Stochastic Modelling of Random Variables with an Application in Financial Risk Management

Author:

Max V. Moldovan Manager (Institute of Management, Moscow State University of Economics, Statistics and Informatics)

Principal supervisor:

A/Prof Andrew C. Worthington (School of Economics and Finance, Queensland University of Technology)

School of Economics and Finance Faculty of Business

Queensland University of Technology August 2003

i

Abstract

The problem of determining whether or not a theoretical model is an accurate

representation of an empirically observed phenomenon is one of the most challenging

in the empirical scientific investigation. The following study explores the problem of

stochastic model validation. Special attention is devoted to the unusual two-peaked

shape of the empirically observed distributions of the conditional on realised volatility

financial returns. The application of statistical hypothesis testing and simulation

techniques leads to the conclusion that the conditional on realised volatility returns are

distributed with a specific previously undocumented distribution. The probability

density that represents this distribution is derived, characterised and applied for

validation of the financial model.

Keywords: model validation; realised volatility; high-frequency data; two-component

effect; modelling of random variables; simple test for normality; change-point

detection; small sample; end-of-sample problem

ii

Acknowledgments

A number of people have contributed to the following thesis. Special thanks to my

scientific mentor Nicholas Nechval for introducing me to science and to Andrew

Worthington for the excellent supervision of the most difficult stages of the research.

Many thanks to Alan Layton for his efforts and talent in leading the School. Many

thanks to Stan Hurn for his sensitivity and professionalism in coordinating the

research. Many thanks to Helen Higgs for her friendly support. Thanks to Ralf Becker

and Shakila Aruman for constructive criticism and helpful suggestions. Thanks to all

High Performance Computing and Research Support staff and especially to Mark

Barry who gave me the actual guidance in using the supercomputing facilities. Many

thanks to people from QUT Document Delivery Service who managed to deliver even

the most ‘hopeless’ documents. Thanks to all Business Faculty Technical Service staff

and especially to Marty Wade for solving problems with computer software. Many

thanks to all staff and students of School of Economics and Finance for providing the

creative work environment. The list of people who have contributed to this research is

not exhaustive and I would like to extend acknowledgments to everyone who

accompanied me over the last three years.

iii

Candidature’s Declaration

I hereby declare that this work has not previously been submitted for a degree or

diploma in any university or other tertiary institution. To the best of my knowledge

and belief, this dissertation contains no materials previously published or written by

another person, except where due reference is made. Any errors, omissions or

inaccuracies are entirely my responsibility.

Max V. Moldovan

iv

Table of Contents Abstract i Acknowledgments ii Candidature’s Declaration iii Table of Contents iv List of Numerical Examples vi List of Tables vii List of Figures viii Chapter 1 Introduction 1 Chapter 2 Stochastic Properties of Financial Time Series 3

2.1 Introduction 3 2.2 Properties of Price Time Series 3 2.3 Properties of Return Time Series 10 2.4 Properties of Financial Return Volatility

and Some Volatility Estimators 15 2.5 Concluding Remarks 18

Chapter 3 Realised Volatility 19

3.1 Introduction 19 3.2 The Main Concepts of Realised Volatility 19 3.3 Modelling Realised Volatility 31 3.4 Concluding Remarks 37

v

Chapter 4 The Two-Component Effect 38

4.1 Introduction 38 4.2 Likelihood Ratio Test 40 4.3 Pearson Goodness of Fit Test 44 4.4 Simple Test for Normality 47 4.5 Simulation-Based Test 57 4.6 Concluding Remarks 62

Chapter 5 Modelling of Random Variables 63

5.1 Introduction 63 5.2 Derivation and Characterisation of ),( αγJ 63 5.3 Application of ),( αγJ :

Forecasting Probability Quintiles of Future Price Distributions 72 5.4 Concluding Remarks 78

Chapter 6 Conclusion 79

6.1 General Overview 79 6.2 Potential Applications and Limitations 80 6.3 Avenues for Further Research 82

Bibliography 86 Appendix A1 Correlogram of AUD/USD futures exchange rates Appendix A2 Correlogram of JPY/USD futures exchange rates Compact Disk (three Matlab functions): j_rnd.m ),(~ αγJ random number generator j_rnd_multimod.m J~ multimodal random number generator change_detect.m Change-point detection in regression relationship

vi

List of Numerical Examples

Numerical Example 2.1: Testing Series for a Unit Root 6 Numerical Example 3.1: Volatility Signature Plot 28 Numerical Example 3.2: Modelling Optimal Realised Volatility Series 32

vii

List of tables Table 3.1 Descriptive statistics of ktv , , kt ,σ and ktL ,σ series 36 Table 3.2 Estimates from ARMA (1,1) applied to AUD

tL 8,σ and JPYtL 10,σ series 36

Table 3.3 Estimates from GARCH (1,1) applied to AUDt ,ε and JPYt ,ε series 37 Table 4.1 Descriptive statistics of standardised returns 39 Table 4.2 Parameters of a mixture of two normals

model fitted to AUD/USD series 43 Table 4.3 Parameters of a mixture of two normals

model fitted to JPY/USD series 43 Table 4.4 Pearson goodness-of-fit statistics for a mixture of two normals 45 Table 4.5 Pearson goodness-of-fit statistics for a single normal 46 Table 4.6 Yield of hydrogen sulphide from gamma

