H i (Y )) = E(H i (X)H i (γx + γ 2 Z)) i! = E(H i (X) q!q 2! γq γ 2 q 2 H q (X)H q2 (Z)) = q +q 2 =i q +q 2 =i i! q!q 2! γq γ 2 q 2 E(H i (X)H q (X))E(H q2 (Z)) = γ i E(H i (X)H i (X)) = γ i i!, where the last equality follows from equations.4.4 and.4.5 that q = i and q 2 = 0 in order to have q + q 2 = i. Therefore (recall that Y = γx + γ 2 Z), ( ) a i Cov(f(X), f(y )) = E(f(X)f(Y )) = E i! H a j i(x) j! H j(y ) = i=0 j=0 i=0 γ j a ia j i!j! E(H i(x)h j (X)) = j=0 i=0 a 2 i (i!) 2 γi i! = i=0 a 2 i i! γi..5 Stochastic Volatility models Example.5. Let {Z i, < i < } be i.i.d. N(0, ) and {X i, < i < } be an AR() model, i.e. X i = ϕx i + U i, where {U i, < i < } IID(0, σu 2 ). We assume that the sequences {X i, < i < } and {Z i, < i < } are mutually independent. We define Y i = X i Z i, < i <. We simulate 000 observations from Y i, i =,..., 000. Figure.4 shows that the composite series is stationary. The ACF is almost zero for all lags h > 0 which shows that the series is uncorrelated. The squares are

34 .5. Stochastic Volatility models 9 AR() and N(0,) composite process Yi Time Series Yi Series Yi^2 ACF ACF Lag Lag Figure.4: Simulated SV model and its ACF correlated, we conclude that the sequence {Y i, < i < } is not independent. Indeed, for h 0 Cov(Y i, Y i+h ) = E(Y i Y i+h ) = E(X i Z i X i+h Z i+h ) = E(X i X i+h )E(Z i )E(Z i+h ) = 0, which means that the series {Y i, < i < } is uncorrelated. On the other hand Cov(Y 2 i, Y 2 i+h) = E(Y 2 i Y 2 i+h) E(Y 2 i )E(Y 2 i+h) = E(Z 2 i X 2 i Z 2 i+hx 2 i+h) E(Z 2 i X 2 i )E(Z 2 i+hx 2 i+h)

35 20. Time Series Models = E(Z 2 i Z 2 i+h)e(x 2 i X 2 i+h) E(Z 2 i )E(X 2 i )E(Z 2 i+h)e(x 2 i+h) = {E(Z 2 i )E(Z 2 i+h)}cov(x 2 i, X 2 i+h) = (VarZ ) 2 Cov(X 2 i, X 2 i+h) where, in particular, Cov(X 2 i, X 2 i+) = Cov(X 2 i, (ϕx i + U i+ ) 2 ) = Cov(X 2 i, ϕ 2 X 2 i + U 2 i+ + 2ϕX i U i+ ) = ϕ 2 Cov(X 2 i, X 2 i ) + Cov(X 2 i, U 2 i+) + 2ϕCov(X 2 i, X i U i+ ) = ϕ 2 Var(X 2 i ) 0. Above we used the fact that X i and U i+ are independent. A similar computation can be carried out for a general h. We conclude that that {X 2 i, < i < } is correlated. Thus, Cov(Yi 2, Yi+h 2 2 ) 0, which means that {Yi, < i < } is correlated as well. In conclusion, Cov(Y i, Y i+h ) = 0 for h 0 (Y i uncorrelated) Cov(Yi 2, Yi+h 2 ) 0 (Y i 2 correlated) so {Y i, < i < } is not an independent sequence and inherits covariance structure from {X 2 i, i 0}. model. The above example suggests the following definition of a stochastic volatility Definition.5.2 Let σ( ) be a positive function. Without loss of generality, we will assume that σ is positive. Let {X i, < i < } be a stationary sequence, and {Z i, < i < } be a sequence of i.i.d. random variables such that E(Z i ) = 0. Define a stochastic volatility (SV) process {Y i, < i < } by Y i = σ(x i )Z i. (.5.) Remark.5.3 For theoretical purposes, there is no need to assume that σ is positive. However, in financial applications σ(x i ) plays a role of a conditional variance. Therefore, we keep this assumption throughout the thesis.

36 .5. Stochastic Volatility models 2 We note that in the above definition we do not impose any assumption about independence between the sequences {X i, < i < } and {Z i, < i < }. However, in practice we need to consider certain models in order to be able to work with such processes. We shall assume that {X i, < i < } is a Gaussian linear process (infinite order moving average) X i = c k η i k, < i <, k= where {η i, < i < } is a sequence of independent standard normal random variables and k= c2 k =. In particular, {X i, < i < } are dependent standard normal random variables. Two special cases which we are going to deal with are: Pure Stochastic Volatility (SV) model; where {η i, < i < } and {Z i, < i < }, are independent. EGARCH model; where {(η i, Z i ), < i < } is a sequence of i.i.d. random vectors. Thus, for fixed i, Z i and X i are independent. Of course, the EGARCH model includes SV. In what follows, we will write stochastic volatility (SV) model for all models described by the Definition.5.2. We will write pure SV for the model described above, where we have independence between the sequences {X i, < i < } and {Z i, < i < }. The pure SV and EGARCH models are flexible enough to model both long memory and heavy tails. Indeed, as we argued in Example.5., the squares of both models inherit the dependence structure of {X i, < i < }. At the same time, we can choose {Z i, < i < } to be heavy tailed. Let F Z (x) = P (Z x) be the marginal cumulative distribution of the noise sequence {Z i, < i < }.

