the shanghai stock exchange: statistical properties … · the shanghai stock exchange i:...

THE SHANGHAI STOCK EXCHANGE: STATISTICALPROPERTIES AND SIMULATION

by

Xiongzhi Chen

Master of Science in MathematicsSichuan University, China

Submitted to the Department of Mathematicsin Partial Ful�llment of the Degree of

Master of Arts in Mathematics

at the

University of HawaiiApril, 2009

THESIS COMMITTEE:

Thomas Ramsey

George Wilkens

ii

We certify that we have read this document and that, in our opinion, it is satisfactoryin scope and quality as a thesis for the degree of Master of Arts in Mathematics

THESIS COMMITTEE

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� � � � � � � � � � � � � � -

� � � � � � � � � � � � � � �

iii

THE SHANGHAI STOCK EXCHANGE: STATISTICAL

PROPERTIES AND SIMULATION BY GENERALIZED HYPERBOLICDIFFUSION

by

Xiongzhi Chen

Submitted to the Department of Mathematics on April 21, 2009

in partial ful�llment of the requirements for the degree of

Master of Arts in Mathematics

Abstract

Empirical data of asset returns, including that of stock market index series, alwaysexhibit non-stationarity, heavy tails, dependence structures, and other stylized featuresas well, among which the tail indices have wide applications in assessing the distributionof outliers and the Hurst index reveals roughly how strong and how long the correlationstructure retains itself.While estimating these statistics is usually hard, simulating �nancial markets by

continuous-time models is even more challenging since the proposed model should beable to at least generate the empirical distribution, capture the intrinsic denpendencestructure, and simulate the volatilities, of the �nancial data, under appropriate statisticalsettings.This thesis contains two independent papers that summarize the author�s study of the

statistical properties of the returns series of the daily price index series of the ShanghaiStock Exchange (SSE)1 for trading days between 01/01/1991 and 14/11/2007, and theauthor�s simulation of the detrended logarithmic index series by the ergodic, generalizedhyperbolic (GH) di¤usion whose invariant density is the generalized hyperbolic (GH)density.More speci�cally, the �rst paper herein gives estimates of the right tail index, the left

tail index, and Hurst index for the returns series2 of the daily SSE index, all of whichare consistent with what most researchers obtained for stock market index series andtend to argue that extreme care should be taken when making assumptions about theunderlying stochastic process that generates the stock price index series. These estimatesare obtained by the simplest method � the traditional ordinary least squares (OLS)method, to avoid unnecessary or even wrong assumptions on stationarity or dependencestructures of the index series, but some comparative estimates via di¤erent methods arealso provided.

1The "SSE index" will speci�cally denote the "daily price index series of the SSE".2The "returns series" will denote speci�cally the "returns series of the daily price index series of the

SSE ".

iv

The second paper is devoted to testing whether an ergodic GH di¤usion process cansimulate well the linearly detrended logarithmic SSE index series.3 The ergodic GH dif-fusion model was proposed by several researchers in the mid 1990s and has successufullysimulated some stock price index series. Its main advantages are: its capability to gen-erate the generalized autoregressive conditional heteroscadatistic (GARCH) e¤ect thatis compatible with the correlation structure of most �nancial time series, and its capa-bility to approximate the emprical densities of the �nancial time series by the model�sinvariant density which is a generalized hyperbolic (GH) density.This second paper decribes for the detrended index an ergodic GH di¤usion whose in-

variant density is proportional to the GH density and whose parameters were estimatedby Markov chain Monte Carlo (MCMC) methods. Six possible models were found sincesix independent, convergent Markov chains are obtained by the Monte Carlo standarderror (MCSE) convergence diagnostic. Unfortunately the Gelman-Rubin convergencediagnostic failed to converge to one even after 350; 000 iterations of the Metroplolis-Hastings algorithm, since each of the six chains converged to its own stationary distribu-tions at around 50; 000 iterations, (which might be the consequence of trapped iterationsfor the chains in the simulation).After six Monte Carlo estimates of the unknown parameters were obtained, we simu-

lated the detrended index by the six competing models. However, none of them success-fully generated paths that match the data well. Also, a uniform residual test was carriedout for only four of the six possible models due to the computational capability of Math-ematica and Matlab. The test results statistically reject all four selected models, thusrejecting the null hypothesis that the underlying stochastic process for the detrendedindex is an ergodic GH di¤usion.One reason for such a failure might be that the invariant density is inappropriate

for the empirical density of the detrended index, since its empirical density is tri-modalwhereas the GH density is unimodal.

3The "detrended index" will speci�cally denote the "linearly detrended logarithmic SSE index series".

v

Acknowledgements

I am deeply indebted to my adviser, Professor Thomas Ramsey, whose constant guid-ance, inspiring suggestions and supports motivated my study on the statistical propertiesof �nancial markets and on the simulation of �nancial time series by continuous-timemodels. Without his help, this thesis would not have been possible.

I would like to thank Professor George Wilkens for his help on my study in the theoryof stochastic control, and, Professor James B. Nation, Professor Wayne Smith, ProfessorWilliam Lampe for their decisive support to my pursuit of further academic opportunitiesin the near future.

I am grateful to other faculty members and stu¤, particularly, the departmental sec-retaries, for their help that made my life and study here at Hawaii much more easier andenjoyable. I am also grateful to fellow graduate students in the department, particularly,to Shashi Bhushan Singh and Gretel Sia for their kind help.

Finally, I would like to express my appreciation to my family, especially my wife, mybrother, and to my friends, especially Jin Li, whose love and contribution paved the wayfor my dream in mathematics.

Please notice that the two papers incorporated into this thesis have their own pagingschemes. The next page will be the starting page of the �rst paper, and the second startsright after the ending page of the �rst.Please go to the next page for the start of the �rst paper and then to the second paper

after around 23 pages .......

THE SHANGHAI STOCK EXCHANGE I: STATISTICALPROPERTIES

XIONGZHI CHEN

Abstract. This paper studies the statistical properties, mainly, the tail behaviour,sknewness, correlation structure, of returns series of the SSE index for trading daysfrom 01/01/1991 to 14/11/2007, and gives their corresponding statistical estimates.The estimated right tail index is around 4:6, that of the left tail index around 2:19

and the Hurst index around 0:61, for the returns series. These statistics are obtainedby the simplest estimation method: linear regression via ordinary least squres (OLS),to avoid unnecessary or even wrong assumptions about the stationarity or dependencestructures of the SSE index.Results provided herein comply with what most researchers have found for stock

market indices (see Cont01 [5] and Mills99 [21]). The tail indices show that statisticallythe underlying stochastic process that genererates the the SSE index can be assumedto be of �nite variance, whereas the Hurst index shows that the this underlying processis not a random walk but has long-range dependence.

Key words and phrases. Hill�s Estimator, Hurst Index, Lo�s Modi�ed R/S Statistic, R/S Statistic,R/S Analysis, Short/Long-range Dependence, Skewness, Tail Index,

1

2 XIONGZHI CHEN

Contents

List of Figures 31. Introduction 42. Tail Index 52.1. Hill�s Estimator 82.2. Linear Regressor 102.3. Mandelbrot�s Visual Check 133. Kurtosis and Skewness 144. Dependence 164.1. Hurst�s R/S Statistic 164.2. Mandelbrot�s R/S Analysis 194.3. Direct Regression 225. Conclusions 23References 23

SSE RRETUNS: TAIL INDICES, SKNEWNESS, HURST INDEX 3

List of Figures

1 Original SSE Price Index 6

2 Returns Series 6

3 Empirical PDF against Normal 10

4 L-L Plot of Right Tail 11

5 L-L Plot of Left Tail 11

6 Regressed L-L Right Tail 12

7 Regressed L-L Left Tail 12

8 Sample Variance 13

9 Asymptotic Sample 4th Moment 13

10 Sample 5th Absolute Moment 14

11 Skewness 15

12 Zoom-in Kernel Density Estimate of EPDF 15

13 Classic R/S Statistics 17

14 L-L Plot of Classic R/S Statistic 17

15 Regressed R/S Statistic for Small Sample Size 18

16 Regressed R/S Statistic for Larger Sample Size 18

17 Pox Plot of R/S Analysis 19

18 Autocorrelation till Lag Order 1000 20

19 Lo�s Modi�ed R/S Statistic 22

20 L-L Plot of Absolute Average Returns 23

4 XIONGZHI CHEN

1. Introduction

It�s widely acknowledged that most asset returns series have the following main styl-ized features revealed by empirical studies: leptokurtosis1, skewness, and power-decayingcorrelations in absolute returns (see Cont01 [5], Mandelbrot01a [18]).As a reference for interested readers, the eleven such features of asset returns given in

Cont01 [5] are listed below:

F1: Absence of autocorrelations: (linear) autocorrelations of asset returns areoften insigni�cant, except for very small intraday time scales (about 20 minutes)for which microstructure e¤ects come into play.

F2: Heavy tails: the (unconditional) distribution of returns seems to display apower-law or Pareto-like tails, with a tail index which is �nite, higher than twobut less than �ve for most data sets studied. In particular this excludes stablelaws with in�nite variance and the normal distribution. However the preciseform of the tails is di¢ cult to determine.

F3: Gain/loss asymmetry: one observes large drawdowns in stock prices andstock index values but not equally large upward movements.

F4: Aggregational Gaussianity: as one increases the time scale �t over whichreturns are calculated, their distribution looks more and more like a normaldistribution. In particular, the shape of the distribution is not the same atdi¤erent time scales.

F5: Intermittency: returns display, at any time scale, a high degree of variability.This is quanti�ed by the presence of irregular bursts in time series of a widevariety of volatility estimators.

F6: Volatility clustering: di¤erent measures of volatility display a positive auto-correlation over several days, which quanti�es the fact that high-volatility eventstend to cluster in time.

F7: Conditional heavy tails: even after correcting returns for volatility clus-tering (e.g., via GARCH-type models), the residual time series exhibit heavytails. However, the tails are less heavy than in the unconditional distribution ofreturns.

F8: Slow decay of autocorrelation in absolute returns2: the autocorrelationfunction of absolute returns decays slowly as a function of the time lag, roughlyas a power law with an exponent � 2 [0:2; 0:4] : This is sometimes interpreted asa sign of long-range dependence.

F9: Leverage e¤ect: most measures of volatility of an asset are negatively cor-related with the returns of that asset.

F10: Volume/volatility correlation: trading volume is correlated with all mea-sures of volatility.

F11: Asymmetry in time scales: coarse-grained measures of volatility predict�ne-scale volatility better than the other way.

1A leptokurtic distribution is one that is peaked near the center and has fat tails2Absolute returns are the absolute values of the returns


These characteristics are extremely helpful for researchers who seek to understandasset returns3 statistically and simulate them e¢ ciently, but they seem to have frus-trated any attempts to build mathematically and computationally feasible continuous-time models that are able to incorporate almost all of them.To verify and better understand these empirical features and construct good simu-

lations to stock market index series, we take the returns series of the SSE index4 andstudy its empirical statistical properties, such as sknewness, kurtosis, tail index, andlong-term dependence structure. These statistics provide statistical support for simulat-ing this returns series by a generalized hyperbolic (GH) di¤usion model (for a hyperbolicdi¤usion model for stock prices, see Bibby97 [2] and Rydberg99 [27]). In order to avoidunnecessary, even wrong assumptions about this returns series, all statistical estimatesherein are obtained nonparametrically by linear regression with ordinary least squares(OLS). We also provide comparisons with other popular estimation techniques of thesestatistics that would be valid under the independent and identically distributed (i.i.d.)assumption. However no non-parametric check on the stationarity of this return series isdone because of the lack of assumptions and the reluctance of the author to impose i.i.dor a normality assumption on the background noise of this series as proposed by most re-searchers, even though it is widely agreed that most �nancial time series are not strictlystationary or even non-covariance stationary (Mills95 [20], Mills99 [21], Loretan94 [11]).The statistical properties estimated in this paper comply with their corresponding

stylized empirical features and tend to argue against the common tradition of assumingstationarity for returns series of stock price indices and against the possibly inappropriatestable-distribution assumption for �nancial time series.

2. Tail Index

Figure 1 and Figure 2 depict the SSE index series ~P = fPt : 1 � t � 4129g and itsreturn series ~R = fRt : 1 � t � 4129g, where the horizontal axis denotes the number oftrading days from Jan 1, 1991 and the vertical axis the price index and returns of theprice index, respectively. It is clear that the original stock index series has an increasingtrend for most part of the period considered, while the log-return series looks more liketrendless noise but displays more volatility during the period roughly from Jan 1991 toAug 2006.Since under most circumstances only observations of a stochastic process are available,

it is reasonable to start from the empirical cumulative distribution function (ecdf) of aprocess to obtain some of the required statistics. Under the traditional assumption thatall random variables of the underlying stochastic process that generate the observationsare identically distributed, the ecdf is even more crucial to statistical inference about theprocess. Studies show that the epdf of most �nancial time series display power-decaying

3Here "asset returns" denotes "the asset returns series", and "return" means conventionally "logreturn".