radiolysis as a function of krypton pressure 56 Table 4.7 Simple standard deviations of intra-day returns 58 Table 5.1 Numbers of exceptions of (5.7) with )1,0(~ Nzt

in (5.7b) applied to AUD/USD series 76 Table 5.2 Numbers of exceptions of (5.7) with )1,0(~ Nzt

in (5.7b) applied to JPY/USD series 76 Table 5.3 Numbers of exceptions of (5.7) with )8.14,08.0(~ J

in (5.7b) applied to AUD/USD series 77 Table 5.4 Numbers of exceptions of (5.7) with )5.15,15.0(~ J

in (5.7b) applied to JPY/USD series 77 Table 5.5 Numbers of exceptions of (5.7) with [ )5.15,15.0(~ J – 0.03]

in (5.7b) applied to JPY/USD series 78

viii

List of figures Figure 2.1 Simulated data points: Non-stationary process 5 Figure 2.2 Simulated data points: Stationary process 5 Figure 2.3 AUD/USD futures exchange rates 7 Figure 2.4 JPY/USD futures exchange rates 7 Figure 3.1 Histogram of 1,iv 24 Figure 3.2 Histogram of 5,iv 25 Figure 3.3 Histogram of 25,iv 26 Figure 3.4 Realised volatility signature plot for AUD/USD series 29 Figure 3.5 Realised volatility signature plot for JPY/USD series 29 Figure 3.6 Histogram of AUD/USD realised volatility series AUD

tv 8, 33 Figure 3.7 Histogram of JPY/USD realised volatility series JPY

tv 10, 33 Figure 3.8 Histogram of AUD/USD realised standard deviation series AUD

t 8,σ 34 Figure 3.9 Histogram of JPY/USD realised standard deviation series JPY

t 10,σ 34 Figure 3.10 Histogram of AUD/USD log realised

standard deviation series AUDtL 8,σ 35

Figure 3.11 Histogram of AUD/USD log realised

standard deviation series JPYtL 10,σ 35

Figure 4.1 Standardised by 8,tσ AUD/USD returns 39 Figure 4.2 Standardised by 10,tσ JPY/USD returns 40 Figure 4.3 AUD

tz 40, : Theoretical against empirical frequencies 45 Figure 4.4 AUD

tz 80, : Theoretical against empirical frequencies 46

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Figure 4.5 Histogram of simulated variables jtz 61

Figure 5.1 )1,0(~ J 66 Figure 5.2 )2,0(~ J 66 Figure 5.3 )20,0(~ J 67 Figure 5.4 )200,0(~ J 67 Figure 5.5 Relationship between d and α 68 Figure 5.6 )5,1(~ −J 69 Figure 5.7 )5,1(~ J 70 Figure 5.8 The left component of )5,0(~ J : Right skewness 71 Figure 5.9 The right component of )5,0(~ J : Left skewness 71 Figure 5.10 Illustration of VaR 73

1

Science may be described as the art of systematic over-simplification – the art of discerning what we may with advantage omit. (K. Popper, 1982, p. 44, The open universe).

Chapter 1

Introduction

Everyday individuals, companies and government departments are exposed to a

wide range of risks. Recognising this, a constructive individual or organisation will

identify practical ways of measuring the risk exposure and reducing it to an

acceptable level in the most efficient manner. This is particularly true for financial

organisations that naturally operate in risk-reward conditions.

In order to optimise the control of financial operations and minimise risks,

stochastic models of financial prices are often used. One of the most important steps

in the development of a stochastic model is determining whether the model is an

accurate representation of the empirical process being studied. The calibration of a

model to the certain empirically observed phenomenon is usually referred to as model

validation. The problem of stochastic model validation remains today perhaps the

most elusive of all the unresolved methodological problems associated with modelling

techniques. It is generally preferable to use some form of objective analysis to

perform the model validation. In this study the stochastic process that underlies the

dynamics of financial variables is examined with the application of statistical

hypothesis testing and simulation techniques. The results will be applied in validation

of the stochastic model that measures the financial risk.

The thesis is structured as follows:

Chapter 2 Stochastic Properties of Financial Time Series presents some

characteristics of the stochastic process that generates financial variables. Special

emphasis is placed on aspects that allow a description of the dynamics of financial

variables in probabilistic terms.

2

Chapter 3 Realised Volatility explores the main ideas underlying the realised

volatility estimator. Specifically, the concept of integrated volatility is introduced and

the asymptotic properties of the realised volatility with respect to the integrated

volatility are examined.

Chapter 4 The Two-Component Effect documents and examines the unusual

two-peaked shape of the empirical distributions of conditional on realised volatility

returns. A group of tests is presented and applied in order to find the cause of the

observed effect.

Chapter 5 Modelling of Random Variables illustrates how modelling techniques

can be applied directly to random variables in the example of derivation of the new

probability density. This chapter also gives some characteristics of this density and

demonstrates how it can be used in the stochastic model for financial risk

management.

Chapter 6 Conclusion summarises all results of the thesis, indicates limitations

and potential applications of these results, and suggests avenues for further research.

3

Chapter 2

Stochastic Properties of Financial Time Series 2.1 Introduction

For successful modelling of the financial time series it is important to

understand the character of the stochastic process that underlies the dynamics of

financial variables. The combination of profit-maximising actions of market

participants with physical structure (microstructure) of a marketplace and the

continuous sequence of events (informational shocks) creates conditions with specific

and somewhat distinct characteristics. In the following chapter these characteristics

are summarised in probabilistic terms.