37 22. Time Series Models Definition.5.4 A function L is called slowly varying at infinity if for all c > 0, L(cx) lim x L(x) = Example.5.5 If we let L(x) = x, then L(cx) L(x) = cx x varying. Example.5.6 If we let L(x) = log x, then log(cx) log x Thus, L(x) = log x is slowly varying. = c, thus L(x) = x is not slowly = log c + log x log x log x, as x. We will assume that for some α (, ), the tail of Z is F Z (x) = P (Z > x) = x α L(x), x (.5.2) where L is a slowly varying function. The parameter α is called the tail index. In particular, for each c > 0, P (Z > cx) lim x P (Z > x) = lim (cx) α L(cx) x x α L(x) = c α. (.5.3) In fact, F Z (x) is called regularly varying at infinity with index α (see Resnick 2007). Lemma.5.7 (Breiman Lemma, Resnick 2007, p. 23) Define Y i = σ(x i )Z i, assume that for fixed i the random variables X i and Z i are independent and that (.5.2) holds. If E[σ α+ϵ (X i )] <, for some ϵ > 0, then P (Y i > x) E[σ α (X i )]P (Z i > x). Breiman Lemma shows that the tail behaviour of Z is inherited by Y..5. Long Memory in Stochastic Volatility Example.5. gives us some idea how to introduce long memory in stochastic volatility. The following example shows how the long memory structure of σ 2 (X i ) is inherited by Y 2 i.

38 .5. Stochastic Volatility models 23 Example.5.8 Let {Z i, i 0} be IID(0, σ 2 Z ), and let {X i, t } be a long memory Gaussian sequence as in Section.4. Assume that {Z i, < i < } and {X i, < i < } are independent. Define Y i = σ(x i )Z i. Then for h 0, E(Y i ) = E(Z i σ(x i )) = E(Z i )E(σ(X i )) = 0, Cov(Y i, Y i+h ) = E(Y i Y i+h ) E(Y i )E(Y i+h ) = E(Y i Y i+h ) 0 = E(Z i Z i+h )E(σ(X i )σ(x i+h )) = 0E(σ(X i )σ(x i+h )) = 0 Cov(Y 2 i, Y 2 i+h) = (VarZ ) 2 Cov(σ 2 (X i ), σ 2 (X i+h )) which shows that the SV model is uncorrelated, but Y 2 i are correlated, hence the sequence {Y i, < i < } is not independent and inherits covariance structure from {σ 2 (X i ), < i < }. Take in particular σ(x) = e x. Then using Lemma.4.7 with f(x) = σ 2 (x) = e 2x and Equation (.4.3) for γ X, where Cov(σ 2 (X 0 ), σ 2 (X h )) = Cov(f(X 0 ), f(x h )) = = i= a 2 i i! a = E(exp 2X X) 0. i= a 2 i (i!) 2 i!γi X(h) ( h (2d ) ) i a 2! γ X(h) a 2 h (2d ) (.5.4) In other words, the covariance of σ 2 (X i ), up to a constant, is the same as that of {X i, < i < }. Lemma.5.9 Let {X i, < i < } be a stationary LRD Gaussian process as defined in Section.4, with covariance defined in (.4.3). If d (0, /2), then V ar( X n ) 2n 2d 2d(2d + ) C n ( 2d).

39 24. Time Series Models The above result is standard in the long memory literature, see for example page 6 in Beran (994). For a comparison with iid or short memory sequences notice that V ar( X n ) C n..5.2 Limit theorems under long memory Recall the classical central limit theorem. If {X i, < i < } are iid with mean zero and unit variance, then n n X d i N(0, ), t= The situation changes completely if long memory is involved. See Chapter 3 in Beran (994). Lemma.5.0 Let {X i, < i < } be a stationary LRD Gaussian process with mean zero and unit variance, as defined in Section.4, with covariance defined in (.4.3). If d (0, /2), then n /2+d d(2d+) t= n X d i N(0, ). From equation (.5.4), Cov(f(X 0 ), f(x i )) a 2 γ X (h) and ( n ) ( n ) V ar f(x i ) a 2 V ar X i. i= It suggests the following lemma. For details see Chapter 3 in Beran (994). Lemma.5. If f has the Hermite expansion given in Lemma.4.7, then the limiting behaviour of i= n {f(x i ) E(f(X i ))} t= is the same as that of a n t= X i, where a = E(f(X)X) 0: n {f(x i ) E(f(X i ))} d a N(0, ) N(0, a 2 ). n /2+d d(2d+) t=

40 .5. Stochastic Volatility models 25 Note: If a = 0, the limiting behaviour is more complicated and it is beyond the scope of this thesis.