4Unless otherwise speci�ed, notationally we will not distinguish a random process X = fXt; t 2 Igfrom an observation X = fXt; t 2 I0g ; I0 � I of X; where I is a nonempty index set and I0 is usuallynonempty, that is, fXt; t 2 I0g also denotes the observation X.

6 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 45000

1000

2000

3000

4000

5000

6000

7000

Number of trading days from 01/01/1991

Pri

ce

In

de

x

Series of Daily Price Index of Shanghai Stock Exchange

Figure 1. Original SSE Price Index

0 500 1000 1500 2000 2500 3000 3500 4000 45000.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Trading Days from 01/01/1991

Re

turn

s

Returns Series of the Daily SSE Index

Figure 2. Returns Series

or semi-heavy tails (see, for example, Cont01 [5], Mantegal95 [22], Loretan [11]) withtail index � ranging between 0 and 5 (see, for example, Mills99 [21]).Given a stochastic process5 X = fXt; t 2 Tg de�ned on some underlying probability

space (;F ; P ) such that all Xt; t 2 T are identically distributed (i.d.)6, the tail index� of X is usually de�ned as (see, Samorodnistky94 [28])

� = supn� > 0 : EP

�jXtj�

�<1

o5For more details on the de�nition of stochastic process, please see Karatzas98[10]6For most �nancial time series, the random variables of the underlying stochastic process that gener-

ates the observation is not identically distributed.


where EP denotes the mathematical expectation with respect to the probability measureP . Tail index is extremely important in risk management since it helps measure theprobability of outliers.Several methods for tail index estimation have been proposed (see, for example, Hill75

[8], Loretan94 [11], Samorodnitsky94 [28], Taqqu97 [30]), and their properties and perfor-mances have been compared (see, Taqqu97 [30], McCulloch [19])). But it seems that theclassical Hill�s estimator, the linear regression method (based on ordinary least squares(OLS) ) and maximum likelihood estimation (MLE) perform better in an overall fashionthan the others (It should be noted that for stable distributions Hill�s estimator behavessomewhat badly ( see, Taqqu97 [30], McCulloch [19] ). Based on these results, we basi-cally base our estimation of the tail index � on linear regression by OLS but also providethe estimate obtained by the modi�ed Hill�s estimator introduced in Loretan94 [11].Before estimating the tail index for the ecdf of returns series ~R = fRt : 1 � t � 4129g

(of P ), we provide a slightly di¤erent version of the de�nition for the generalized tailindex (compared to Loretan94 [11]):

De�nition 1. Let X = fXt : t 2 Tg be a stochastic process such that Xt; t 2 T isidentically distributed (i.d.) with F (x) = P (Xt0 < x) where t0 2 T is �xed. If

limx!1

P (Xt0 > x) = Cx�� (1 +BR (x)) ; x > 0 (2.1)

holds for some constants � > 0; C and for some BR (x) with

lim�!1

BR (x) = 0

Then � is called the right tail index of X:

The left tail index � is de�ned similarly. The above de�nition actually relaxes the i.i.d.assumption introduced in Loretan94 [11], Mills95 [20] and such relaxation of assumptionsmay be important for analyzing returns series, in that for most �nancial stochasticprocesses X = fXt : t 2 Tg the random variables Xt; t 2 T are not i.i.d or even i.d..Thus in practice we would have to content ourselves with the following interpretationof what is meant by saying "the empirical distributions of a stochastic process havingheavy tails". ( For details on how to construct stochastic process from a consistentfamily of �nite dimensional marginal distributions, please see Karatzas98 [10] for thethe Kolmogorov consistency theorem):

De�nition 2. Let X = fXt : t 2 Tg be a stochastic process with underlying probabilityspace (;F ; P ) and let

ON = fXti (!i) : ti 2 T; !i 2 ; 1 � i � Ng ; N 2 Nbe a discrete observation of X: We say that the epdf of X has right tail index � if andonly if

limx!1

�limN!1

~PN

�[N

i=1fXti (!i) > xg

��= Cx� (1 +BR (x)) ; x > 0 (2.2)

holds for any such observation, for some constants � > 0; C 2 R and for some BR (x) 2R with

limx;N!1

BR (x) = 0

8 XIONGZHI CHEN

where ~PN denotes the empirical density obtained from the observation ON

The above de�nition describes more realistically what we see and do when plottingthe empirical cdf of samples by smoothing techniques, and it is almost equivalent to theassumption that the process X is ergodic.

2.1. Hill�s Estimator. In econometrics, the most popular estimator of the constants Cand � (see, Hill75 [8]) in (2.1) are obtained by using rank statistics of the observationsfXi : 1 � i � Ng, that is, if

�X(i); 1 � i � N

are the order statistics of fXi : 1 � i � Ng

such that X(1) � X(2) � � � � � X(N) then

� =

0@s�1 sXj=1

logX(N�j+1) � logX(N�s)

1A�1 (2.3)

=

0@s�1 sXj=1

�logX(N�j+1) � logX(N�s)

�1A�1is the right tail index estimator for fXt : t 2 Tg (assuming that all Xt; t 2 T are theidentically distributed). Further, the estimator for C is given by

C =s

NX �(N�s) (2.4)

Empirical study shows that the tails of ecdf of asset returns are not asymptotically soheavy as those of stable distributions but mostly are power-law decaying or semi-heavy.McCulloch97 [19] has pointed that the Hill�s estimator is usually up-biased (in that theestimate can be around 2 even for stable distributions with tail index � = 1:65) andbehaves badly for stable distributions with tail index � close to 2, and that an estimate� > 2 obtained by Hill�s estimator only rejects the joint hypothesis that the underlyinginnovations7 are i.i.d. and have stable distributions.Due to the appeal of the Hill�s estimator, we still carried the estimate for � by using

(2.3), (2.4) and followed the suggestions from Philips96 [24] that, according to minimizingthe mean squared error (MSE) of the limiting distribution of �; we should set

s (N) =h�N2=3

i(2.5)

while � is adaptively estimated by

� =

�� 1p2Ns2 (�1 � �2)��2=3 (2.6)

and �1; �2 are preliminary estimates of � using truncates s1 = [N�] 8; s2 = [N � ] with0 < � < 2=3 < � < 1 and � = 0:6; � = 0:9 for the actual implementation. Note thatthese estimates are for the right or the left tail of the distribution respectively.The most devastating shortcoming of the above method is that it is based on the i.i.d.

assumption on the innovations of the sample. Since su¢ cient evidence show that most

7"Innovations" is used by several authors to mean the "underlying noise that contaminates the sampleor observation".

8[x] ; x 2 R denotes the maximal interger not exceeding x


returns series are not i.i.d., we can assume that this is also true for ~R and consequentlyexpect that the estimator � will be bigger than 2.Hereunder we provide the detailed estimation procedure for the Hill�s estimator for

~R: by setting s1 =�41290:6

�= 147, s2 =

�41290:9

�= 1795, we have

�1 =

240@s�11 s1Xj=1

logX(N�j+1)

1A� logX�1(N�s1)

35�1

=

P147j=1 logX(4129�j+1)

147� logX(4129�147)

!�1

=

�1237:917

147� 8:1118

��1= 3:2320

and

�2 =

0@s�12 s2Xj=1

log�X(N�j+1) � logX(N�s1)

�1A�1

=

P1795j=1 logX(4129�j+1)

1795� logX(4129�1795)

!�1

=

�13473:64

1795� 7:1903

��1= 3:1655

which means

� =

��3:2320p2

4129

1795(3:2320� 3:1655)

��2=3= 0:4963

and

s (N) =h0:4963 � (4129)2=3

i= [1366:1] = 1366

Thus

� =

0@s�1 sXj=1

logX(N�j+1) � logX(N�s)

1A�1

=

P1366j=1 logX(4139�j+1)

1366� logX(4129�1366)

!�1

=

�10366:64

1366� 7:2958

��1= 3:4100

and

C =s

NX �(N�s) =

1366

4129(7:2958)3:4100 = 290: 19

10 XIONGZHI CHEN

0.4 0.2 0 0.2 0.4 0.6 0.80

3

6

9

12

15

18

21

24

27

29

Returns

Pro

ba

bility

Empirical Pdf against Nomal Density

Empirical PDFN(0.00091,0.0289)

Figure 3. Empirical PDF against Normal

Consequently, assuming ~Rt; 1 � t � 4139 are identically distributed (which might alsonot be true) and the right tail index is constant in the sense that the sample size for ~Ris large enough to stablize the estimate (see, Mills99 [21]), we have asymptotically

P (R1 > x) = 290: 193:4100x�3:4100 (1 + �R (x))

where limx!+1 �R (x) = 0; i.e.,

logP (X > x) � 19: 337� 3:41 log x

Thus the Hill estimator of the right tail index is 3:41.

2.2. Linear Regressor. Now we will use visual examination and linear regression toestimate the tail index � without assumptions on the underlying innovations for thereturn series ~R. The following Figure 3 shows the comparison between the empiricalprobability density function (epdf) obtained by a kernel estimator and the normal densitywhose mean and standard deviation are the same as those of ~R,which shows clearlythat the epdf has heavier tail, bigger kurtosis, compared to that of a normal densityN (0:00091; 0:00289).We also provide in Figure 4 and Figure 5 the log-log plots of both right and left

tails. A visual check on these two �gures reveals that there are some points for xright beyond which logFN (x) and log (1� FN (x)) are strongly linear against log x,respectively. These points are x � 0:024 for the left tail and x � 0:03 for the right tail,where FN (�) is the empirical cdf of ~R and N is the size of ~R.Since all we want are the asymptotic properties (assuming that a sample sizeN = 4129

is su¢ ciently large for the estimation of asymptotic statistics), we can safely choosex � 0:024 and x � 0:03 respectively and then use linear regression by OLS to estimatethe right and left tail index. The following Figure 6 and Figure 7 display where thelog-linearity are the strongest for su¢ ciently large jxj with reference linear regressors.Both of the �gures show that both ends of the two tails diverge a lot from their linear


14 12 10 8 6 4 2 08

7

6

5

4

3

2

1

0

log(t),t>0

log(1

P(X>

t))

Right Loglog Tail of Daily Return Proces(Shanghai Stock Exchange:01/01/1991~14/11/2007)

Figure 4. L-L Plot of Right Tail

12 10 8 6 4 2 08

7

6

5

4

3

2

1

0

log(x),x>0

log(P

(X<

x))

Left Loglog Tail of Daily Return Process(Shanghai Stock Exchange: 01/01/1991~14/11/2007)

Figure 5. L-L Plot of Left Tail

regressors This is the consequence of a few outliers in the sample ~R due to relativelylarge gains or losses that happened approximately in Oct 2007.The regression equation for the right tail is

ln (1� FN (x)) = �0:9766� 4:670 lnx; x � 0:030061and that of left tail is

lnFN (x) = lnP (X < �x) = �10:370� 2:192 lnx; x � 0:024045Consequently, the OLS estimate for the right tail index is

�R = 4:670

and that for left tail index is�L = 2:192

12 XIONGZHI CHEN

1.6 1.52 1.44 1.36 1.28 1.2 1.12 1.04 0.96 0.88 0.8 0.72 0.64 0.56 0.48 0.4 0.32 0.24 0.29

8.5

8

7.5

7

6.5

6

5.5

5

4.5

4

3.5

3

2.5

2LogLog Plot of Right Tail with Line of Linear Regression

log(x), x>=0.03

log

(P(X

>x))

Loglog Plot of Right TailLine of Regressiony=9.764.67*log(x)

Figure 6. Regressed L-L Right Tail

4 3.8 3.6 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 110

9.2

8.4

7.6

6.8

6

5.2

4.4

3.6

2.8

2

log(x),x>0.024

log(

P(X

<x)

)

Loglog Plot of Left Tail with Reference Line of Linear Regression

Loglog Plot of Left TailLine of Regressiony= 10.372.19*log(x)

Figure 7. Regressed L-L Left Tail

These indices are consistent with both the fact that usually the ecdf of stock returnseries are not stable distributions but Paretian type with power-decaying tails, andthe abundant evidence against assuming in�nite variance for stock return series whoseinnovations are identically distributed (see, Mills99 [21]). They are also in accordancewith the tail range most researcher have found empirically and the observation that theecdf has a relatively heavier left tail than the right tail (see Mills99 [21]). (The reasonwhy the linear regression estimator is smaller than Hill�s estimator is unknown to meand will be examined in future work.)