This chapter assembles some empirical facts relevant to this study and the

underlying theoretical hypotheses that attempt to explain these facts. Emphasis is

placed not on contradictions between theories and reality but on points that allow

approximating the complex behaviour of financial markets by an appropriate

stochastic model. Section 2.2 describes the non-stationary memory-free character of

financial prices. How non-stationary price series can be transformed to stationary

return series is shown in Section 2.3. Linear serial dependencies in return series are

also considered in this section. The properties of the financial return volatility are

examined in Section 2.4. Some most widely used volatility estimators are introduced

and criticised in this section. Section 2.5 concludes the chapter with a brief summary

and remarks.

2.2 Properties of Price Time Series

Time series variables can be characterised by the parameters of a stochastic

process that they have been most likely generated from. If the parameters of the

process are not constant over time, the process is called non-stationary. In dealing

with a non-stationary stochastic process, one can face difficulties with making

inferences about future outcomes of variables. For example, Alexander (1961, 1964),

Cootner (1964), Cowles (1960), Fama (1965), Mandelbrot (1963), and Osborne

4

(1959, 1962), amongst many others, found that financial price time series are

generally non-stationary. If the process is non-stationary, it is difficult and sometimes

impossible to make inferences about future realisations of variables generated by the

process. For example, Fama (1965) described the behaviour of financial price series

as having “…no memory, that is, the past cannot be used to predict the future in any

meaningful way”. In this regard, the only relevant information about a future outcome

of a variable is the current value. Campbell et al. (1997) wrote that “…the ‘best’

forecast of tomorrow’s price is simply today’s price”. Fama (1965), Mandelbrot

(1963), Lo and MacKinlay (1988), amongst many others, use the random walk model

to describe the dynamics of financial prices:

ttt PP εµ ++= −1 (2.1)

where tP is the price at time t , ),0(~ 2tt N σε is the stationary random disturbance

term and µ is the drift term.

Note that the coefficient on 1−tP in (2.1) is equal to 1. This means that the shock

in the variable does not disappear over time and affects all consequent realisations. A

process of this type is known as containing a unit root. The presence of a unit root in

the series implies that a stochastic process that generates this series does not have a

constant unconditional expectation. If the regression coefficient on 1−tP in (2.1) is less

than 1 but greater than –1, the process becomes mean reverting and therefore

stationary. Two simulation examples demonstrate the difference between the process

with and without a unit root. Figure 2.1 shows five hundred randomly simulated data

points for the random walk with drift.

5

Figure 2.1 Simulated data points: Non-stationary process

-10

0

10

20

30

40

50

Non-stationary series Unconditional mean

Note: ttt yy ε++= −103.0 with )1,0(~ Ntε

In this example, variables ty start from a point close to 0.03 and move regardless of

the unconditional mean. This indicates that the process lacks the unconditional first

moment (the mean) and therefore is non-stationary. However, as soon as the

coefficient on 1−ty enters the unity circle (-1,1), the process becomes stationary.

Figure 2.2 shows five hundred randomly simulated data points, now without a unit

root.

Figure 2.2 Simulated data points: Stationary process

-5

-4

-3

-2

-1

0

1

2

3

4

Stationary series Unconditional mean

Note: ttt yy ε++= −15.003.0 , with )1,0(~ Ntε

6

The presence of a unit root in financial price time series has an underlying

theoretical explanation in the notion of market efficiency, as exposed by Bachelier

(1900), Samuelson (1965) and Fama (1970). Market efficiency relies on the idea that

if resources are limited and all market participants act rationally in order to maximise

their wealth, then the profit margin will tend to be zero. More formally, the Efficient

Market Hypothesis (EMH) states that if the market fully reflects all relevant

information, then all knowledge about the history of the price dynamics is irrelevant

and the best guess about the future price is the current price. Thus, in the context of

the EMH, the only relevant information for forecasting the future price is the current

price level, which is identical to the presence of a unit root in the time series of

interest. The following numerical example demonstrates how price time series can be

tested for the presence of a unit root.

Numerical Example 2.1: Testing Series for a Unit Root

Two main datasets are used in this study. These are the foreign futures exchange

rates between the Australian dollar (AUD) and US dollar (USD) and between the

Japanese yen (JPY) and US dollar (USD). AUD/USD and JPY/USD futures exchange

rates series are obtained from Tick Data, Inc (www.tickdata.com) and cover the

period from 2 January 1990 to 31 March 2000 ( 2586=T trading days). The duration

of each trading day t is 400 minutes. Trading is open Monday to Friday, from 7.20

a.m. to 2.00 p.m. Data are recorded tick by tick and contains 234,905 and 3,494,384

observations of prices on AUD/USD and JPY/USD futures contracts respectively.

However, lower sampling frequencies are used in the study. Specifically, in this

example a daily sampling frequency is considered. The price of the asset in day t is

the value of the last transaction in this day. Thus, 2586=T prices on AUD/USD and

JPY/USD futures exchange rates contracts are selected.

If a time series contains a unit root, then a shock does not disappear over time

and permanently affects the distributional parameters of the series. In this example,

the basic methodology for testing series for a unit root is presented. Modern

econometric software allows the entire procedure to be completed in just a few

minutes. However, the conclusion is not always obvious and therefore it is important

7

to understand the main ideas that underlie the test. A three-step testing procedure

applied to the AUD/USD and JPY/USD futures exchange rates is shown below.