41 Chapter 2 High quantiles estimation for long memory stochastic volatility models In this section we establish asymptotic theory for high quantile estimation of EGARCH and SV processes. Specifically, in section 2. we will establish the asymptotic behavior of a tail empirical process with deterministic levels. In section 2.2 we will obtain a corresponding behavior of a tail empirical process with random levels. The latter result will be used in section 2.3 to obtain asymptotic normality of the Hill estimator of the tail index. Recall the Definition.5.2 of the EGARCH process. Let u n be a sequence of constants such that: u n ; n F (u n ). The associated conditional tail distribution function is defined for x > 0: T n (x) = P (Y > u n ( + x) Y > u n ) = F (u n( + x)). F (u n ) 26

42 2.. Tail empirical process with deterministic levels 27 Under the assumption (.5.3), we have T n (x) ( + x) α = T (x) as u n. (2.0.) We also need the following assumption. Assumption (A): If i j, then for any sets A and B, P (Y i u n A, Y j u n B) C i,j P (Y i u n A)P (Y j u n B), as n, where u n A = {u n x : x A}. Also, sup i,j C i,j C. This condition holds for example when Y i is the SV model. It is also verified for EGARCH model, when Z i = Z iψ (η i ) + Ψ 2 (η i ), where {Z i, i } is the sequence of i.i.d. random variables with the same distribution as Z i and {Z i, i } is independent of {η i, < i < }. Example If Y i = σ(x i )Z i is a pure SV model, by using Breiman s Lemma.5.7, we have P (Y i > u n y, Y j > u n y) = E[P (Y i > u n y, Y j > u n y X i, X j )] E[σ α (X i )σ α (X j )]P (Z i > u n y)p (Z j > u n y) E[σ α (X i )σ α (X j )] E[σ α (X i )]E[σ α (X j )] P (Y i > u n y)p (Y j > u n y), so C i,j = E[σα (X i )σ α (X j )]. In particular, if σ(x) = E[σ α (X i )]E[σ α (X j )] ex and using X i + X j N(0, 2 + 2ρ j i ), so C i, j = E[eα(X i +X j ) ] = e α 2 2 (2+2ρ (j i) ). Since ρ, we have C E[e αx i]e[e αxj ] i,j e α2 = C. e α2 2 e α Tail empirical process with deterministic levels Define the empirical tail distribution function T n (s) = nf (u n ) n {Yi >u n(+s)}, (2..) i=

43 28 2. High quantiles estimation for long memory stochastic volatility models and the tail empirical process Note that E[ T n (s)] = e n (s) = T n (s) T n (s), s > 0. n nf (u n ) P (Y > u n( + s)) = F (u n( + s)) F (u n ) = T n (s), therefore, T n (s) is the proper centering. Our main result of this section is the following theorem. Theorem 2.. Let Y i = σ(x i )Z i be the EGARCH model as in the definition.5.2. Assume that the tail of Z is regularly varying with index α (see (.5.2)) and E[σ α (X)] <. Furthermore, assume that X i is a long memory Gaussian process with covariance given by (.4.2). Assume that (A) holds. If n 2d F (un ) 0, then n F (u n )e n (s) W (T (s)), where W is a standard Brownian motion. If n 2d F (un ), then n d /2 d(2d+) where denotes weak convergence in D[0, ). e n (s) T (s) E(X iσ α (X i )) N(0, ), E(σ α (X i )) Remark 2..2 If n 2d F (un ) c (0, ), then the limiting process is the sum of two processes which appear in Theorem 2.., however, their joint distribution may be complicated. Remark 2..3 We note that in the definition of T n ( ) we use the unknown quantity F (u n ). Consequently, the result for Theorem 2.. is not applicable in practice. However, this result is used to conclude a limiting behaviour of a practical tail empirical process considered in the next section.

44 2.. Tail empirical process with deterministic levels 29 Remark 2..4 The space D[0, ) is defined in the Appendix. The proof of this theorem will be divided into several steps. Decompose the tail empirical process into e n (s) = T n (s) T n (s) = n ( {Yi >u n (+s)} E[ {Yi >u n (+s)} F i ]) nf (u n ) i= + n E[ {Yi >u n(+s)} F i ] T n (s) =: M n (s) + R n (s), nf (u n ) i= where F i = σ(η i, η i,, Z i, Z i, ). Note that X i is F i -measurable, since X i = k= c kη i k and Z i is independent of F i. In what follows, we will show that n F (u n )M n (s) W (T (s)). This follows from Lemma 2..6 (finite dimensional convergence) and Lemma 2..8 (tightness). Furthermore, in Lemma 2..9 we will show that n d /2 d(2d+) From these two expressions, the theorem follows. R n (s) T (s) E(X iσ α (X i )) N(0, ) E(σ α (X i )) Note that M n (s), n, is a martingale. Indeed if we let V i = {Yi >u n (+s)} and M i = nf (u n ) (V i E[V i F i ]), then E(M i F i ) = = = E{(V i E[V i F i ]) F i } nf (u n ) {E(V i F i ) E[E(V i F i ) F i ]} nf (u n ) {E(V i F i ) E(V i F i )} = 0. nf (u n )