0 500 1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5x 103

Sample size

Sa

mp

le V

ari

an

ce

Sample Variance against sample size with reference line

Figure 8. Sample Variance

2.3. Mandelbrot�s Visual Check. Supplementary to the tail estimations conductedabove, we plot the sample moments computed by

�p (n) =1

n

nXi=1

0@Ri � 1

n

nXj=1

Rj

1Aof order p = 2; 4 for ~R for relatively large sample size n; 1000 � n � 4129 in Figure 8and Figure 9 .

1000 1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

3

3.5x 1013

Sample size n>=1000

Sam

ple 4

th m

omen

t

Sample 4th moment for relatively larger sample size

Figure 9. Asymptotic Sample 4th Moment

These graphs seem to show that as the sample size increases to in�nity, the samplevariance and 4th order sample moment stabilize themselves around certain values re-spectively. Thus by the idea of Mandelbrot63 [15], we seem to be able to infer that ~R

14 XIONGZHI CHEN

1500 2000 2500 3000 3500 4000 45000

0.5

1

1.5

2

2.5

x 1017

sample size n>=1500

Sam

ple 5

th a

bsolu

te m

omen

t

Sample 5th absolute moment against sample size

Figure 10. Sample 5th Absolute Moment

does have �nite variance and 4th order moment 9(if ~R has a unique law of probability10),which supports the estimates of the tail indices here. Actually we also plotted the sam-ple 5th absolute sample moment for ~R1 = fRt; 4129 � t � 1500g in Figure 10 to checkwhether the right-tail index is low-biased or not (since if it�s low-biased then the 5thabosulte sample moment might diverge eventually) and it seems that the 5th absolutemoment is also �nite by Mandelbrot�s idea.It should be pointed out that this visually checked convergence might be false due to

insu¢ cient sample size. But proof or disproof of this seems to be impossible here in thispaper since we are not able estimate the exact asymptotic properties of the tail of theempirical cumulative distribution function (ecdf) without assuming much, much moreon the underlying noise for ~R. Also visual check might be misleading particularly whenthe sample is from a distribution with tail index 2; as justi�ed by the sample variance ofthe Cauchy distribution. However under very stringent assumptions about the mixingproperties and the order of moments of the innovations, the �niteness of 4th momentcan be tested as done in Loretan94 [11].

3. Kurtosis and Skewness

Kurtosis is easily recognized from the �gure of epdf (Figure 3). To check the skewness,the easiest way is plot X(N�p) �Xmed against Xmed �X(p) since for symmetric epdf�s

(X(N�p) �Xmed; Xmed �X(p))

will be points one a line of slope �1; where X(i) are rank statistics of the samplefXi : 1 � i � Ng and Xmed is the median and 1 � p � [N=2] + 1. Empirical evidenceon �nancial time series show that within a fairly wide range around its mean the epdf

9We say that "a stochastic process X = fXt; t 2 Tg has some property" if (it is assumed that) allXt; t 2 T have the same property.

10A stochastic process X = fXt; t 2 Tg is said to have as unique law of probability if all Xt; t 2 Thave the same law of probability.


0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0.050.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Lower quantiles of returns

Hig

her

quan

tiles

of

retu

rns

Higher quantiles agains lower ones

dataHypothetical line of slope 1linear regressor

Figure 11. Skewness

0.1 0.05 0 0.05 0.1 0.150

5

10

15

20

25

30

Log Returns

Pro

ba

bility

Kernel density estimate of the epdf of returns

Figure 12. Zoom-in Kernel Density Estimate of EPDF

is symmetric, and skewness appears seemingly in the tails (see, Mills99 [21], Mantega95[22]). Moreover, most epdf�s are skewed to the left (see, Mills95 [20], Badrinath98 [1]).Figure 11 plots R(T�p) � Rmed against Rmed � R(p) for ~R = fRt : 1 � t � 4129g An

OSL linear regression estimate of R(p) agains R(T�p) gives

R(N�p) = �1:0629 �R(p) + 0:0004Surprisingly it shows that the epdf is skewed to more to the right than to the left

(also, see the "zoom in" Figure 12 of a kerneal density estimate of the epdf), althoughthe epdf of the returns ~R is symmetric in a relatively wide range around its median. Thisstrange observation might help explain why the ergodic generalized hyperbolic di¤usionscan not �t the SSE index in the second paper, but it complies with the fact that formost of the period from July 1994 to Oct 2007 the SSE index was increasing, and that

16 XIONGZHI CHEN

the probability of a large decease in the returns is relatively lower than that of a largeincrease in the returns.

4. Dependence

Dependence, long-term or short-term, exists in time series observed from a lot of nat-ural phenomena (see, Cont01 [5], Mandelbrot62, [14], Mandelbrot01a [18]) and has beenpaid a lot of attention in recent years. For discrete time model that incorporates suchdependence, FIGARCH (i.e., fraciontal itegrated GARCH) model has been introduced,whereas for continuous model, fractional Brownian motion (fBm) (see, Mandelbrot62[14], Mandelbrot01a [18]) has been constructed, both of which have wide applications.For a discretely sampled stochastic process, the Hurst Index H can be used to measure

the strength and duration of its dependence structure. Hereunder we decribe threemethods to estimate this index and provide the estimates for for ~R.

4.1. Hurst�s R/S Statistic. Hurst (see, Hurst51 [9]), Kolmogorov and Mandelbrot arethe �rst few pioneeers who discovered the long-term momery e¤ects in natural phenom-ena, de�ned the scaling properties of probability distribution, and applied self-similaritymathematically. Actually, what Hurst found out empirically or how the Hurst index isde�ned is that

E (R=S(n)) / CnH (4.1)

holds for su¢ ciently large sample size n for some constants C > 0 and someH 2 (1=2; 1) ;where E is the expectation with repect to the probability measure P of the underlyingprobability space (;F ; P ) on which the random process that generates the sample~X = fxi; i = 1; 2; :::; ng is de�ned and

R=S (n) =R (n)

S (n)(4.2)

is called the classic rescaled range statistic or R=S statistic with

� =1

n

nXi=1

xi; �2 =

1

n

nXi=1

(xi � �)2 ; Sn =

vuut 1

n

nXi=1

(xi � �)2

and

R (n) = max1�j�n

(jXi=1

xi �j

n

nXi=1

xi

)� max1�n�N

(jXi=1

xi �j

n

nXi=1

xi

)One major disadvangate of the classic Hurst R=S estimator is that it is too sensitive to

short-range dependence11, but one key advantage it has is that, if we de�ne a long-rangedependence parameter d by

d = H � �with � = 1=2 for �nite variance processes or 1=� for in�nite variance processes with tailindex 0 < � < 2; then R=S statistic consistently provides an estimate of

H = d+1

2

11See Mandelbrot01a [18] for details on "short/long-range dependence".


0 500 1000 1500 2000 2500 3000 3500 4000 45000

20

40

60

80

100

120

Sample Size

(R/S

)(n

)

Rescaled Range Statistics against Sample Size

Figure 13. Classic R/S Statistics

0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

log(n)

log((R

/S)(n

))

Loglog Plot of Rescaled Range Statistic against Sample Size

Figure 14. L-L Plot of Classic R/S Statistic

independent of the value of � since

R (n) / nH = nd+1=�

andS (n) / n1=��1=2

asymptotically (see, Taqqu98 [31], Mandelbrot75 [17]).Figure 13 is a plot of the estimated classic R=S (n) against n for

~Rn := fRt : t = 1; ::; ng ; 1 � n � 402512and the corresponding log-log plot in provided in Figure 14 .As can been seen from the log-log plot Figure 14, linearity is very strong for two

sections: exp (0:8) � n � exp (6) and 8 � n � exp(6): Consequently it�s reasonable to

12" := " means " de�ned as "

18 XIONGZHI CHEN

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

sample size in log scale

R/S

statis

tic

R/S statistic against sample size with regressor

ln(R/S) vs ln(n)regressor

Figure 15. Regressed R/S Statistic for Small Sample Size

6 6.5 7 7.5 8 8.53.4

3.6

3.8

4

4.2

4.4

4.6

4.8

5

log(n), 410<= n <=4129

log

((R

/S)(

n))

Loglog Plot of R/S Statistic against Sample Size with Regressor

log(R/S(n))0.4265*log(n)+0.9152

Figure 16. Regressed R/S Statistic for Larger Sample Size

estimate H for each section. An OLS regressor for (logR=S (n) ; log n) for 2 � n � 410is given as

L1 (n) = 0:3366 � lnn+ 1:6223and that for 410 � n � 4025 is given as

L2 (2) = 0:4265 � lnn+ 0:9152which are combined into

logR=S (n) =

�1:0833 � lnn+ 3: 219 1 if 4 � n � 4100:4265 � lnn+ 0:9152 if 410 � n � 4025

The corresponding plots of logR=S (n) against log n with regressors are provided inFigure 15 and Figure 16Even though for the last 100 sample points, the linearity is strong with a much larger

slope of regression than have been previously given, but they contribute little whenincorporated into the second group for regression.


0 1 2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

log(n)

(RS)

(n)

Pox Plot of R/S with Reference Lines

Red line:slope 0.62

Green LineSlope 0.5

Figure 17. Pox Plot of R/S Analysis

Thus focusing on large sample properties, we can just take 0:4265 to be the estimate ofd+1=2 since we showed that ~R has �nite variance by its tail index estimate (by explicitlyassuming that ~R has a unique probability law). This suggests that the estimated d =�0:08 and that ~R is a process with short-range dependence.However, the R=S analysis and modi�ed R=S statistic in the following subsection

both show the existence of long-range dependence in ~R, contradicting the short-rangedependence revealed by classic R=S statistics. This might be a consequence of theexcessive sensitivity of the classic R=S statistic to short-range dependences in variousdurations. We present the R=S analysis next.

4.2. Mandelbrot�s R/S Analysis. The actual and more realistic implementaion of theclassic R=S statistic is through the so called R=S analysis (see, Mandelbrot69 [16]) andis decribed brie�y as follows (see, Rachev00 [26]): let N be the sample size. Divide the

sample in K blocks each of size N=K and de�ne km = 1+(m� 1)N

Kwithm = 1; � � � ;K:

Then estimate Rn (km) and Sn (km) for each lag n such that km + n < N: Finally, plot

logRn (Km)

Sn (km)against log n for each m = 1; � � � ;K; in one �gure. The resulting graph is

called "rescaled adjusted range plot" or "pox plot of R=S (see, Rachev00 [26])".To allow �exibiliy in grouping, we only selected the �rst 4000 entries from ~R with

group size S = 400 and provide the pox plot of R=S in Figure 17Note that in the Figure 17 the red line has slope 0:62 and the blue one has slope 0:5,

the latter of which corresponds to the Hurst index of a standard Brownian motion. As

can be seen clearly from the plot that for large sample size n � exp (5) ; allRn (Km)

Sn (km)stays within the cone formed by L1 = 0:62� log x and L1 = 0:5� log x (which is relativelysu¢ cient for asymptotic property). The most striking feature is that the line

L = 0:57 � log n

20 XIONGZHI CHEN

100 200 300 400 500 600 700 800 900 10000.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

Lag Order

Sam

ple

Aut

ocor

rela

tion

Sample Autocorrelation Function (ACF)

Figure 18. Autocorrelation till Lag Order 1000

gives a good least squares �t to these scattered R=S statistics, which suggests the esti-mated Hurst index is

H = 0:57 (4.3)

, in consistency with what most researchers have found as the Hurst index estimates for�nanical time series (see, Cont01 [5]).Thus we can reject the hypothesis that the returns seris of the SSE index follows

a standard Brownian motion. Moreover, if the process ~R has �nite variance we thencan conclude that the returns ~R has a relatively shorter memory among long-rangedependence processes as justi�ed by the estimated long-range dependence parameter

d = 0:07 (4.4)

(Note that this is also supported by the autocorrelations of ~R for the maximal order oflag L = 1000 in Figure 18To reduce the excessive sensitivity of R=S analysis to short-range dependences (see

Figure 14 ), Lo91 ([12]) introduced the so called "Lo�s modi�ed R=S statistic" to detectoverall long-rang dependence without giving any estimation of the Hurst index H andprovided the aymptotic distribution for this statistic. Lo�s modi�ed R=S statistic isde�ned as

VN (q) =1pN

RNSN (q)

(4.5)

where

� =1

N

NXi=1

Xi; !j (q) = 1�j

q + 1; q < N (4.6)


and

RN = max1�q�N

0@ qXj=1

Xj � q�

1A� min1�q�N

0@ qXj=1

Xj � q�

1A (4.7)