First, a visual examination of the time series aims to detect whether the

behaviour of the time series appears to follow a random walk (see Figure 2.1). If a

series follows a random walk, each successive change in a variable is drawn

independently from previous changes and therefore variables move regardless of the

unconditional mean. Two line graphs of futures exchange rates in Figures 2.3 and 2.4

illustrate that both AUD/USD and JPY/USD series display a pattern typical at a

random walk.

Figure 2.3 AUD/USD futures exchange rates

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

AUD/USD Unconditional mean

Figure 2.4 JPY/USD futures exchange rates

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

JPY/USD Unconditional mean

8

Second, a unit root can be detected by examination of autocorrelation functions.

A time series can be characterised by two autocorrelation functions. Firstly, the simple

autocorrelation function indicates correlation between lagged variables in the series.

The simple autocorrelation function is:

∑

∑

=

−

=−

−

−−= T

tt

kT

tktt

k

yy

yyyy

1

2

1

)(

))((ρ̂ (2.2)

where kρ̂ is equivalent to the regression coefficient in regression ty on kty − . The

second way to describe a linear dependence is obtained by defining the partial

autocorrelation function. Partial autocorrelation indicates dependence between two

lagged variables in the series when dependencies of variables with shorter lags are

removed. For calculating the partial autocorrelation function, all intermediate lags

between ty and kty − are included in the regression:

ktktt ycyccy −− +++= ...110 (2.3)

This construction ensures that kc reflects the linear relationship between ty and kty −

only.

Typically, non-stationary time series produce a simple autocorrelation function

that is close to one at the first lag and decays very slowly with an increase in the

number of lags. This indicates that the shock does not disappear over time and affects

most of the consequent observations. The partial autocorrelation function is

commonly close to one at the first lag, and close to zero for the second and

consequent lags. This means that the first lag absorbs the entire shock and

observations are not related to each other for lags longer than one. Appendices A1 and

A2 present correlograms of daily AUD/USD and JPY/USD futures exchange rates. Q-

Statistics suggested by Box and Pierce (1970), and associated p-values indicate that a

hypothesis of not jointly different from zero simple autocorrelations can be easily

9

rejected for all lags. Both the simple and partial autocorrelations confirm that the time

series potentially contain a unit root.

Finally, the statistical testing of a time series aims to find the objective criteria

for rejection of the null hypothesis of a unit root. The most popular statistical test is

suggested by Dickey and Fuller (1979, 1981). The test is designed to check if the

regression coefficient in the regression of ty on 1−ty is statistically equal to unity.

Nonetheless, the test is not as straightforward as it first appears because the

potentially non-stationary behaviour of the time series can violate the assumptions

behind the Ordinary Least Squares (OLS) estimator. The Dickey-Fuller test is

constructed as follows:

ttt yay ε+= −11 (2.4)

tttt yayy η+−=− −− 111 )1( (2.5)

ttt yby η+=∆ −11 (2.6)

where in (2.6) 0: 10 =bH is tested against 0: 1 ≠bH A . The rejection of 0H is a

rejection of a unit root in the series. In addition to (2.6) Dickey and Fuller (1979)

considered two more specifications:

ttt ybby η++=∆ −110 (2.7)

ttt tbybby η+++=∆ − 2110 (2.8)

Regressions (2.7) and (2.8) above both include deterministic components. The first

regression includes the intercept 0b while the second in addition to the intercept 0b

includes the time trend tb2 . Importantly, coefficient estimates in the Dickey-Fuller

test have a distribution of statistics that differ from usual OLS regressions. Therefore

the critical values simulated by Dickey and Fuller (1979) must be used in the test.

Additionally, in the Dickey-Fuller test errors, tη are assumed to be independent with

constant variance. However, return series are often serially correlated and this can

distort standard errors of regression coefficients. To account for these features, the

following augmented Dickey-Fuller (ADF) test is normally used:

10

t

p

iititt yyby ηγ +∆+=∆ ∑

=+−−

2111 (2.9)

where as before 0: 10 =bH is tested against 0: 1 ≠bH A . The ADF test can also

include deterministic terms. The same critical values as for the original Dickey-Fuller

test are used.

In this particular example, it is found that 0H of a unit root cannot be rejected

for any number of autoregressive lags from 0 to 200. This allows concluding that the

AUD/USD and JPY/USD exchange rate series contain a unit root.

2.3 Properties of Return Time Series

Although price time series are generally non-stationary, their first difference is

usually stationary. The differentiating of non-stationary variables leads to the

stationary series as follows:

ttt PP εµ ++= −1 (2.10)

ttt PP εµ +=− −1 (2.11)

ttP εµ +=∆ (2.12)

Here the non-stationary variables tP after differentiating become stationary tP∆ with

unconditional mean µ . This transformation is called time series integration. Series is

integrated of order d, if the d-th difference of series is stationary. Most financial price

time series are integrated of order one or I(1) [see, for instance, Campbell and Perron

(1991) and Campbell et al. (1997)]. For financial price time series, it is convenient to

use log differences that are equivalent to continuously compounded returns:

ttt pp εµ ++= −1 (2.13)

ttt pp εµ +=− −1 (2.14)

ttr εµ += (2.15)

11

where tr is a log return at time t , )log( tt Pp = , µ is an unconditional mean of the log

returns and ),0(~ 2tt N σε .