45 30 2. High quantiles estimation for long memory stochastic volatility models 2.. Weak convergence of the martingale part M n First, we will evaluate the variance of the martingale part (see Lemma 2..5). Then, we will prove a central limit theorem for the martingale part (see Lemma 2..6). That convergence is easily generalized to a finite dimensional convergence. Furthermore, we will show that the martingale part is tight (see Lemma 2..8). Lemma 2..5 Under the conditions of Theorem 2.., we have ( ) n Var { {Yi >u n(+s)} E( {Yi >u n(+s)} F i )} nf (u n ) ( + s) α nf (u n ) i= ( + s) α ne[σ α (X i )]F Z (u n ). Proof: Let V i = {Yi >u n(+s)}, then using the martingale property ( ) n Var { {Yi >u n (+s)} E( {Yi >u n (+s)} F i )} nf (u n ) i= = n n n 2 F 2 Var[V i E(V i F i )] = (u n ) i= n 2 F 2 (u n ) Var[V i E(V i F i )] = nf 2 (u n ) Var[V i E(V i F i )] = nf 2 (u n ) {E(V i E(V i F i )) 2 0} = nf 2 (u n ) {E(V i 2 ) + E[(E(V i F i )) 2 ] 2E[V i E(V i F i )]}. (2..2) Thus, we split Var[V i E(V i F i )] into 3 terms; the first term A = E(V 2 i ), the second one B = E[(E(V i F i )) 2 ] and the third term C = E[V i E(V i F i )]. For the first term, we have A = E(V 2 i ) = E(V i ) = E[ {Yi >u n(+s)}] = P (Y i > u n ( + s)) = F (u n ( + s)). By Breiman s Lemma.5.7, A E[σ α (X i )]F Z (u n ( + s)) as u n,

46 2.. Tail empirical process with deterministic levels 3 and by (.5.3), Thus, P (Z > u n ( + s)) lim u n P (Z > u n ) = ( + s) α. A E[σ α (X i )]( + s) α F Z (u n ) ( + s) α F (u n ) as u n. (2..3) For the second term, Using (.5.3) we have, E(V i F i ) lim u n P (Z > u n ) = lim E( {σ(xi )Z i >u n(+s)} X i ) u n P (Z > u n ) P (Z i > un(+s) = lim ) σ(x i ) u n P (Z > u n ) = ( + s) α σ α (X i ). (2..4) Thus, B lim u n P 2 (Z > u n ) [ ] = lim E E(Vi F i ) E(V i F i ) u n P (Z > u n ) P (Z > u n ) [ ] E(V i F i ) = E lim u n P (Z > u n ) lim E(V i F i ) u n P (Z > u n ) = E[( + s) 2α σ 2α (X i )] = ( + s) 2α E[σ 2α (X i )]. Consequently, B ( + s) 2α E[σ 2α (X i )]F 2 Z(u n ) as u n. (2..5) Here and in the sequel, we are allowed to interchange the limit with integration since E[V i F i ] is bounded. lim u n For the third term, using (.5.3) and (2..4), we have, C P 2 (Z > u n ) = lim E[V i E(V i F i )] u n P 2 (Z > u n ) ( ) ( = lim E V i E(V i F i ) = E u n P (Z > u n ) P (Z > u n ) ( ) V i = E lim u n P (Z > u n ) ( + s) α σ α (X i ) ( ) = ( + s) α lim E V i u n P (Z > u n ) σα (X i ). lim u n = lim u n E V i P (Z > u n ) lim ( u n ) E(V i F i ) P (Z > u n ) V i P (Z > u n ) ( + s) α σ α (X i ) )

47 32 2. High quantiles estimation for long memory stochastic volatility models Furthermore, ( ) ( + s) α lim E V i u n P (Z > u n ) σα (X i ) ( ) = ( + s) α lim E V i E[ u n P (Z > u n ) σα (X i ) X i ] ( ) = ( + s) α lim E σ α V i (X i )E[ u n P (Z > u n ) X i] ( = ( + s) α lim E σ α (X i )E[ ) {σ(x i )Z i >u n (+s)} X i ] u n P (Z > u n ) = ( + s) α lim (σ E α (X i ) P (Z i > un(+s) X ) σ(x i ) i) u n P (Z > u n ) = ( + s) α E[σ α (X i )( + s) α σ α (X i )] = ( + s) 2α E[σ 2α (X i )] Thus, C ( + s) 2α E[σ 2α (X i )]F 2 Z(u n ) as u n. (2..6) After comparing the results (2..3), (2..5) and (2..6), we find that A dominates B and C, so terms B and C are negligible. Consequently, Var[V i E(V i F i )] A E[σ α (X i )]( + s) α F Z (u n ), (2..7) and n Var [V i E(V i F i )] = n Var[V i E(V i F i )] = nvar[v i E(V i F i )] i= i= ne[σ α (X i )]( + s) α F Z (u n ) n( + s) α F (u n ). By (2..2), (2..3) and (2..7), we get ( ) n Var { {Yi >u n (+s)} E( {Yi >u n (+s)} F i )} nf (u n ) i= ( + s) α nf (u n ). Since n i= [V i E(V i F i )] is a martingale, it suggests the following result.