= max1�q�N

0@ qXj=1

(Xj � �)

1A� min1�q�N

0@ qXj=1

(Xj � �)

1Aand

S2N (q) =1

N

NXi=1

(Xi � �)2 +2

N

qXj=1

!j (q)

0@ NXi=j+1

(Xi � �) (Xi�j � �)

1A (4.8)

= �2 + 2

qXj=1

!j (q) j

To use the Lo�s modi�ed R=S statistic, we cite from in Lo91 [12] the following theorem

Theorem 1 (Lo91 [12]). Suppose the random process " = f"t : t � 0g satis�es the fol-lowing conditions

a1): E ("t) = 0a2): suptE

�j"tj2�

�<1 for some � > 2

a3): 0 < �2 = limn!1E�1

n

Pnj=1 "

2j

�<1

a4): f"tg is strongly mixing with mixing coe¢ cients �k that satisfy1Xj=

�1�2=�j <1

a5): q � o�n1=4

�as both n and q approch to +1

ThenVN (q)!d W1 = maxW0 (t)�minW0 (t) (4.9)

if f"tg has no long-range dependence, where W0 (t) is the standard Brownian bridge and

P (W1 2 [0:809; 1:862]) = 0:95

Note that in the above theorem, at least uniformly the fourth moment is required tobe �nite in order to get the asymptotic distribution. Assumption a3) is a weak conditionon the ergodicity of the variance and assumption a4) requires that the random variable"t can not be so dependent.From the estimates already obtained, we know that ~R = fRt : 1 � t � 4129g does not

satisfy assumption a2) even though it might satisfy assumption a4) since the daily returnsseries have very weak dependence (see, Cont01 [5]). Based on the previous tail indexestimates and running the risk that ~R might violate the fourth moment requirementa4), we still plotted the Lo�s R=S statistic VN (q) in Figure 19 for the detrended �R =

fRt � � : 1 � t � 4139g where � is the sample mean of ~R

22 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 45000.5

1

1.5

2

2.5

3

3.5

modified lags q

V N(q)

Lo's RS Statistic

Figure 19. Lo�s Modi�ed R/S Statistic

Note in Figure 19 the con�dence interval of the Lo�s R=S statistic are given bythe red and green horizontal lines (under the assumption of the previous theorem).Obivously, approximately for q � 1500; Lo�s R=S statistic stays outside the 95% con�-dentce interval and thus suggests the existence of long-range dependence. (It should benoted that Lo�s modi�ed R=S statistic is widely used even though not all assumptionsin the above theorem are satis�ed. See Rachev00 [26].)

4.3. Direct Regression. As Rachev00 [26] has pointed out that the R=S analysis isstill sensitive to the presence of short-range dependence, we adopt the method of di-rect regression therein to estimate the Hurst index, that is, for discrete observationsn~Xi : 1 � i � N

oof a randrom process X = fXt : t 2 Tg we can regress

�k = log E (jXi+k �Xij) 13

against log k to estimat the Hurst indexH; where E is the sample mean and i = 1; � � � ; N .For the returns ~R = fRt : 1 � t � 4129g, the results are presented in Figure 20in which the linear regressor is given by

�k = log E (jRt+k �Rtj) = 0:6133 log k � 4:6021

Thus the estimated Hurst index for the returns series ~R is

H� = 0:6133

This result con�rms the existence of long-range dependence in ~R and is consistent withsimilar results obtained in Oh06 [23].

13If fXt; t = 1; 2; ::; Ng is the logarithmic SSE index, then E (jXt+k �Xtj) is called the average ab-solute returns over k quotes. (See, Rachev00 [26]) .


0 1 2 3 4 5 6 7 8 95

4

3

2

1

0

1

2

Lag order

Log

Abs

olut

e A

vera

ge R

etur

n

LL Plot of Absolute Average Returns againstLag Order with Linear Regressor

log absolute average returnsLinear Regressor

Figure 20. L-L Plot of Absolute Average Returns

5. Conclusions

We have mainly estimated the tail indices, Hurst index of the returns series of the SSEindex, by non-parametric linear regression. These statistics are consistent with empiricalestimates obtained by most researchers, except that returns series of SSE index skewsmore to the right. An imperfection of this paper is that no statistical tests have beencarried out for the estimates, since we do not pre-assume any speci�c properties of thestochastic process that generates this index series. However, the results will de�nitelyhelp to measure the appropriateness of the ergodic generalized hyperbolic di¤usion as amodel for the detrended logarithmic SSE index series in the next paper.

Please continue to the second part of the thesis after the references................

References

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[2] Bibby, B. M. & Sorensen, M. (1997): A Hyperbolic Di¤usion Model for Stock Prices. Finance andStochastics 1(1), 25�41.

[3] Chang Xuejiang, et al. (1993): Time series analysis. China Higher Education Press[4] Rama Cont, Peter Tankov (2004): Financial modelling with jump processes. Chapman & Hall[5] Rama Cont (2001), Empirical properties of asset returns: stylized facts and statistical issues. Quan-

titative Finance Volume 1 (2001), 223-236[6] Peter Hall and A. H. Welsh (1985): Adaptive Estimates of Parameters of Regular Variation. The

Annals of Statistics, Vol. 13, No. 1 (Mar., 1985), pp. 331-341[7] J.D. Hamilton, R. Susmel (1994): Autoregressive condtional heteroskedasticity and changes in

regime. Journal of econometrics, 64, 307-33[8] B. M Hill (1975): A simple general approach to inference about the tail of a distribution. Annal of

statistics, 3, 163-74

24 XIONGZHI CHEN

[9] H. Hurst (1951): Long term storage capacity of reservoirs. Transactions of the Americal Society ofCivil Engineering, 116, 770-99

[10] I. Karatzas, S.E. Shreve 1998, Brownian Motion and Stochastic Calculus. Springe-Verlag[11] Mico Loretan, Peter C.B. Philips (1994): Testing the covariance stationarity of heavy-tailed time

series. Journal of empirical �nance 1 (1994) 211-248 North-Holland[12] Andrew W. Lo (1991), Long-term memory in stock market prices. Econometrica, Vol. 59, No. 5

(September, 1991), 1279-1313[13] Andrwe W. Lo, A. Craig MacKinlay (1999): A non-random walk down wall street. Princeton Uni-

veristy Press[14] Benoit B. Mandelbrot (1962): New methods in statistic economics. Journal of political economy, 71,

421-40[15] Benoit B. Mandelbrot (1963): The variation of certain speculative prices. Journal of Business

XXXVI, 392-417.[16] BB Mandelbrot, JM Wallis (1969), Robustness of the rescaled range R/S in the measurement of

noncyclic long run statistical dependence, Water Resources Research, 5:967-988.[17] Benoit B. Mandelbrot(1975): Limit theorems on self-normalized range for weakly and strongly de-

pendent processes. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebieter, 31:271-285[18] Benoit .B. Mandelbrot (2001): Scaling in �nancial prices I: tails and dependence. Quantitative

�nance Volume 1 (2001) 113-123[19] J. Huston McCulloch (1997): Measuring Tail Thickness to Estimate the Stable Index �: A Critique.

Journal of Business & Economic Statistics, Vol. 15, No. 1 (Jan., 1997), pp. 74-81[20] Terence C. Mills (1995): Modelling Skewness and Kurtosis in the London Stock Exchange FT-SE

Index Return Distributions. The Statistician, Vol. 44, No. 3 (1995), pp. 323-332[21] Terence C Mills (1999): The econometric modelling of �nancial time series. Cambridge Univeristy

Press[22] Rosario N. Mantenga, H. Eugene Stanley (1995): Scaling behaviour in the dynamics of an economic

index. Nature Vol 376, July 1995[23] Gabjin Oh, Seunghwan Kim (2006): Statitical properties of the returns of stock prices of internatial

markest. Journal of the Korean Physical Society, Vol 48, Feburuary 2006, pp S197-S201.[24] Peter C. B. Phillips, James W. McFarland, Patrick C. McMahon (1996): Robust Tests of Forward

Exchange Market E¢ ciency with Empirical Evidence From the 1920. Journal of Applied Economet-rics, Vol. 11, No. 1 (Jan. - Feb., 1996), pp. 1-22

[25] C.-K. Peng, et al (1994): Mosaic organization of DNA nucleotides. Phys. Rev. E 49, 1685 (1994).[26] Svetlozar T. Rachev, Stefan Mittnik (2000): Stable Paretian Models in Finance. Wiley[27] T. H. Rydberg (1999): Generalized hyperbolic di¤usion processes with applications in �nance. Math-

ematical Finance, Vol 9, 183-2001[28] Gennady Samorodnitsky, Murad S. Taqqu (1994): Stable non-gaussion random proceses. Chapman

& Hall[29] Michael F. Shlesinger (1994): Comment on "stochastic process with ultraslow convergence to a

Gaussian: the truncated Levy �ight". Physcal Review Letters, Volume 74 Number 24[30] Murad Taqqu, Estimators for long-range dependence: an empirical study, working paper[31] Murad S. Taqqu, Vadim Teverovsky (1998): On Estimating the Intensity of Long-Range Dependence

in Finite and In�nite Variance Time Series.[32] Xiao-tian Wang, et al. (2006): Fractional poisson process. Chaos, solitons and fractals 28 (2006)

143-147

SHANGHAI STOCK EXCHANGE II: NOT AN ERGODICGENERALIZED HYPERBOLIC DIFFUSION

XIONGZHI CHEN

Abstract. Motivated by recent successes of modeling stock prices byhyperbolic distributions, (generalized) hyperbolic di¤usions (Bibby97[5], Elerian98 [18], Rydberg99 [37]), this paper attempts to simulate thelinearly detrended logarithmic SSE index series by an ergodic general-ized hyperbolic (GH) di¤usion which induces a returns series whose tailindex and correlation structure match the estimates obtained in partone of this thesis and whose invariant density is proportional to somegeneralized hyperbolic (GH) density.We adopt methodologies from Bibby97 [5], Bibby04 [7], and Sorensen02

[40] to build such a model and estimate the six paramteres involvedin the model by Markov chain Monte Carlo (MCMC) methods whoseconvergence criteria are the Monte carlo standard error (MCSE) (see,Roberts96 [35]) and the Gelman-Rubin (GR) diagnostic (see, Gelman92[20], Brooks98 [10], Brooks00 [11]), while retaining a good acceptancerate (see, Chib95 [12], Johannes03 [25]).We run 6 independent chains in the simulation whose initial values

are widely dispersed but centered around the nonlinear least squares es-timate of the paramters of a GH density based on the empirical squareddi¤usion function of the linearly detrended logarithmic SSE index se-ries. MCSEs suggests that all six independent chains have converged totheir own stable distributions at around 50; 000 iterations and remainsat around 10�3 afterwards. Unfortunately the GR statistic does not con-verge to 1 even after 350; 000 iterations and the acceptance rates becametoo low for six chains after around 50; 000 iterations, so the simulationhas to be terminated since further iterations will not give statistcallysigni�cant moves of the chains.We take the last 10% of the �rst 50; 000 simulated samples from

each chain in the simulation to obtain 6 Monte Carlo estimates for theparameter vector which suggest 6 possible models. Due to the compu-tational inability of both Mathematica and Matlab resulted from the 2badly mixed chains, only 4 chains are selected for the uniform residualtests (see, Bibby97 [5]).Tests results show that all 4 possible models are rejected statistically.

Thus under the scheme of approximating the unknown posterior for theparameter vector by the likelihood function obtained from the Milstein�sdiscretization, we reject the hypothesis that " the underlying stochasticprocess for the linearly detrended logarithm SSE index in the periodfrom 01/01/1991 to 14/11/2007 is an ergodic, GH di¤usion", i.e., thelinearly detrended logarithmic SSE index is not an ergodic GH di¤usion.

Key words and phrases. Generalized Hyperbolic Distribution, Ergodicity, InvariantMeasure, Markov chain Monte Carlo Methods, Metropolis-Hastings Algorithm, Gelman-Rubin Convergence Diagnostic/Statistic, Uniformal Residual Test.