There is much empirical evidence that the unconditional mean µ for short-term

(eg. daily) returns is statistically indistinguishable from zero [see Rossi (1996) and

Campbell et al. (1997) amongst others]. Therefore the unconditional mean in (2.15)

can be disregarded for daily and intra-daily returns. In the absence of the

unconditional mean, the additive property of log returns can be demonstrated as

follows:

∑∑==

==k

nnt

k

nntt rr

1,

1, ε (2.16)

where tr is a log return over the period t , intra-period log returns ntr , are sampled k

times over the period t and ),0(~ 2,, ntnt N σε . It should be noted here that if intra-

period innovations are not serially correlated, then the variance of tr is equal to the

sum of intra-period variances: ∑=

=k

nntt

1

2,

2 σσ . This holds because the variance of

uncorrelated normal random variables is additive.

A question that naturally arises is why returns are assumed to be normally

distributed. To answer this question it is important to recognise that if the EMH holds,

then price changes should be independently identically distributed (i.i.d.) variables.

Osborne (1959) argued that if there are sufficient transactions per time interval (say,

30 or more per hour) and price changes are i.i.d. random variables with finite

variance, then accumulated within each time interval returns according to the Central

Limit Theorem (CLT) should be approximately normally distributed. However,

Mandelbrot (1963b), Osborn (1959), Fama (1963, 1965), Lo and MacKinlay (1988),

amongst others, noted that asset returns generally have a leptokurtic distribution that

is inconsistent with the assumption of normality. Moreover, Fama (1965) analysed

return time series and concluded that any serial dependencies are usually not strong

enough to explain deviations from normality. Recognising that the volatility of returns

12

is not constant over time, Clark (1973) suggested the Mixture of Distributions

Hypothesis (MDH). The MDH assumes that returns can be presented as drawn from a

family of normal distributions with different variance parameters, rather than from

single distribution with constant variance. The dispersion of returns around the mean

varies over time and this results in empirically observed ‘fat’ tails of unconditional

distribution. This view allows avoiding the violation of the CLT because returns are

expected to be normally distributed conditional on the underlying volatility.

If the EMH holds then nobody can make profits from the history of time series,

as equivalent to the non-arbitrage conditions. This means that returns, at least in the

short run, must be unpredictable. However, the absence of predictability does not

imply a serial independence, since variables can be serially dependent nonlinearly.

The following expression illustrates this point:

0)](),([ =+ktt rgrfCov (2.17)

for all t and for 0≠k where )(⋅f and )(⋅g are arbitrary functional transformations of

the variable. If returns are i.i.d. variables, then (2.17) holds for all and any )(⋅f and

)(⋅g . Since the number of variable transformations (functional forms) is virtually

infinite, one can never be sure that observations are i.i.d. In fact, even untransformed

returns are often serially correlated. One of the potential reasons for the linear

predictability of returns is the physical structure of financial markets or market

microstructure. Originally a financial market is an exchange mechanism that brings

together sellers and buyers of financial assets. For providing the efficient exchange,

any financial market has trading rules that commonly affect price dynamics in one

way or another. While market microstructure effects may be safely disregarded in low

frequency financial series (eg. weekly or monthly), they become increasingly more

important in intraday returns. There are three main microstructure effects: non-

synchronous trading, bid-ask bounce and price discreteness. First, the non-

synchronous trading or non-trading effects may arise because prices are usually

recorded at time intervals of fixed length while actual trades can occur with different

often irregular frequencies. Moreover, as the sampling frequency increases, the

number of intervals where price changes cannot be observed is also likely to increase.

13

As a result, returns in such intervals move from the previously recorded values

towards zero inducing spurious negative serial correlation. In multivariate time series,

spurious cross-autocorrelation may arise because different assets are trading with

different intensities. This happens if information arrives close to the sampling time.

The asset with higher trading frequency is then likely to reflect this information while

the asset with lower trading frequency often reflects the same information only in the

next period. As a consequence, returns will show spurious positive cross-

autocorrelation. There are several modelling approaches to deal with non-synchronous

trading. For example, Andersen et al. (2001a) suggested regressing return series on

the constant and moving average term:

ttt ccr εε ++= −110 (2.18)

Authors argued that the residuals tε are free from spurious serial dependences caused

by non-synchronous trading. Campbell et al. (1997) noted that all approaches to

modelling the non-synchronous trading effect are aimed at controlling the incorrect

assumption of the equally spaced trading.

The next microstructure effect arises due to the existence of a bid-ask spread in

asset prices. The bid-ask spread can be seen as compensation to market-makers for

providing liquidity. For example, Copeland and Galai (1983) subdivided market

participants into informed traders who have a clear and quantitative reason for

trading, and uninformed or liquidity traders who trade because of reasons other than

profit from information asymmetry. C and G (1983) interpreted the bid-ask spread as

compensation to dealers for taking the risk of dealing with informed traders. Although

the bid-ask spread is typically no larger than one or two ticks (tick is an artificially

imposed minimal amount of one transaction), there is some evidence of its economic

significance in a trading mechanism. Blume and Stambaugh (1983) noted that the bid-

ask spread can induce a significant upward bias in expected returns. Moreover, Keim

(1989) argued that bid-ask spread partly explains the January effect, which is an

empirical fact that smaller capitalisation stocks show better performance over the few

days at the end of one and beginning of the next year. Additionally, regardless of the

bid-ask spread size, market-makers are earning enough compensation for their

14

services and this alone can confirm the economic significance of the bid-ask spread.