48 2.. Tail empirical process with deterministic levels 33 Lemma 2..6 Under the conditions of Theorem 2.. we have nf (u n )M n (s) = nf (u n ) n d [V i E(V i F i )] N(0, ( + s) α ). i= To prove the above lemma, we recall the following martingale CLT (see Hall and Heyde (980)). Lemma 2..7 (Martingale CLT) Let ((X ni, A n,i ) : n ; i =,, k n ) be a martingale difference array; that is, for each fixed n, X ni is a martingale difference sequence. If k n i= E(X2 ni A n,i ) p σ 2, and k n i= E[X2 p nii( X ni > ϵ) A n,i ] 0 for each ϵ > 0 then Z n = k n i= X ni d Z N(0, σ 2 ). We apply the above theorem to the scaled martingale part: Z n = nf (u n )M n (s) and X ni = M i = nf (u n) [V i E(V i F i )] and A n,i = F i. Let M i = V i E(V i F i ), so that M 2 i = nf (u n ) [V i E(V i F i )] 2 = nf (u n ) M 2 i. Then E(M 2 i F i ) = E[(V i E(V i F i )) 2 F i ] = E[[V 2 i + (E(V i F i )) 2 2V i E(V i F i )] F i )] = E(V 2 i F i ) + E[(E(V i F i )) 2 F i ] 2E[V i E(V i F i ) F i ],

49 34 2. High quantiles estimation for long memory stochastic volatility models where E(V 2 i F i ) = E(V i F i ), E[(E(V i F i )) 2 F i ] = (E(V i F i )) 2, E[V i E(V i F i ) F i ] = E(V i F i )E(V i F i ) = (E(V i F i )) 2. Thus, E(M 2 i F i ) = E(V i F i ) (E(V i F i )) 2, (2..8) and recall that, We compute that E(V i F i ) lim u n P (Z > u n ) = lim u n P (Z i > un(+s) σ(x i ) ) P (Z > u n ) = ( + s) α σ α (X i ). E(M 2 i F i ) P (Z > u n ) = E(V i F i ) P (Z > u n ) ( ) 2 E(Vi F i ) P (Z > u n ) P (Z > u n ) E(V i F i ) P (Z > u n ) ( + s) α σ α (X i ). Thus, E(M 2 i F i ) ( + s) α σ α (X i )F Z (u n ), (2..9) consequently, using (2..9) and Breiman s Lemma (.5.7) n E(Mi 2 F i ) = i= nf (u n ) ( + s)α n n E(M 2 i F i ) ( + s) α F Z (u n ) nf (u n ) n (σ α F Z (u n ) (X i )) E(σ α (X i ))F Z (u n ) i= i= n (σ α (X i )) i= P ( + s) α. where lim n n i= (σα (X i )) n = E(σ α (X i )) follows directly from the Ergodic Theorem and the first condition in the martingale CLT is verified. We note that the sequence X i is ergodic, since X i = φ(ε i, ε i,..., ), where ε i are i.i.d. normal random variables.

50 2.. Tail empirical process with deterministic levels 35 For the second condition, ( ) M 2 i E nf (u n ) F { M i >ε nf (u n )} i = ([V nf (u n ) E i E(V i F i )] 2 { M i >ε nf (u n )} F i ) = 0 since M i is bounded and 2 and nf (u n ) as n, the indicator function returns a value 0 for n sufficiently large. Thus, the second condition in the martingale CLT is Recall that M n (s) = = n E ( ) Mi 2 P { Mi >ε} F i 0. i= Lemma 2..8 Define M n (s) = 2.., the process M n (s) is tight. Proof: n ( {Yi >u n(+s)} E[ {Yi >u n(+s)} F i ] ) nf Z (u n ) i= n (V i (s) E(V i (s) F i )) nf (u n ) i= nf (u n )M n (s). Under the conditions of Theorem Using Theorem 3.5 in Billingsley (968), in order to check tightness we have to verify the following condition: for s s 2 s 3, we have to bound E[ M n (s 2 ) M n (s ) 2 M n (s 3 ) M n (s 2 ) 2 ]. (2..0) By Hölder s inequality, E[ M n (s 2 ) M n (s ) 2 M n (s 3 ) M n (s 2 ) 2 ] [ E ( M n (s 2 ) M )] [ n (s ) 4 2 E ( M n (s 3 ) M )] n (s 2 ) 4 2,

51 36 2. High quantiles estimation for long memory stochastic volatility models where E( M n (s 2 ) M n (s ) 4 ) ( n 4 = (nf (u n )) E [V i (s 2 ) V i (s )] [E(V i (s 2 ) F i ) E(V i (s ) F i )]). 2 i= If we denote V i (s) E(V i (s) F i ) = X i (s) and X i (s 2 ) X i (s ) = X i, then, [V i (s 2 ) V i (s )] [E(V i (s 2 ) F i ) E(V i (s ) F i )] = X i (s 2 ) X i (s ) = X i. then, E( M n (s 2 ) M n (s ) 4 ) n = (nf (u n )) E( X i ) 4 2 = = (nf (u n )) 2 E( (nf (u n )) 2 ( (nf (u n )) 2 i= n i= X 4 i + n E X i 4 + i= ( ne( X 4 ) + n (nf (u n )) E( X 2 i 2 ) 2 i= n ( X i 2 X j 2 )) i,j=,i j n i,j=,i j n i,j=,i j E( X 2 i X 2 j )) E( X 2 i X 2 j ) ) (Stationarity). The first inequality is the standard inequality for martingales. (Hall, Heyde 980) Furthermore, E( X 4 ) CE(V (s 2 ) V (s )) 4 = CP (u n ( + s ) < Y < u n ( + s 2 )) for some constant C and note that P (u n ( + s ) < Y < u n ( + s 2 )) lim n F (u n ) Furthermore, if assumption A holds, = T (s ) T (s 2 ) T (s ) T (s 3 ). E( X 2 i X 2 j ) CE[V i (s 2 ) V i (s )] 2 [V j (s 2 ) V j (s )] 2