1

2 XIONGZHI CHEN

Contents

List of Tables 3List of Figures 41. INTRODUCTION 52. CONSTRUCTION OF ERGODIC, GH DIFFUSION 62.1. Genenal Theory on Solutions of One Dimensional SDEs 62.2. Ergodic Di¤usion with GH Invariant Density 83. MCMC ESTIMATION OF PARAMETERS 103.1. Milstein Discretization Scheme 113.2. MCMC Methods and Metropolis-Hastings Algorithm 153.3. Convergence Checking 173.4. Implementation and Estimates 193.5. Test of Model 224. CONCLUSION AND DISCUSSION 24References 245. APPENDIX 275.1. Numerical Results 27

SSE: NON-GENERALIZED HYPERBOLIC DIFFUSION 3

List of Tables

1 Monte Carlo Estimates for 6 Chains 22

2 MCSEs 22

3 Test Statistics for 4 Models 23

4 XIONGZHI CHEN

List of Figures

1 Linearly Detrended Logarithmic SSE Index 11

2 Empirical Density of Detrended Index 20

3 Squared Volatility from Model 2 20

4 Maximal MCSEs for Individual Chains 21

5 Acceptance Rates for Individual Chains 21

6 QQ Plot of Uniform Residuals for Model 2 23

7 Sample Path for � for 6 Chains 27






13 Autocorrelation for � for 6 Chains 33






19 Model Based Squared Volatilities 39

20 Simulated Detrended Index from Models 1, 2, 3 40

21 Simulated Detrended Index from Models 4, 5, 6 41

22 QQ Plot of Uniform Residuals for 4 Tested Models 42


1. INTRODUCTION

In view of recent successful applications of generalized hyperbolic (GH)distributions in modeling asset returns, stock prices and option prices (seeBibby97 [5], Karsten99 [27], and Eberlein95 [16]), we tentatively assumethat the underlying stochastic process X = fXt : t � 0g for the linearly de-trended logarithmic SSE index1 is an ergodic GH di¤usion2 whose invariantdensity is some GH density and which for the trading days considered hereininduces a returns series that has the tail indices and dependence structuresas previously estimated in part one of this thesis. To be more speci�c, wetry to solve following simulation

Problem 1. Construct an ergodic, weak solution X = fXt : t � 0g to theparametric stochastic di¤erential equation (SDE)

dXt = b (Xt; �) dt+ v (Xt; �) dBt (1.1)

with appropriate drift function b (�) ; di¤usion function v (�) ; where the pa-rameter � lies in some bounded, open subset of Rm for some m 2 N,B = fBt; t � 0g is a standard Brownian motion, such that

(1) X has the generalized hyperbolic (GH) density

fgh (x;�; �; �; �; �) (1.2)

=

��2 � �2

��=2K��1=2

��q�2 + (x� �)2

�p2��1=2��K�

��p�2 � �2

��2 + (x� �)2

�(��1=2)=2exp (� (x� �))

as its invariant or initial or asymptotic density, where the domainof variation of the parameters is � 2 R and8<: � � 0; j�j < � if � > 0

� > 0; j�j < � if � = 0� > 0; j�j � � if � < 0

(1.3)

and the modi�ed Bessel function K� of the second kind is given by

K� (x) =1

2

Z 1

0y��1 exp

��x2

�y + y�1

��dy; x > 0 (1.4)

(2) and discretized on the trading days considered in part one of this the-sis, X induces a returns series that has tail indices and dependencestructures as estimated in part one of this thesis.

We adopt the traditional notations and rewrite fgh compactly as

fgh (x; �) = a (�; �; �; �)K��1=2

��

q�2 + (x� �)2

�(1.5)

��2 + (x� �)2

�(��1=2)=2exp (� (x� �))

1"Detrended index" will denote " the linearly detrended logarithmic SSE index series",and the exact linear detrending will be explained in later sections.

2For a de�nition of "di¤usion", please see Shreve98 [38]. For a de�nition of "generalizedhyperbolic di¤usion", please see Rydberg99 [37]

6 XIONGZHI CHEN

with parameter vector � = (�; �; �; �; �) by letting

a (�; �; �; �) =

��2 � �2

��=2p2��1=2��K�

��p�2 � �2

� (1.6)

Further, we simply denote the unnormalized GH density fgh (x; �) by

f (x; �) (1.7)

= K��1=2

��

q�2 + (x� �)2

��2 + (x� �)2

�(��1=2)=2exp (� (x� �))

We adopt the method in Bibby95 [4] and Bibby97 [5] to build such anergodic GH di¤usion for the detrended index. As for parameter estimationwe use Markov chain Monte Carlo (MCMC) methods driven by Metropolis-Hastings (MH) algorithm to sample from the unknown posterior of the pa-rameter vector (see, Johannes03 [25] and Tes04 [42]. Note however that inTse04 [42] at least two mistakes are found by the author of this thesis).It is somehow sad that for the detrended index, due to the intense com-

putational complexity, the method of martingale estimating function ( see,Bibby95 [4] and Bibby97 [5]) has not been successfully implemented, andthat due to lack of time, estimation methods introduced in Ait02 [3] 3 havenot been applied.

2. CONSTRUCTION OF ERGODIC, GH DIFFUSION

There are many references to building di¤usion-type models with givenmarginal distributions and (exponentially-decreasing) autocorrelation func-tions (see Bibby97 [5], Bibby04 [7], Madan02 [30] and Ait02 [3]). All modelsconstructed fall into three categories: (i) X is a di¤usion with a given initialdensity; (ii) X is an ergodic di¤usion whose asymptotic density or invari-ant density is the given density; (iii) X is a di¤usion with a given familyof marginal densities. Herein, we only concentrate on the construction of adriftless, ergodic GH di¤usion with a GH invariant density.

2.1. Genenal Theory on Solutions of One Dimensional SDEs. Firstof all, we would like to cite some powerful theorems on the existence anduniqueness of weak solutions to one-dimensional stochastic di¤erential equa-tions (SDEs). These theorems give the su¢ cient or necessary and su¢ cientconditions on the existence, uniqueness, recurrence and ergodicity propertiesof solutions to such SDE�s.From Kutoyants04 [26] comes the following su¢ cient condition for the

existence and ergodicity of a weak solution as a di¤usion process.

Theorem 1 (Kutoyants04[26]). Suppose b is locally bounded, v is continuousand positive, and

xb (x) + v2 (x) � A�1 + x2

�3Ait�s method is the only one that gives explicit approximation to the transition den-

sities of su¢ ciently good di¤usions and estimates the unknown parameters only after agood approximation of the transition densities are obtained, thus distinguishing itself fromall other methods that estimate the unknown parameters with a pre-speci�ed model. Fora comparison of estimation methods on discretely sampled di¤usions, please see Jessen99[24].


for some constant A > 0; then the stochastic di¤erential equation

dXt = b (Xs) dt+ v (Xt) dWt

has a unique weak solution.Moreover, if

V (b; x) =

Z x

0exp

��2Z y

0

b (s)

v2 (s)ds

�dy ! �1; as x! �1

and

G (b) =

Z +1

�1v�2 (x) exp

�2

Z x

0

b (s)

v2 (s)ds

�dx <1

then the solution is ergodic with invariant density

fb (x) = G�1 (b) v�2 (x) exp

�2

Z x

0

b (s)

v2 (s)ds

�Remark 1. The conditions on existence of weak solutions in the above the-orem is relatively strong but necessary for statistical inference.

From Shreve98 ([38]), we cite the elegant theorem proved by H. J. Engel-bert and W. Schmidt (1984).

Theorem 2 (Engelbert and Schmidt 1984). Equation

dXt = v (Xt) dWt

has a non-exploding weak solution unique in probability law for every initialdistribution �0 if and only if for all real x

v (x) = 0, (8" > 0)Z "

�"

dy

[v (x+ y)]2=1

Finally, we cite from Skorokhod89 [39] this powerful theorem on ergodic-ity:

Theorem 3 (Skorokhod89). If r1 = �1; r2 = +1;andR +1�1 b�2 (x) dx <

1; then dXt = b (Xt) dWt is ergodic with ergodic distribution

� (A) =

RA v

�2 (z) dzRR v

�2 (z) dz

Remark 2. In Theorem 3, r1; r2 denote respectively the left, right boundaryfor the state space of X: For an ergodic di¤usion, its ergodic distributioncorresponds to its invariant distribution. For more details on this, please seeSkorokhod89 [39]. Here we brie�y cite from Skorokhod89 [39] a de�nitionthat, a �-�nite measure � is said to be invariant if �Tt = � for all t �0; where �Tt (A) =

RP (t; x;A)� (dx) = E� (IA (x (t))) and P (t; x; A) is

the underlying transition density. The relationship between the di¤usioncoe¢ cient and the invariant density can be also found in Bibby97 [5], wherethe above theorem is used.

8 XIONGZHI CHEN

2.2. Ergodic Di¤usion with GH Invariant Density. With these the-orems at hand, we are now ready to construct the proposed ergodic GHdi¤usion. We borrow the model from Bibby97 [5] and Rydberg99 [37] andassume that the underlying stochastic process S for the SSE index satis�es

St = exp (Xt + �t) ; t � 0 (2.1)

where � is some constant to be determined and

Xt = X0 +

Z t

0v (Xs) dWs (2.2)

for some v (�) satisfying the conditions which give existence and uniquenessin probability law of a weak solution to (2.2).

2.2.1. Model I. With such a speci�cation in (2.1) � (2.2), we have twochoices: one is to estimate � by the ordinary least squares linear regres-sion to obtain � based on the observed SSE index4 ~P = fPt; t = 1; :::; 4131gand assume that

lnSt � �t; t = 1; :::; 4131 (2.3)

forms an observation of the ergodic di¤usion

Xt = X0 +

Z t

0v (Xs) dWs; t � 0 (2.4)

whose invariant density is proportional to some unnormalized GH densityf (�; �) as de�ned in (1.7), then estimate the unknown parameter � based on~P ; the other is to assume that

lnSt = Xt + �t = X0 +

Z t

0v (Xs) dWs +

Z t

0�ds (2.5)

is itself an ergodic di¤usion whose invariant density is some GH densityde�ned in (1.2), then estimate the involved unknown parameter vector �� =(�; �; �; �; �; �; �) based on the observed SSE index ~P .Hereunder we call (2.3) � (2.4) the simpli�ed Model I. Setting the di¤u-

sion function (or, di¤usion coe¢ cient) v (�) to be

v�x; ~�

�=p�2=f (x; �) (2.6)

with the augmented parameter vector ~� = (�; �) yields

Xt = X0 +

Z t

0

r�2=f

�x; ~�

�dWs (2.7)

Since v�x; ~�

�> 0 for x 2 R; � in (1.5), � 6= 0 and f has �nite second order

moment by the following moment generating function for the GH density.

M (t) = et��

�2 � �2

�2 � (� + t)2

��=2 K��q�2 � (� + t)2�K�

��p�2 � �2

� ; j� + tj < � (2.8)

4Please see the �rst part of the thesis for the observed SSE index.


(see, Karsten99 [27]), the conditions of Theorem 2, Theorem 3 are satistis�edand X is weakly uniquely de�ned by (2.7) with ergodic density proportionalto f (�; �).

2.2.2. Model II. However, if we take (2.5) � (2.6) as Model II and wantto ensure that the invariant density is the GH density, then according toTheorem 1 we have to �nd v (�; �) such that

V (x; �) =

Z x

0exp

��2Z y

0

�

v (s; �)2ds

�dy ! �1; as x! �1 (2.9)

and

G =

Z +1

�1v (x)�2 exp

�2

Z x

0

�

v (s; �)2ds

�dx <1 (2.10)

so that the invariant density is

f (x; �) =

v (x; �)�2 exp

�2R x0

�

v (s; �)2ds

�G

(2.11)

Remark 3. If the initial X0 has the density f (�; �) then X is stationary.

More speci�cally, we have to solve for v (�; �) the following equation8>><>>:fgh (x; �) = G

�1v (x; �)�2 exp

�2R x0

�

v (s; �)2ds

�R x0 exp

��2R y0

�

v (s; �)2ds

�dy ! �1; as x! �1

(2.12)

from which we know that G <1 since fgh has �nite second moment.From (2.12) we have

v (x; �)2d ln fgh (x; �)

dx+ 2v0 (x; �) v (x; �) = 2� (2.13)

where the superscript 0 denotes di¤erentiation with respect to x: Settingh (x; �) = v2 (x; �) and solving the equivalent

d

dxffgh (x; �)h (x; �)g = 2�fgh (x; �)

yields

h (x; �) = f�1gh (x; �)

�2�

Z x

0fgh (s; �) ds+G

�1�

(2.14)

and

v (x; �) =ph (x; �) (2.15)

= f�1=2gh (x; �)

�2�

Z x

0fgh (s; �) ds+G

�1�1=2

Next, we still have to show

limx!+1

Z x

0exp

��2Z y

0

1

v (s; �)2ds

�dy = +1

10 XIONGZHI CHEN

By exploiting fgh (x; �) = G�1v (x; �)�2 exp

�2R x0

1

v (s; �)2ds

�;we have

I1 : =

Z x

0

dy

exp�2R y0 v

�2 (s; �) ds (2.16)

= G�1Z x

0

dy

fgh (y; �) v2 (y; �)

= G�1Z x

0

dy�2�R y0 fgh (s; �) ds+G

�1�

such thatlim

x!�1I1 = �1 for � � 0

Thus, the proposed ergodic di¤usion model in (2.5) is

St = exp (�t+Xt)

and

Xt = X0 +

Z t

0�ds+

Z t

0v (Xs; �) dWs (2.17)

= X0 +

Z t

0�ds+

Z t

0f�1=2gh (Xs; �)

�2�

Z Xs

0fgh (r; �) dr +G

�1�1=2

dWs

with invariant density fgh (�; �) for � � 0:Finally, applying Ito�s formula to H (t; x) = exp (b (t) + x) to the model,

we obtain St as

dSt = St

��b0 (t) +

1

2v2 (lnSt � b (t))

�dt+ v (lnSt � b (t)) dWt

�(2.18)

Remark 4. It is well known that under certain non-degeneracy and lo-cally integrability conditions, a stochastic integral-di¤erential equation canbe transformed into a driftless equation (see, Shreve98 [38]) or a constantdi¤usion equation (see, Rogers79 [36] ). But in most cases with the presenceof drift functions, we will have to use the methods in Ait02 [3] or in Lo88[28].