The presence of the bid-ask spread creates several potential problems for the

characterisation of returns. Firstly, a price can bounce between bid and ask while the

economic value of the asset remains constant. Therefore the bid-ask bounce can create

volatility, which is clearly spurious since it does not reflect underlying informational

shocks that force prices to move. Additionally, the bid-ask bounce can induce

spurious negative serial correlation in return time series. Serial correlation arises

because if the fundamental value of an asset is constant and a current price is an ask

(bid) price, then a price value can either remain fixed or move to a bid (ask) price

inducing negative serial correlation. Secondly, since at one point in time there are at

least two prices on one asset, it is not clear which price should be used for return

calculation. There are some approaches to price calculation from bid and ask prices.

For example, Müller et al. (1990) suggested using the logarithmic average of bid and

ask prices:

2/)]log()[log( ,, ibidiaski PPx += (2.19)

The third market microstructure effect, which is caused by the artificially

imposed minimal amount of one transaction, or tick, is price discreteness. As a

consequence of this effect, a return on a particular asset cannot move less than the

tick-to-price ratio, inducing autocorrelation in return time series:

ii P

r ξ= (2.20)

where the return r on asset i cannot move less than the ratio of the minimal price

movement (tick) ξ to the current price P . Although a tick is usually fixed for a

particular market, the magnitude of the asset price affects the minimal movement of

return. For example, a tick at 125.0 imposes the %25.0 minimal movement on the

return of the asset with a current price at fifty dollars and %25.1 on the return of the

asset that currently costs ten dollars. Note that as price moves, the minimal return

moves as well. As a result, price discreteness is less evident in returns on more

volatile assets with high prices.

15

2.4 Properties of Financial Return Volatility and Some Volatility Estimators

Although independent variables are never correlated, uncorrelated variables can

be dependent. When returns are properly adjusted for linear serial dependencies

caused by the market microstructure, they are likely to be linearly serially

independent. However, even the simple non-linear transformation of returns, such as

square [ 2)( tt rrf = and 2)( ktkt rrg ++ = in (2.17)], indicates the high level of serial

dependence. Predictability of squared returns is related to volatility clustering or the

Autoregressive Conditional Heteroscedasticity (ARCH) effect. The ARCH effect does

not mean that a future return can be predicted from the series history and therefore

does not violate the EMH. However, serial correlation in squared return series means

that the magnitude of price changes can be expected. Describing the ARCH effect,

Mandelbrot (1963) wrote that “…large changes tend to be followed by large changes,

of either sign, and small changes tend to be followed by small changes”. Both Diebold

and Nerlove (1989) and Engle, Ito, and Lin (1990) attributed the ARCH effect to the

non-uniform arrival of news. Indeed, while news is unpredictable by definition [see

Shannon (1993)], each particular event often provokes the sequence of related events.

Interpreting each event as an informational shock that forces prices to move, it is

reasonable to expect that the chain of related events will be reflected on a line graph

of returns. Lamoureux and Lastrapes (1990) found that clustering disappears when

volume (the number of transactions per time interval), which is a traditional proxy for

the amount of information arrivals, is used for explaining variations of returns. L and

L (1990) interpreted this result as support of the hypothesis that the ARCH effect

arises due to time dependencies in information arrivals.

While returns can be directly observed and examined, the volatility or second

moment of returns is not directly observable. To understand why, it should be noted

that there is much empirical evidence that the volatility of financial returns is not

constant over time [see Schwert (1989) and Nelson (1990) amongst others]. In other

words, returns can be viewed as generated by a stochastic process with the

continuously changing second moment. To observe the volatility, the exact properties

of this process must be known, which is not possible, at least at the present time.

16

Therefore financial volatility can only be expressed by series that are believed to be a

reasonable approximation of return variability.

There are a number of approaches to volatility measurement in financial time

series. First, the Exponentially Weighted Moving Average (EWMA) is a simple but

effective method for estimating the variations of returns. The EWMA has the

following specification:

2

12

12 )1( −− −+= ttt rλλσσ (2.21)

where the volatility 2tσ of return at time t is conditional on the previous period’s

squared return 21−tr and the previous period’s conditional volatility 2

1−tσ with a decay

factor λ . The obvious weakness of the EWMA estimator is the dependence of

volatility estimates on the parameter λ (decay factor), which is selected arbitrarily

and assumed to be constant for different periods. A more detailed analysis of the

EWMA estimator can be found in the RiskMetrics Group technical documentation

that is freely available from the web site (www.riskmetrics.com).

The second volatility estimator that should be noted is implied volatility [see

Corrado and Miller (1996) and Bali et al. (2002) amongst others]. Implied volatility is

the market’s forecast of volatility of an underlining financial instrument for the

duration of the option. Knowing the current option price, current and strike prices of

the underlying financial instrument, the risk-free rate and the time to expire in the

Black-Scholes option-pricing model, means that implied volatility can be calculated

as the single value that yields to the observed parameters. Since the implied volatility

value is directly related to the asset price, some practitioners mistakenly argue that it

is the true volatility estimate. However, it is important to recognise that to satisfy this

title, implied volatility must be a parameter of a true model. At the same time, a

model is only an approximation of reality and humans can never know the nature of

things a priori [‘things-in-themselves’ in Kant (1781)]. For instance, the empirical

evidence that suggests misspecification of the Black-Scholes option-pricing model,

and thus places under a big question the ‘true’ nature of implied volatility estimates, is

volatility smiles, which are different values of implied volatility for the same asset at

17

a given point of time but calculated from options with different dates of expiry [see,

for instance, Brown et al. (1999) and Skiadopoulos et al. (1998)].