52 2.. Tail empirical process with deterministic levels 37 and so = CP (u n ( + s ) < Y i < u n ( + s 2 ), u n ( + s ) < Y j < u n ( + s 2 )) C i,j P (u n ( + s ) < Y i < u n ( + s 2 )) P (u n ( + s ) < Y j < u n ( + s 2 )), C i,j P (u n ( + s ) < Y i < u n ( + s 2 )) P (u n ( + s ) < Y j < u n ( + s 2 )) lim n F (u n ) F (u n ) = C i,j (T (s ) T (s 2 ))(T (s ) T (s 2 )) = C i,j (T (s ) T (s 2 )) 2 C i,j (T (s ) T (s 3 )) 2. Thus, E( M n (s 2 ) M n (s ) 4 ) = Analogously, n F (u n ) (T (s ) T (s 3 )) + n 2 ( n i,j=,i j C i,j ) ( n(n ) n F (u n ) (T (s ) T (s 3 )) + C n 2 n F (u n ) (T (s ) T (s 3 )) + C(T (s ) T (s 3 )) 2. (T (s ) T (s 3 )) 2 ) (T (s ) T (s 3 )) 2 Thus, E( M n (s 3 ) M n (s 2 ) 4 ) n F (u n ) (T (s ) T (s 3 )) + C(T (s ) T (s 3 )) 2. E[ M n (s ) M n (s 2 ) 2 M n (s 2 ) M n (s 3 ) 2 ] This guarantees tightness. n F (u n ) (T (s ) T (s 3 ))+C(T (s ) T (s 3 )) Weak convergence of long memory part R n The term R n is treated using Lemma.5.. Note that R n (s) = n where n G(X i ) i= G(X i ) = G(X i ) E(G(X i ))

53 38 2. High quantiles estimation for long memory stochastic volatility models and G(X i ) = F (u n ) E[ {Y i >u n (+s)} F i ] = F (u n ) E[ {σ(x i )Z i >u n (+s)} F i ]. Note that function G depends on n and s. However, R n (s) has a form similar to the expression in Lemma.5.. We have to compute a = E(G(X i )X i ) = E{ F (u n ) E[ {σ(x i )Z i >u n(+s)} F i ]X i } = F (u n ) E{E[ {σ(x i )Z i >u n(+s)}x i F i ]} = F (u n ) E( {σ(x i )Z i >u n(+s)}x i ) ( ) ( ) {σ(xi )Z = E X i >u n(+s)} {σ(xi )Z i = E E[X i >u n(+s)} i X i ] F (u n ) F (u n ) ( = E X i E[ ) {σ(x i )Z i >u n P (Z (+s)} i > X i ] = E (X u n(+s) X ) σ(x i ) i) i F (u n ) F (u n ) ( ) ( + s) α σ α (X i ) E X i = ( + s) α E(X iσ α (X i )) E(σ α (X i )) E(σ α (X i )). This suggests the following limiting result. The proof is not given here, see Kulik and Soulier (20). We note that their proof for the long memory part R n (s) does not involve SV assumption and thus it holds for EGARCH model as well. Lemma 2..9 Under the conditions of Theorem 2.., n d /2 d(2d+) R n (s) T (s) E(X iσ α (X i )) N(0, ) E(σ α (X i )) We note that the limiting process is degenerate, i.e. it is a standard normal random variable, multiplied by a deterministic function.

54 2.2. Tail empirical process with random levels Tail empirical process with random levels Recall the definition (2..). We replace there n F (u n ) with k n and u n with Y n k:n, the (k + )th largest observation, in order to define the following process: ˆT n (s) = k n n j= {Yj >Y n k n:n(+s)}, T (s) = ( + s) γ, where γ = α, ê n(s) = ˆT n (s) T (s). It is assumed that k n and as n. The process ê n(s) is the modification of the process e n considered in Section 2.: instead of the deterministic level u n, we have a random level. We will show below that this process has different convergence properties than e n. The statement of the Theorem 2.. can be re-written as a n e n (s) = a n ( T n (s) T n (s)) Ψ(s), where the scaling a n and the limiting process Ψ(s) have two different forms. We will argue below that a n ê n(s) Ψ(s) T (s)ψ(0). First, we have to replace the centering T n (s) with T (s). This is allowed under the so-called second order regular variation condition (see Kulik and Soulier (20) for details). In particular, if F (x) = x α + x β, x >, β > α, then the second order regular variation holds. Lemma 2.2. From Theorem 2.. and under second order regular variation,

55 40 2. High quantiles estimation for long memory stochastic volatility models In the short memory case, if a n = n F (u n ) = k n, Ψ(s) = W (T (s)) then, kn ê n(s) W (T (s)) T (s)w () = B(T (s)), (2.2.) where B( ) is the Brownian bridge. In the long memory case, if a n = const.n /2 d, Ψ(s) = E[Xσα (X)] E[σ α (X)] T (s)n(0, ), then, a n ê n(s) E[Xσα (X)] E[σ α (X)] T (s)n(0, ) E[Xσα (X)] T (s)n(0, ) = 0. E[σ α (X)] Therefore, in the case of random levels, the long memory scaling is too big. It is possible to show that (2.2.) holds for arbitrary d (0, /2), see Kulik and Soulier (20). Generally, for any choice of d (0, /2), we will have kn ê n(s) B(T (s)). (2.2.2) Lemma (Vervaat s Lemma) Assume that s n is a sequence of constants, and x n ( ), ρ( ) are deterministic functions and x n ( ), ρ ( ) are the generated inverse function, If s n (x n (s) ρ(s)) Y (s), then s n (x n (s) ρ (s)) (ρ (s)) Y (ρ (s)) Proof of lemma 2.2.: Define δ n = Y n kn:n u n u n. We have T n (s + δ n ( + s)) = ˆT n (s), and ê n(s) = T n (s + δ n ( + s)) T (s) = e n (s + δ n ( + s)) + T n (s + δ n ( + s)) T (s). Under second order regular variation we can replace T n with T. Thus, heuristically, ê n(s) e n (s + δ n ( + s)) + T (s + δ n ( + s)) T (s).