3. MCMC ESTIMATION OF PARAMETERS

To reduce the computational complexity resulted from computing thenumerical integral

R s0 fgh (r; �) dr involved in Model II, we will only estimate

the model

lnSt � �t = Xt = X0 +Z t

0v�Xs; ~�

�dWs; t � 0 (3.1)

de�ned in (2.1) � (2.4) with

v��; ~��=

s�2

f (�; �) (3.2)

where S = fSt; t � 0g here denotes the underlying stochastic process for theobserved SSE index ~P and X = fXt; t �g denotes that for the detrended in-dex ~D = flnSt � �t; t = 1; :::; 4131g which is also denoted by X as a discreteobservation of X. (See Figure 1 for the plot of ~D. )


0 500 1000 1500 2000 2500 3000 3500 4000 45003

2

1

0

1

2

3

4

5

6

7

Number of Trading Days

Lin

ea

rly D

etr

en

ed

In

de

x

Linearly Detrended Logarithmic SSE Index

Figure 1. Linearly Detrended Logarithmic SSE Index

3.1. Milstein Discretization Scheme. To approximate the likelihood func-tion and the unknown posterior p

�~�jX

�for the parameter vector ~� =

(�; �; �; �; �; �) given the discrete observation X = fXt; t = 1; 2; :::; ng, westart from the Milstein approximation scheme (of strong convergence of or-der one) by discretizing model (3.1) for a small change �t in t into

Xt+�t = Xt + v�Xt; ~�

��Wt +

1

2v�Xt; ~�

� @v �Xt; ~��@Xt

h(�Wt)

2 ��ti

which can be rewritten into

Xt+�t �Xt +1

2v�Xt; ~�

� @v �Xt; ~��@Xt

�t+v�Xt; ~�

�2@@Xtv

�Xt; ~�

� (3.3)

=�v�Xt; ~�

�p�t��+

0@12v�Xt; ~�

� @v �Xt; ~��@Xt

�t

1A �2 + v�Xt; ~�

�2@@Xtv

�Xt; ~�

�

where@v�Xt; ~�

�@Xt

:=@v�x; ~�

�@x

��x=Xt

:= @@Xtv�Xt; ~�

�and � is standard

normal.

12 XIONGZHI CHEN

In order to obtain the conditional density after discretization, we use theresults from Elerian98 [18] and set8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

At =1

2v�Xt; ~�

� @v �Xt; ~��@Xt

�t

Bt = �v�Xt; ~�

�2@@Xtv

�Xt; ~�

� +Xt �At�t =

1

�t�@@Xtv

�Xt; ~�

��2yt+1;�t =

Xt+1 �BtAt

(3.4)

to conclude that yt+1;�t =��+

p�t�2 and

h (Xt+�tjXt) =1

jAtj�2�Xt+1 �Bt

At; 1;�t

�where �2 (�; d; l) is a non-central chi-squared density function with d degreesof freedom and location parameter l. More speci�cally

�2 (z; d; l) =1

2exp

�� l + z

2

��zl

�d=4�1=2I�1=2

�plz�

where I�1=2 (w) =

r2

�wcosh (w).

In our case with d = 1, we have

�2 (z; 1; l) =1

2exp

�� l + z

2

��zl

��1=4I�1=2

�plz�

=1

2exp

�� l + z

2

�z�1=4l1=4

p2

p�pp

lzcosh

�plz�

=1p2�exp

�� l + z

2

�z�1=2 cosh

�plz�

and

h�Xt+�tjXt; ~�

�(3.5)

=

exp

��0:5

��t +

Xt+�t �BtAt

��cosh

�r�t (Xt+�t �Bt)

At

�p2� jAtj

s�Xt+�t �Bt

At

�which saves a bit of computation time in computing the uniform residualsfor model test in later subsection.With the Milstein�s discretization, the likelihood function ~LM

�~�jX

�of ~�

given observation X is approximated by

LM

�~�jX

�:=

nQt=1h�Xt+�tjXt; ~�

�(3.6)


which further approximates the unknown p�~�jX

�(by the Bayesian rule).

However, based on the observation X (of the detrended index), the ap-

proximate likelihood funtion ~LM

�~�jX

�generates huge quantities which

are even beyond the computational capabilities of Mathematica and Mat-lab. Thus we use another scheme by approximating the unknown posterior

p�~�jY

�based on the transformed observation Y as follows.

We set

8>>>>>>>>>><>>>>>>>>>>:

g�Xt; ~�

�=1

2v�Xt; ~�

� @v �Xt; ~��@Xt

ct = v�Xt; ~�

�p�t

dt = g�Xt; ~�

��t =

1

2v�Xt; ~�

� @v �Xt; ~��@Xt

Yt;�t = Xt+�t �Xt + g�Xt; ~�

��t = Xt+�t �Xt + dt

to obtain

Yt;�t = ct�+ dt�2 = dt

"��+

ct2dt

�2� c2t4d2t

#(3.7)

whereand � is standard normal. Since Z =��+

ct2dt

�2has density

h�z; ~��=1

2exp

��rt + z

2

��z

rt

��1=4I�1=2 (

prtz) (3.8)

where rt = c2t =�4d2t�and I�1=2 (w) =

r2

�wcosh (w) (see, Elerian98 [18]),

then the density of Yt;�t is

h��y; t; ~�

�=

1

jdtjh

�y

dt+c2t4d2t

�(3.9)

and the approximate likelihood given the observationsY = fYt;�t; t = 1; :::; ngis

LM

�~�jY

�=

nQt=1h��y; t; ~�

�(3.10)

14 XIONGZHI CHEN

Next, we compute the functions needed in the Milstein�s discretization5.Firstly we have

@Xtv�Xt; ~�

�= @Xt

8><>:vuut �2

f�Xt; ~�

�9>=>; (3.11)

= �12

�2f�2�Xt; ~�

� d

dXtf�Xt; ~�

�v�Xt; ~�

�= �0:5v

�Xt; ~�

� d

dXtln f

�Xt; ~�

�and secondly we have

d

dxln fgh

�x; ~�

�=

d

dx

�lnK��1=2

��

q�2 + (x� �)2

�+�� 1=22

ln��2 + (x� �)2

�+ � (x� �)

�= � +

(�� 1=2) (x� �)�2 + (x� �)2

+d

dxlnK��1=2

��

q�2 + (x� �)2

�= � +

(�� 1=2) (x� �)l (x; �; �)

��K�+1=2

��pl (x; �; �)

�+K��3=2

��pl (x; �; �)

��2�K��1=2

��pl (x; �; �)

� � � (x� �)pl (x; �; �)

from the fact that

d

dxlnK��1=2

��

q�2 + (x� �)2

�(3.12)

=

�12

�K�+1=2

��q�2 + (x� �)2

�+K��3=2

��q�2 + (x� �)2

��K��1=2

��q�2 + (x� �)2

�� (x� �)q

�2 + (x� �)2

=�12

�K�+1=2

��pl (x; �; �)

�+K��3=2

��pl (x; �; �)

��K��1=2

��pl (x; �; �)

� � � (x� �)pl (x; �; �)

wherel (x; �; �) = �2 + (x� �)2 (3.13)

(see Karsten99 [27] for properties of the modi�ed Bessel function of the thirdkind).

5It is obvious that for the observed detrended index, we have �t = 1.


Consequently

@

@Xtv�Xt; ~�

�= �0:5v

�Xt; ~�

� d

dxln f

�Xt; ~�

�(3.14)

= �0:5v�Xt; ~�

�� +

(�� 1=2) (Xt � �)l (Xt; �; �)

��K�+1=2

��pl (Xt; �; �)

�+K��3=2

��pl (Xt; �; �)

��2K��1=2

��pl (Xt; �; �)

� � (Xt � �)pl (Xt; �; �)

9=;

3.2. MCMC Methods and Metropolis-Hastings Algorithm. To es-timate the parameter vector ~� in Model I by Markov chain Monte Carlo( MCMC ) methods we �rst remove the constraints on the ranges of thecomponent paramters (except for j�j < �) by a batch of transformations8>>>><>>>>:

� = exp (��) ; �� 2 R� = exp (��) ; �� 2 R� = exp (��) ; �� 2 R

�� = ln� + �

� � � , � =exp (��)� 1exp (��) + 1

exp (��) ; �� 2 R

(3.15)

Let the new parameter vector be denoted by ~� = (��; ��; ��; �; �; ��) =�~��1;~��2;~��3;~��4;~��5;~��6

�2 R6 and still let � =

�~��1;~��2;~��3;~��4;~��5

�for notational

simplicity, we then have

v�x; ~�

�=

sexp (��)

f (x; �)

By the Bayesian formula, the posterior density ~p�~�jY

�for the parameter

vector ~� given the observation Y is approximated by

p�~�jY

�=

p�Yj~�

��~��

Rp�Yj~�

��~��d�/ �

�~��LM

�~�jY

�(3.16)

where ��~��is the prior density for ~�, LM

�~�jY

�is as in (3.10).

To implement the MCMC methods to estimate ~�, we set the prior ��~��to

be a radially symmetric uniform distribution centered at zero with compactsupport as large as required, and propose the next value of parameter vector~�0by generating it from the multivariate normal proposal N

�~�n;�

�with

mean vector ~�n and a pre-assigned constant covariance matrix �, where ~�nis the current value of the parameter vector ~� at the n-th iteration in theMetropolis-Hastings (MH) algorithm that drives the proposed Markov chain.

16 XIONGZHI CHEN

Under such settings, the acceptance probability for ~�0given ~�n is simply

��~�0j~�n

�= min

8<:1; p�~�0jY�q�~�nj~�

0�p�~�njY

�q�~�0j~�n

�9=; (3.17)

= min

8<:1; p�Yj~�0

��~�0�q�~�nj~�

0�p�Yj~�n

��~�n

�q�~�0j~�n

�9=;

� min

8<:1; LM�~�0jY��~�0�q�~�nj~�

0�LM

�~�njY

��~��q�~�0j~�n

�9=;

= min

8<:1; LM�~�0jY�

LM

�~�njY

�9=;

where p�Yj~�

�is the joint posterior for the transformed observationY given

~� and q�~�0j~�n

�= N

�~�n+1; ~�n;�

�(Note that N

��; ~�n;�

�here also denotes

a multivariate normal density with mean vector ~�n and covariance matrix�).The Metropolis-Hastings (MH) algorithm (see, for example, Chib95 [12])

is summarized below:

Step 1: Given the current state ~�n, generate a candidate ~�0from the

proposal density q��j~�n

�Step 2: Calculate the acceptance probability �

�~�0j~�n

�Step 3: 6Generate a random number u from the uniform distribution

on (0; 1) : If u � ��~�0j~�n

�, set ~�n+1 = ~�

0. Otherwise, set ~�n+1 = ~�n.

Step 4: Repeat the previous steps to obtain a chainn~�0; ~�2; :::

o; where

~�0 denotes the initial state of ~�. Discard the burn-in valuesn~�0; ~�0; :::~�m

ofor some preassigned m 2 N (probably depending on the acceptancerates) obtained while the chain converges in distribution . Then the

remaining values,n~�m+1; ~�m+1; :::

o; form a correlated Markov chain

sampled from its stationary density which is p�~�jY

�.