The next group of volatility estimators are range-based estimators. The range-

based estimators recognise that prices which are sampled in a discrete time can lie

close together or far apart simply by chance, without reflecting the actual level of

volatility within the period. To account for this drawback, the range-based estimators

measure return variations relying on the amplitude of price movements. To exploit the

amplitude, in addition to the open and close prices, the highest and lowest prices

within a period are also taken into account. The limitation of the range-based

estimators is the weakly studied asymptotic properties, which indicate how fast

estimates converge to true volatility values. Additionally, the range-based estimators

are sensitive to outliers. Yang and Zhang (2000) and Alizadeh et al. (2001) further

discuss the recent developments in range-based volatility estimation.

The ARCH-based estimators, introduced by Engle (1982) and generalised by

Bollerslev (1986), have been probably the most popular volatility estimators over the

last two decades. The underlying idea of the ARCH volatility estimation techniques is

based on the estimation of return variances conditional on previous return realisations.

The ARCH estimators are designed to replicate empirically observed clustering

patterns of the financial volatility. The generalised autoregressive conditional

heteroscedasticity (GARCH) model is shown below:

ttt cxr ε+= ' (2.22a)

∑∑=

−=

− ++=q

jjtj

p

iitit hbawh

11

2ε (2.22b)

where tr is the return in time t, tx' is a matrix of predetermined explanatory variables,

c is a vector of parameters, tε is the linearly serially independent part of return, th is

conditional variance of returns and ia , pi ,...,1= and jb , qj ,...,1= are the ARCH

and GARCH parameters respectively. The traditional method for parameter

estimation in GARCH models is the maximum likelihood estimation (MLE). While

log likelihood values can give some guidance for selecting the optimal number of

18

autoregressive lags p and q , using series of different lengths, one will almost always

arrive at different volatility estimates that correspond to the same time intervals.

Remembering that the actual volatility is fixed, it is difficult to justify why volatility

estimates over the period should be dependent on the length of series outside this

period. The additional argument against the ARCH volatility estimator is that of

generally highly leptokurtic distributions of conditional on ARCH-estimated volatility

returns [see Hsieh (1989)].

2.5 Concluding Remarks

This chapter has presented some properties of financial time series relevant for

this study. In Section 2.2, the stochastic properties of financial price time series were

reviewed. It was noted that most financial price time series are non-stationary. The

EMH was suggested as one of the most plausible reasons for the non-stationary

character of the process that generates financial prices. Numerical Example 2.1 at the

end of the section provided the objective evidence that series studied in this thesis

contain a unit root. Section 2.3 shifted the attention from the non-stationary price

series to returns, which are the stationary first difference of prices. Specifically, it was

noted that if price changes are i.i.d. variables, then, according to the CLT, returns

should be normally distributed. The empirically observed leptokurtosis of return

series was attributed to the changing second moment of the return generation process.

Thus, returns were assumed to be normally distributed conditional on underlying

volatility. In Section 2.4, the volatility of returns was reviewed. It was shown that

serially uncorrelated returns can still be serially dependent due to the ARCH effect.

The presence of the ARCH effect was attributed to specifics of information arrivals

(‘event generating’ process). Noting that volatility of financial returns is not directly

observable, some volatility estimators were introduced and criticised.

19

Chapter 3

Realised Volatility

3.1 Introduction

With the increased availability of high quality datasets, the interest of

researchers is gradually shifting towards the use of high-frequency financial series.

The recent developments in financial econometrics allow estimating the variability of

financial returns using more frequently sampled intra-period data in place of

traditional multi-period datasets. The idea of estimating return volatility over a period

from intra-period data can be traced back to Merton (1980). Merton (1980) argued

that estimates of return variance over a fixed period can be obtained without

information outside this period provided that returns can be sampled over intervals

that approach zero. Following this argument, French, Schwert and Stambaugh (1987)

estimated monthly stock return volatility using daily returns. Schwert (1990), Hsieh

(1991) and Fung and Hsieh (1991) have already used 15 minute returns to estimate

daily volatility. However, only recently Andersen and Bollerslev (1997a) and

Andersen et al. (2001b) related intra-period accumulation of squared returns to the

mathematical concept of quadratic variation and transformed it to the realised

volatility estimator.

The chapter is structured as follows. In Section 3.2, the main ideas behind the

realised volatility estimator are introduced. The Numerical Example 3.1 at the end of

the section demonstrates how the optimal sampling frequency for estimation realised

volatility series can be selected. In Section 3.3, realised volatility series are modelled

with application of traditional ARMA and ARCH time series modelling techniques.

The chapter ends with some brief concluding remarks.