56 2.2. Tail empirical process with random levels 4 The first term in the Taylor s expansion of T (s + δ n ( + s)) T (s) is T (s)δ n ( + s) = αt (s)δ n = T (0)T (s)δ n. Thus, we expect ê n(s) ê n (s + δ( + s)) + T (0)T (s)δ n. (2.2.3) Now, we want to show that kn T (0)δ n d W (). Using Vervaat s Lemma 2.2.2, if we take x n (s) = T n (s), ρ(s) = T (s) and s n = k n = n F (u n ) and we have from Theorem 2.., for the weakly dependence case, kn ( T n (s) T (s)) d Ψ(s) = W (T (s)). Then kn ( T n (s) T (s)) d (T (s)) W (T T (s)) = (T (s)) W (s). Take s =, we have kn ( T n () T ()) d (T ()) W (). Note that T (s) = s γ, (T (s)) = γs γ, Thus (T ()) = γ = /α. We conclude kn ( T n () T ()) d /αw () and α = T (0). Equivalently, kn T (0)( T n () T ()) d W (). Note also that T n () = Y n k n :n u n = δ n

57 42 2. High quantiles estimation for long memory stochastic volatility models and T () = 0. Thus, we conclude kn T (0)δ n d W (), which also ensures that δ n 0 as k n. Therefore, kn ê n(s) k n e n (s) + (k n )T (0)T (s)δ n Ψ(s) T (s)ψ(0). and the proof is complete. 2.3 Asymptotic normality of the Hill estimator Using results from Section 2.2, in this section we establish asymptotic normality of Hill estimator based on a stationary EGARCH model. Let Y :n Y 2:n... Y n :n Y n:n be the increasing order statistics of a stationary sequence Y,..., Y n. The Hill estimator is defined as H kn,n = ˆγ n = k n k n j= log Y n j+:n. Y n kn :n The tail empirical distribution ˆT n (s) and the tail empirical process ê n(s) play a crucial role in the study the Hill estimator. Indeed: Lemma 2.3. We have and 0 α = γ = ˆγ n = B(T (s)) + s where B is a standard Brownian bridge. 0 0 ds d = γ T (s) ds, (2.3.) + s ˆT n (s) ds (2.3.2) + s 0 B(t) t dt, (2.3.3)

58 2.3. Asymptotic normality of the Hill estimator 43 Proof: For. notice that T (s). 0 + s ds = ( + s) γ ds = ( + s) γ ds 0 + s 0 = γ( + s) γ = γ( + 0) γ = γ. For 2, define ˆT n(s) = k n n j= {Y j >Y n kn:n (s)}, where s. 0 ˆT n (s) + s ds = = k n n ˆT n(s) ds = s j= 0 k n n j= {Yj >sy n k:n } s ds {Yj >sy n kn:n } ds. (2.3.4) s Note that {Yj >Y n k = Y n:n(s)} and s, Y { j Y >s} j must be bigger than Y n kn:n. n k n:n This means that Y j = Y n kn+:n or Y j = Y n kn+2:n,, or Y j = Y n:n. Therefore, the sum in (2.3.4) becomes k n n j= Y j Y n kn:n s ds = k n k n j= Y n j+:n Y n kn:n s ds = k n k n j= log( Y n j+:n ) = ˆγ n. Y n kn :n For 3, using the substitution rule, by letting t = T (s) = ( + s) γ, then dt = ( )( + γ s) γ ds, so ds = ( γ)( + s) γ +s dt = ( γ) dt. Therefore t 0 B(T (s)) + s ds = 0 B(t)( γ) t dt = ( γ) 0 B(t) t dt d = γ 0 B(t) t (Since Brownian motion increments are normally distributed, therefore the integral is symmetric.) From (2.3.)-(2.3.2) and (2.2.2) we conclude kn (ˆγ n γ) = k n 0 ê n(s) + s ds d 0 dt. B(T (S)) + s ds = d B(t) γ dt. 0 t We note that the latter integral is a standard normal random variable. Therefore, we obtain the following result.

59 44 2. High quantiles estimation for long memory stochastic volatility models Recall that k = k n as n. We note also that k n is the user-chosen number of extreme observations. Theorem Under the assumptions of Theorem 2.. and the appropriate secondorder condition, k n (ˆγ n γ) converges weakly to the centered Gaussian distribution with variance γ 2.