As long as the chain converges, the post burn-in values, say,n~�m+1; ~�m+1; :::; ~�n1

oare used to estimate the unknown parameters by Monte Carlo methods. Inthis paper, the ergodic mean

~��=

1

n1 �m

n1Xi=m+1

~�i ! Ep(~�jX)

�~�jY

�; a:s: (3.18)

6Note that the acceptance criterion adopted here is the most widely used one.


which is exactly the Monte Carlo estimate7 of the unknown ~�, is used as anestimate of the unknown ~�; where "! a:s:" denotes "strong convergence"

and the expectation is taken with respect to the posterior density p�~�jY

�:

(For more details on Monte Carlo methods in �nance, see Glasserman04[32]).

3.3. Convergence Checking. In the actual implementation of the MCMCmethods, the sampled path, denoted by

n~�i : i = 1; 2; � � � ; N

o;forms a con-

vergent Markov chain whose stationary density is the unknown posterior

p�~�jY

�; and the target quantity Ep(~�jY)

h'�~��iis estimated by the er-

godic average, i.e., the Monte Carlo (MC) estimate

~'N =1

N

NXi=1

'�~�(i)�

(3.19)

where ' (�) is a (pre-assigned) real-valued function, Ep(~�jY) [�] again denotes

the expectation with respect to p�~�jY

�. In this paper, ' is just the identity

function.Yet the most important part in MCMC simulation is convergence checking

(since the best we can do is not to reject the null hypotheis that the chain hasconverged), which can be done by a combination of checking Monte Carlostandard errors (MCSEs) and the Gelman-Rubin convergence diagnostic.

3.3.1. MCSE Convergence Criterion. In Roberts96 [35] is stated the follow-ing convergence theorem

pN�~'N � Ep(~�jY)

h'�~��i�

d! N�0; s2'

�(3.20)

for some s2' � 0; whered! denotes convergence in distribution and N (a; b)

denotes a normal density with mean a and variance b. Thus the accuracy of

the ergodic average as an estimate of Ep(~�jY)

h'�~��iis essentially measured

by s2': a quantity that can only be empirically estimated from the sample.In practice, s2' is estimated by the so called " batch mean ", where theMCMC algorithm is run for N = m� n iterations for su¢ ciently large n sothat

yk =1

n

knXi=(k�1)n+1

'�~�(i)�; k = 1; 2; :::;m (3.21)

are approximately independently distributed as

N�Ep(~�jX)

h'�~��i; s2'=n

�7Monte Carlo estimate is essentially just the avarage of the sample from the target

quantity, regardless of ergodicity. Theoretical convergence of the Markov chains in thesimulation and ergodicity of the means obtained from the post burn-in samples here areensured by the detailed balance condition built into the Metroplolis-Hastings algorithm.

18 XIONGZHI CHEN

and then s2' is estimated by

s2' =n

m� 1

mXk=1

(yk � ~'N ) (3.22)

Subsequently the standard error of ~'N can be estimated byqs2'=N; which

is called the Monte Carlo standard error (MCSE) and is used to measure themixing performance of the MH algorithm and as a convergence diagnosticfor the MCMC algorithm .

3.3.2. Gelman-Rubin Statistic. Since a single Markov chain may convergelocally to some mode of the unknown posterior and the MSCE is not ableto diagnose whether the proposed unknown posterior has been su¢ cientlysampled, we have to run multiple chains in the MCMC simulation to ensurea large proportion of the unknown posterior is to be sampled and checkwhether all chains have converged to the same stationary distribution. Con-sequently, we adopt the famous Gelman-Rubin (GR) statistic8 (see Gel-man92 [20], Brooks00 [11], and Brooks98 [10]) for convergence diagnosis ofmultiple-sequence simulations.Now we decribe how to construct the GR statistic in both univariate

and multivariate cases. We independently simulate I � 2 sequences oflength 2N; each beginning with di¤erent initial values that are su¢ ciently9

widely dispersed with respect to the unknown yet stationary distribution (towhich the I independent Markov chains are supposed to converge). Thenwe discard the �rst N iterations of each simulated sequence and retain onlythe last N . For the purpose of our simulation here, we compute variance

between the I sequence means, which are denoted by �(i)

� , and de�ne

B =1

I � 1

IXi=1

��(i)

� � ��

(3.23)

where ~�(i)t denotes the t-th observation of ~� from the i-th chain and

�(i)

� =1

N

2NXt=N+1

~�(i)t ; �

�� =

1

I

IXi=1

�(i)

�

Further, we de�ne W; the mean of the I within-sequence variances, s2(i), by

W =1

I

IXi=1

s2(i) (3.24)

where

s2(i) =1

N � 1

2NXt=N+1

�~�(i)t � �(i)�

�28In the literature, the GR statistic is often called the "Potential Scale Reduction Ractor

(PSRF)".9It is possible to measure how "su¢ ciently" the initial values are dispersed, see Demp-

ster77 [14]. Also a method to choose su¢ ciently widely dispered initials are given in thispaper. However due to the time constraints for the project, the author was not able touse the method suggested therein.


Finally if ~� is univariate, then we de�ne

V =N � 1N

W +

�1 +

1

I

�B (3.25)

and

R =V

W(3.26)

Otherwise, we de�ne

Rp =N � 1N

+

�1 +

1

I

�� (3.27)

where � is the largest eigenvalue of the positive de�nite matrix W�1B. (SeeBrooks98 [10] for more details on (3.26), (3.27).)The key property of R, Rp is that they both converge to 1 when all individ-

ual Markov chains in the MCMC simulation converge to the same stationarydensity. Consequently they can be used as convergence diagnostics. Anotherinteresting relationship between them is that, if R (k) denotes the GR statis-tic for the k-th component of a d-dimensional multivariate parameter vector~� in multiple-chain MCMC simulation, thenmax1�k�d

nR (k)

o� Rp. Since

each R (k) � 1; k = 1; :::; d if the initial values for the Markov chains in thesimulation are su¢ ciently widely dispersed (see Brooks00 [11]), we do nothave to check the convergence for each R (k) ; k = 1; :::; d when convergenceof Rp is con�rmed.

3.3.3. Con�dence Interval under GR Scheme. Under the Gelman-Rubin di-agnostic scheme, a 100 (1� p)% central posterior con�dence interval for theparameter ~� is obtained based upon the �nal N of 2N iterations as decribedbelow (see, Brooks98 [10]):

S1: From each individual chain, take the empirical 100 (1� p)% in-

terval, that is, the 100p

2% and the 100 (1� p)% points of the N

simulation draws to form I within-sequence interval length estimatesS2: From the entire set of IN observations gained from all chains, com-pute the empricial 100 (1� p)% interval to obtain a total-sequenceinterval length estimate

S3: Evaluate R de�ned as

Rinterval =length of total-sequence interval

mean length of within-sequence intervals

3.4. Implementation and Estimates. Since the approximate posteriorp�~�jY

�is propotional to LM

�~�jY

�and the prior �

�~��is chosen to be a

radially symmetric uniform distribution centered at zero, we see that the

modes of p�~�jY

�depends on those of LM

�~�jY

�. In order to choose ini-

tial values that are su¢ ciently widely dispersed with respect to p�~�jY

�, we

estimated all 3 modes of the empirical density of the detrended index (see,Figure 2 for this density and its nonlinear least squares �t by a GH density)and for each mode a group of two independent chains are simulated. Unfor-

20 XIONGZHI CHEN

4 2 0 2 4 6 80

0.02

0.04

0.06

0.08

0.1

0.12

0.14


Pro

ba

bility

Empirical Density of Linearly Detrended Logarithmic SSE Indexagainst Its Nonlinear Least Squares Fit

Empirical Density of the Detrended IndexNonlinear Least Squares Fit

Figure 2. Empirical Density of Detrended Index

3 2 1 0 1 2 3 4 5 6 70

1

2

3

4

5

6

7x 103 Compared Squared Diffusions for Chain 2


Sq

ua

red

Vo

latil

ity

Squared Empirical Diffusion FunctionModel Predicted Squared Diffusion Function

Figure 3. Squared Volatility from Model 2

tunately this simulation scheme gives negative z�s in (3.8), which forces theauthor to choose the following initialization strategy.Firstly by �tting the empirical squared di¤usion of the detrended index

by the v��; ~��in (2.6) in the least square sense (within the working precision

of Matlab), we obtain the initial estimate

~�0 = (�0; �0; �0; �0; �0; �0)

= (�0:3090;�2:7787;�19:2004; 3:2323; 3:6693;�3:7583)(Figure 3 plots the squared di¤usion function for X = fXt; 1 � t � 4131gbased on the estimated parameter vector from chain 2 in the simulation)Then we simply propose 6 independent Markov chains whose initial states

are the 6 di¤erent vectors generated by a symmetric uniform distributioncentered at ~�0 and start the Metropolis-Hastings algorithm. The tuning


0 10 20 30 40 50 60 70 80 90 1000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

Number of Groups in Iterations (Group size 500)

Ma

xim

um

Co

mp

on

en

t M

on

te C

arl

o S

tan

da

rd E

rro

r

Maximum of Monte Carlo Standard Errors for 6 Component Paramters for Individual Chains

Figure 4. Maximal MCSEs for Individual Chains

� � � � � �Chain 1 �0:633729 �3:003546 �19:157304 3:041060 4:578130 �3:509746Chain 2 �0:358308 �2:672644 �20:055300 2:611961 4:382647 �5:049936Chain 3 �0:412318 �1:712247 �18:745429 2:597826 3:120015 �5:662547Chain 4 �0:423069 �1:950044 �18:693538 2:999596 3:721765 �4:576278Chain 5 �0:277084 �1:368601 �19:774321 3:635318 3:143871 �4:097122Chain 6 �0:389012 �2:301538 �19:248898 2:943287 4:338962 �4:459357

Table 1. Monte Carlo Estimates for 6 Chains

paramter is set to be � = 0:0015; partially based on the dimension of theparameter space (see, Haario01 [23], Roberts08 [34]) and by the requirementto control the acceptance rates to be between 0:2 and 0:5 (see, Johannes03[25], Chib95 [12], and Tse04 [42]).At around 50; 000 iterations MCSEs suggests that all six independent

chains have converged to their own stationary distributions and remainsat around 10�3 afterwards. (See Figure 4 for the plot of the maximumcomponent MSCEs for 6 individual chains for the �rst 50; 000 iterations.)But the GR statistic does not convergence to 1 even after 350; 000 it-

erations and the avarage acceptance rates are too low for six chains afteraround 50; 000 iterations, so the simulation has to be terminated since the 6chains might have converged to di¤erent stationary distributions and furtheriterations will not statistcally move the chains forward. (See Figure 5 forthe acceprance rates and the �gures for the badly mixed sample paths inthe appendix)We take the last 10% of the �rst 50; 000 simulated samples (which we

call the "e¤ective samples" ) from each chain in the simulation to reduceor remove the e¤ects of initilization in the Markov chains, and obtain 6MC estimates for the parameter vector ~�; which suggests 6 possible models.Table 1 summarizes the MC estimates for each component parameter foreach individual chain amd the corresponding MSCEs for the 100-th batch are

22 XIONGZHI CHEN

0 0.5 1 1.5 2 2.5 3 3.5

x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Iterations

Acce

pta

nce

Ra

tes

Acceptance Rates Against Number of Iterations for 6 Chains

Figure 5. Acceptance Rates for Individual Chains

� � � � � �Chain 1 0.0482 0.0635 0.0278 0.0544 0.2010 0.2914Chain 2 0.0002 0.0006 0.0001 0.0001 0.0004 0.0012Chain 3 0.0003 0.0012 0 0.0001 0.0001 0.0018Chain 4 0.0004 0.0016 0.0001 0.0001 0.0007 0.0024Chain 5 0.0843 0.4031 0.0190 0.0774 0.1154 0.1500Chain 6 0.1628 0.9247 0.1062 0.0403 0.2049 0.7743

Table 2. MCSEs

provided in Table 2 (Note that all numbers in Table 2 should be multipliedby 10�3).From Table 2, the 95% Bayesian con�dence interval (CI) for each estimate

in Table 1 can be computed by (3.20), but is unnecessary to be computedhere because the next subsection rejects all 6 possible models based on theseestimates given in Table 1, even though the MCSEs in Table 2 must givegood estimates for the CIs,.