60 Chapter 3 Numerical experiments 3. Estimators for the tail index α The tail index α dominates the asymptotic behaviour of a distribution and indicates the heaviness of the tail. There are lots of estimators for α, we particularly focus on the POT estimator, Hill estimator and Pickands estimator. 3.. Peaks Over Threshold (POT) Maximum Likelihood (ML) estimator The POT method analyzes exceedances over a given high thresholds. Suppose {Y i, < i < } is an i.i.d. sequence and u is a threshold. Define the exceedance times {τ r, r } by τ = inf{j : Y j > u}, τ 2 = inf{j τ : Y j > u},. τ r = inf{j τ r : Y j > u}, 45

61 46 3. Numerical experiments The sequence {Y τr, r } are called the exceedances. If {Y i, < i < } is i.i.d. with common distribution F, then {Y τr, r 0} is also i.i.d., and for F [u] (x) = P (Y τr > x) = P (Y > x Y > u), P (Y >x,y >u) P (Y >x) F [u] = P (Y >u) (x) = = F (x) if x > u, P (Y >u) F (u) if x < u. Assume that P (Y > x) = F Y (x) = x α L(x), then for x > u, P (Y > xu) P (Y > u) = F Y (xu) F Y (u) = (xu) α L(ux) u α L(u) Assume that Y is Pareto, then we have, x α as u. The maximum-likelihood estimator of α P (Y > x) = x α, x >, F [u] x α = ( x u (x) = α u ) α if x > u, if x < u. Consider likelihood function for Y τ,..., Y τr : Taking logarithms yields, L = r α i= can be found as follows: ( ) (α+) Yτi u log L = r log α (α + ) α (P OT ) = ˆγ r,n = r r log i= ( Yτi r log i= ( Yτi u u ) ) (3..) 3..2 Hill estimator Consider equation 3.., if we choose u = Y (kn+) and Y () > u, Y (2) > u,, Y (kn) > u, then we have the Hill Estimator of /α based on k n upper-order statistics: ˆγ (Hill) k n,n = k n k n i= log Y (i) Y (kn+). (3..2)

62 3.. Estimators for the tail index α 47 There is a theoretical formula which gives an optimal choice of k n. However, that formula is useless for practical purposes. In practice, one looks at stability region of the Hill plot and then one picks the biggest possible k n to have the shortest confidence interval. For example, on Figure 3.3 I would choose k n = Pickands estimator Suppose Z i, i is i.i.d. with common distribution F. The Pickands estimator of γ = α uses differences of quantiles and is based on using three upper-order statistics Z (k), Z (2k), Z (4k), from a sample of size n. The estimator is defined as (recall that k = k n depends on n) ( ) ( ) (P ickands) Z(k) Z (2k) ˆγ k,n = log. (3..3) log 2 Z (2k) Z (4k) Using the result in Resnick(2007), p.93, where a n = n /α, we have Z ([k/y]) a n/k yγ, 0 y <, a n/k γ Z (k) Z (2k) Z (2k) Z (4k) = P (Z (k) a n/k ) a n/k (Z (2k) a n/k ) a n/k (Z (2k) a n/k ) a n/k (Z (4k) a n/k ) a n/k ( 0 γ (( 2 )γ ) γ (( 2 )γ ) γ (( 4 )γ ) = 2 γ. By the continuous mapping theorem and taking logarithms yields ( ) Z(k) Z (2k) P log log 2 γ Z (2k) Z (4k) ) and dividing by log 2 ( ) ( log Z ) (k) Z (2k) P γ log 2 Z (2k) Z (4k)

63 48 3. Numerical experiments 3.2 Simulations: POT method for different choices of threshold u We simulate Y i = e 0.2X i Z i, i =,..., n, where Z i are independent Pareto with parameter α = 2 and X i is ARFIMA with standard normal innovations and d = 0.2. For a sample of size n = 000, we obtain Y,..., Y 000 and we estimate α using the maximum likelihood estimator based on exceedences over threshold u, α = r r log i= ( Yτi where r = number of exceedences over u. For the deterministic levels, we choose u = max(y,..., Y 000 )/2 based on the first sample, and then keep this value for Monte Carlo simulation. Also in simulation, we could let u = 5, since we assumed α = 2. Indeed, in a sample of size n, we expect np (Y > u ) values which are bigger than u. We have u ) np (Y > u ) nu 2 = k n, so that u = n k n. Consequently, for n = 000 and k n = 200, we obtain u = 5. We choose u = Y (n kn,n) for random levels. The following simulated results indicate that when u = Y (n kn,n), the estimator is better. Sample Mean of /ˆα when u = max(y,..., Y 000 )/2: ; Sample Mean of /ˆα when u = Y (n kn,n): ; Sample standard deviation of /ˆα when u = max(y,..., Y 000 )/2: ; Sample standard deviation of /ˆα when u = Y (n kn,n):

64 3.2. Simulations: POT method for different choices of threshold u 49 Histogram of estim Histogram of estim_rando Frequency 0 50 Frequency estim estim_random Normal Q Q Plot Normal Q Q Plot Sample Quantiles Sample Quantiles Theoretical Quantiles Theoretical Quantiles Figure 3.: Statistics for POT simulation. The graphs illustrate empirical distribution of ˆα,..., ˆα M based on M = 000 Monte Carlo runs. The Figure 3. illustrates the above results visually. The empirical distribution of ˆα,..., ˆα 000 at random levels is symmetric and has a mean equal to which is very close to the true value 0.5. The QQ plot also shows that it is normally distributed. The Histogram and QQ plot of the estimator at deterministic levels are not appropriate.