3.5. Test of Model. To test the proposed model

Xt = X0 +

Z t

0v�Xs; ~�

�dWs; t � 0

for the detrendex indexnln ~Pt � �t; t = 1; :::; 4131

o, we simply follow the

methods in Bibby97 [5] and compute the uniform residuals

Ut

�~�(j)est

�= P~�(j)est

(Xt+1 � yjXt = x) ; x; y 2 R; j = 1; ::; 6; t = 1; ::; 4130


Reject Null p-Value Chi2 Stat DFChain 1 Yes 1:8573E � 161 777:0412 9Chain 2 Yes 2:8942E � 149 720:3632 9Chain 4 Yes 9:476E � 152 731:9177 9Chain 6 Yes 8:8394E � 152 732:0581 9

Table 3. Test Statistics for 4 Models

for each estimate ~�(j)est of ~� obtained from the 6 chains, where P~�(j)est

(�j�) isexactly the conditional (transition) density in (3.5). If the models are ap-

propriate, then Ut�~�(j)est

�; t = 1; ::; 4130 for each �xed j should be indepen-

dent and identically uniformly distributed in (0; 1) (which is easily seen fromthe Milstein�s discretization, the property of the standard Brownian motion,and the invariance property of independence (between random vectors) un-der Borel transformations).Due to the computational inability of both Mathematica and Matlab re-

sulted from badly mixed chain 3 and chain 5 (as revealed by the sample pathsgiven in the appendix), we have to discard models obtained from them andtest only the other four models obtained from chain 1, chain 2, chain 4 andchain 6 in the simulation by the uniform residual test (see, Bibby95 [4],

Bibby97 [5]). We directly test the null hypothesis: Ut�~�(j)est

�; t = 1; ::; 4130

for each �xed j are independent and identically uniformly distributed in

(0; 1) 10 (rather than testing �2P4130t=1 ln

�Ut

�~�(j)est

��for �xed j as done in

Bibby97 [5] ).Tests results for these 4 chains are summarized in Table 3 , where "p-

Value" denotes the signi�cance leve at which the null hypothesis is rejected,"Chi2 Stat" the chi-squared statistic, and "DF" the degrees of freedom. Sup-

plimentary to these statistics, the quantile plot of Ut�~�(j)est

�; t = 1; ::; 4130

for j = 2 is provided in Figure 6.These test results clearly show that all 4 possible models are rejected

statistically because of the low chi-squared statistics (compared to the samplsize 4131) and degrees of freedom. Thus under the schme of approximatingthe unknown posterior for the parameter vector by the likelihood functionobtained from the Milstein�s discretization, we reject the null hypothesis that" the underlying stochastic process for the linearly detrended logarithmicSSE in the period from 01/01/1991 to 14/11/2007 is an ergodic, GH di¤usionwith some GH invariant density", i.e., the detrended index is not an ergodicGH di¤usion.

4. CONCLUSION AND DISCUSSION

It is no doubt that the ergodic GH di¤usion can not �t the detrendedindex since the for the ergodic GH di¤usion its invariant density, the GHdensity, is unimodal while the empirical density of the detrended index is

10This is easily done by Matlab command "chi2gof", i.e., chi-squared goodness of �ttest, designed to test whether the sample are independent and identically uniformly dis-tributed in (0; 1) or not.

24 XIONGZHI CHEN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.4

0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Quantiles of Standard Uniform

Qu

an

tile

s o

f U

nifo

rm R

esid

ua

ls

QQ Plot of the Uniform Residuals against Standard Uniform

Figure 6. QQ Plot of Uniform Residuals for Model 2

tri-modal. Uniform residual test results also reject the assumption that theunderlying stochastic process for the detrended index is an ergodic GH dif-fusion since obviously the residuals from the model are not independent andidentically uniformly distributed in (0; 1) and the detrended index still has astrong linear trend. For more evidence on rejecting the ergodic GH di¤usion,please see the appendix for other numerical results such as autocorrelations,simulated detrended index, etc..The high acceptance rates in the �rst 50; 000 iterations might be the result

of correctly estimated initial states for the Markov chains in the simulationor due to the small tuning parameter which results in a narrow scope ofsampling. Since we approximate the unknown posterior by the likelihoodfunction obtained from the Milstein�s discretization scheme and the initialstates for the Markov chains are obtained based on the empirical squareddi¤usion, it might be that these initial states capture the modes of theunknown posterior su¢ ciently well before the chains wandered away fromthe high acceptance region. Also, a small tuning parameter might be thecause of small MCSEs in the simulation, since the proposal states for theMarkov chains in the simulation might be concentrated locally around someregion of the unknown posterior.Finally, due to the high volatility of the SSE index, it unknown to the

author yet whether the posterior of the unknown parameter vector basedon the transformed observation approximates the exact unknown posteriorwell or not.

References

[1] Yacine Aït-Sahalia 1996: Nonparametric pricing of interest rate derivative securities.Econometrica, Vol 64, No. 3 (May, 1996), 527-560

[2] Yacine Aït-Sahalia 1999: Transition densities for interest rate and other nonlineardi¤usions. The journal of �nance, Vol IV, No 4, August 1999.

[3] Yacine Aït-Sahalia: Maximum Likelihood Estimation of Discretely Sampled Di¤u-sions: A Closed-From Approximation Approach, Econometrica, Vol. 70, No. 1 (Jan-uary, 2002), 223�262


[4] B.M. Bibby, M. Sorensen 1995: Martingale estimation functions for discretely ob-served di¤usion processes. Bernoulli 1(2), 1995, 017-039

[5] B.M. Bibby, M. Sorensen 1997: A hyperbolic model for stock prices. Finance andStochastics 1, 25-41, 1997

[6] B.M. Bibby 2001: Simpli�ed estimating functions for di¤usion models with a high-dimensional parameter. Scandinavian Journal of Statistics, Vol 28:99-112, 2001

[7] B.M Bibby, et al 2004: Estimating Functions for Discretely Sampled Di¤usion-TypeModels. Working paper

[8] B.M. Bibby, IB M. Skovgaard, M. Sorensen 2005: Di¤usion-type models with givenmarginal distribution and autocorrelation function. Bernoulli 11(2), 2005, 191-220

[9] Jaya P. N. Bishwal 2008: Parameter Estimation in Stochastic Di¤erential Equations.Springer Verlag 2008

[10] Stephen P. Brooks and Andrew Gelman 1998: General methods for monitoring con-vergence for iterative simulations. Journal of Computational and Graphical Statistics,Vol. 7, Number 4, 1998

[11] S.P. Brooks and P. Giudici 2000: Markov chain Monte Carlo convergence assessmentvia two-way analysis of variance. Journal of Computational and Graphical Statistics,Vol. 9, No 2, 2000

[12] S. Chib, E. Greenberg 1995: Understanding the Metropolis-Hastings Algorithm. TheAmerican Statistician, November 1995, Vol. 49, NO. 4

[13] Rama Cont 2001: Empirical properties of asset returns: stylized facts and statisticalissues. Institute of Physics Publishing 2001 PII:S1469-7688(01)21683-2

[14] A. P. Dempster; N. M. Laird; D. B. Rubin 1977: Maximum Likelihood from Incom-plete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B(Methodological), Vol. 39, No. 1. (1977), pp. 1-38

[15] Bjorn Eraker 2001: MCMC analysis of di¤usion models with application to �nance.Journal of Business and Economic Statistics. April 2001, Vol 19, No. 2

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[19] H.J. Engelbert and W. Schmidt (1984). On solutions of one-dimensional stochasticdi¤erential equations without drift. Probability Theory and Related Fields

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[21] Guang-lu Gong 1979: Introduction to stochastic di¤erential equations (Chinese). Bei-jing University Press

[22] P. Gopikrishnan, H. E. Stanley, et al 1999: Scaling of the distribution of �uctuationsof �nancial market indices. Physical Review E, Vol 60, Number 5, Nov. 1999

[23] H. Haario, et al, 2000: An adaptive metropolis algorithm. Bernoulli 7(2), 2001, 223-242[24] B. Jessen, R Poulsen, 1999: A comparison of approximation techniques for transition

densities of di¤usion process. Working paper, Center for analytical �nance, Universityof Aarhus

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[26] Yury A. Kutoyants 2004: Statistical inference for ergodic di¤usion processes. SpringerVerlag 2004

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Carlo method. Quantitative Finance, Vol 4 (2004) 158-169


5. APPENDIX

5.1. Numerical Results. We present some numerical results in the MCMCsimulation. The e¤ective sample paths are the last 10% of the �rst 50; 000iterations.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.8

0.6

0.4Sample paths for alpha for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.6

0.4


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.5

0.4


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.5

0.4


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.3

0.28


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50000.6

0.4


Figure 7. Sample Path for � for 6 Chains

28 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.2

3

2.8Sample paths for beta for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.8

2.6


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001.8

1.7


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.2

2


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50001.4

1.35


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.5

2Sample paths for beta for chain 6



0 500 1000 1500 2000 2500 3000 3500 4000 4500 500019.4

19.2

19Sample paths for delta for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500020.2

20

19.8Sample paths for delta for chain 2

0 500 1000 1500 2000 2500 3000 3500 4000 4500 500018.8

18.7


0 500 1000 1500 2000 2500 3000 3500 4000 4500 500018.8

18.7


0 500 1000 1500 2000 2500 3000 3500 4000 4500 500019.8

19.78


0 500 1000 1500 2000 2500 3000 3500 4000 4500 500019.4

19.2

19Sample paths for delta for chain 6


30 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003

3.1

3.2Sample paths for lamda for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

2.6


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.55

2.6


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.8

3


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.6

3.65


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50002.8

3




0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5

5Sample paths for miu for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003

3.2

3.4Sample paths for miu for chain 3

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.6

3.8


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.1

3.2

3.3Sample paths for miu for chain 5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004

4.5



32 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50003.6

3.4

3.2Sample paths for sigma for chain 1

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50005.5

5


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50005.7

5.65


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50005

4.5

4Sample paths for sigma for chain 4

0 500 1000 1500 2000 2500 3000 3500 4000 4500 50004.15

4.1


0 500 1000 1500 2000 2500 3000 3500 4000 4500 50005

4.5

4Sample paths for sigma for chain 6



0 100 200 300 400 500 600 700 800 900 10000

0.5

1ACFs for the paramter alpha for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10001

0


0 100 200 300 400 500 600 700 800 900 10000

0.5


Figure 13. Autocorrelation for � for 6 Chains

34 XIONGZHI CHEN

0 100 200 300 400 500 600 700 800 900 10000

0.5

1ACFs for the paramter beta for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10001

0


0 100 200 300 400 500 600 700 800 900 10000

0.5




0 100 200 300 400 500 600 700 800 900 10001

0

1ACFs for the paramter delta for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10001

0


0 100 200 300 400 500 600 700 800 900 10000

0.5



36 XIONGZHI CHEN

0 100 200 300 400 500 600 700 800 900 10001

0

1ACFs for the paramter lamda for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5




0 100 200 300 400 500 600 700 800 900 10000

0.5

1ACFs for the paramter miu for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10001

0


0 100 200 300 400 500 600 700 800 900 10000

0.5



38 XIONGZHI CHEN

0 100 200 300 400 500 600 700 800 900 10000

0.5

1ACFs for the paramter sigma for chain 1

0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10000

0.5


0 100 200 300 400 500 600 700 800 900 10001

0


0 100 200 300 400 500 600 700 800 900 10000

0.5




3 2 1 0 1 2 3 4 5 6 70

0.005

0.01Compared Squared Diffusions for Model 1

3 2 1 0 1 2 3 4 5 6 70

0.005


3 2 1 0 1 2 3 4 5 6 70

0.005


3 2 1 0 1 2 3 4 5 6 70

0.005


3 2 1 0 1 2 3 4 5 6 70

0.005


3 2 1 0 1 2 3 4 5 6 70

0.005


Empirical Squared VolatilityModel Based Squared Volatility






Figure 19. Model Based Squared Volatilities

40 XIONGZHI CHEN

0 500 1000 1500 2000 2500 3000 3500 4000 45004

2

0

2

4

6

8Simulated Detrended Index from Model 1

0 500 1000 1500 2000 2500 3000 3500 4000 45004

2

0

2

4

6


0 500 1000 1500 2000 2500 3000 3500 4000 45004

2

0

2

4

6


Detrended IndexSimulation from Model 3



Figure 20. Simulated Detrended Index from Models 1, 2, 3


0 500 1000 1500 2000 2500 3000 3500 4000 45005

0

5

10Simulated Detrended Idex from Model 4

0 500 1000 1500 2000 2500 3000 3500 4000 45005

0

5


0 500 1000 1500 2000 2500 3000 3500 4000 45005

0

5





Figure 21. Simulated Detrended Index from Models 4, 5, 6

42 XIONGZHI CHEN

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Unifmorm Quantiles

UR

Q

uantiles

QQ Plot of Uniform Residuals against Standard Uniform for Model 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Unifmorm Quantiles

UR

Q

uantiles


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Uniform Quantiles

UR

Q

uantiles


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

Standard Uniform Quantiles

UR

Q

uantiles


Figure 22. QQ Plot of Uniform Residuals for 4 Tested Models

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