3.2 The Main Concepts of Realised Volatility

Realised volatility is a relatively new and rapidly growing research area of

financial econometrics. The underlying assumptions behind the realised volatility

estimation are non-arbitrage conditions in the market and continuous time diffusion of

20

logarithmic prices. The non-arbitrage conditions assumption refers to the absence of

the economically significant predictability in returns. Specifically, under no-arbitrage

conditions the magnitude of unconditional return expectations in general is

substantially smaller than the magnitude of unpredictable innovations and can be

easily disregarded [see Andersen et al. (2002)]. The diffusion assumption implies that

log prices evolved as a Brownian motion and in the absence of the drift can be

expressed as:

ttt dWdp σ= (3.1)

where tp and tσ are a log price and standard deviation of a log price change at time

t respectively and tW denotes a standard Brownian motion [ )( τ−−≡ ttt WWdW with

0→τ ]. It should be noted here that the drift term in (3.1) is omitted because the

unconditional mean in general is not statistically different from zero even for daily

return series [see Rossi (1996) and Campbell et al. (1997) amongst others]. Andersen

et al. (2000a) noted that the absence of the drift term does not affect any properties of

the realised volatility estimator as long as drift is independent of the volatility path

over a period t .

A standard Brownian motion tW in (3.1) is a stochastic process { }0);( ≥ttz with

the following properties [Taylor and Karlin (1998)]:

(a) Every increment )()()( sBtsBnz −+= is normally distributed with mean 0=µ

and variance tta =⋅ (a = 1 since a standard Brownian motion is considered).

(b) For any pair of disjoint time intervals ( ] ( ]4321 ,,, tttt , with 43210 tttt <≤<≤ , the

increments )()()( 122tBtBnz t −= and )()()( 344

tBtBnz t −= are independent

random variables. It is similar for N disjoint intervals, where N is an arbitrary

positive integer.

(c) 0)0( =B , and )(tB is continuous as a function of t .

Under the above assumptions and disregarding the unconditional mean the

process can be rewritten in discrete terms as follows:

21

nttnt zx ,, σ= , )/1,0(~, kNz nt , kn ,...,1= , ∞→k (3.2)

where in a period t the log return ntx , is sampled k times and tσ is the standard

deviation over the period t . Note that the process implies homogenous intra-period

variance k/1 of ntz , and constant standard deviation tσ over a period t . Thus, the

variance of ntx , is assumed to be homogenous as well.

Since log returns are additive and the standard deviation is assumed to be

constant over the period, the following aggregations can be applied:

∑=

=k

nntt xr

1, (3.3)

tt

k

nntt

k

nntt zzz σσσ == ∑∑

== 1,

1, )1,0(~ Nzt (3.4)

Therefore daily returns can be presented as following decomposition:

ttt zr σ= (3.5)

where 1−−= ttt ppr and tσ are a logarithmic return and standard deviation of return

over the time period t respectively and )1,0(~ Nzt . The same decomposition has

been applied in Andersen et al. (2000a) for computation of standardised returns using

realised standard deviations in place of standard deviations tσ in (3.5).

Realised volatility can be estimated from high-frequency intraday returns. The

use of more frequently sampled data ensures that the volatility series reflect most

information about variable dynamics inside a period of interest. Conversely, prices

sampled at daily intervals can fall close together or far apart simply by chance. In this

regard, Andersen and Bollerslev (1997a) noted that squared returns are the unbiased

and consistent, but not efficient volatility estimator. A and B (1997a) argued that the

daily squared returns are contaminated by the measurement (sampling) error, and

using them as a benchmark for volatility forecast evaluation can provide misleading

22

results. To minimise the measurement error, A and B (1997a) suggested computing

the volatility by summing intra-period squared returns and noted that this method

enables estimation of “…increasingly more accurate” ex-post volatility series.

To expose the main aspects of realised volatility estimation better, the relevant

notation are introduced first. Let Tppp ,...,, 21 be a sequence of log daily asset prices

and ttt ppr −= −1 corresponding log returns (returns are calculated from the point of

view of an American investor). Within a day, the price is observed on k evenly

spaced occasions. k log intraday prices for day t are denoted by { }knp nt ,...,2,1,, = . It

should be noted that the assets considered in this study are not traded continuously

(see Numerical Example 2.1). This implies the presence of non-trading (‘overnight’)

periods, and as a result, seasonal irregularity in return sampling. However, Andersen

et al. (2002) noted that the properties of the realised volatility estimator hold even for

unevenly spaced time intervals. Following this argument, overnight returns are added

to the first intraday return. Thus, the n -th intraday return is ntntnt ppx ,1,, −= − , where

ktt pp ,10, −= . A daily log return tr is a sum of intraday log returns, that is, ∑=

=k

nntt xr

1, .

Andersen et al. (2001b, 2002) showed that if a log price follows a continuous-

time process and there are no arbitrage opportunities, then volatility can be expressed

by realised volatility or by a sum of squared intraday returns. Realised volatility in

day t based on k intraday return observations is expressed as follows:

∑=

=k

nntkt xv

1

2,, (3.6)

Thus, if a trading day t lasts for 400 minutes and prices are sampled in 40 minute

intervals, then the realised volatility is the sum of 10=k intraday squared returns

{ }210,

22,

21, ..., ttt xxx :

23

∑=

=10

1

2,10,

nntt xv (3.7)

and the corresponding realised standard deviation and logarithmic realised standard

deviation are ktkt v ,, =σ and )ln( ,, ktktL σσ = respectively.

Andersen et al. (2001b) further demonstrated that as the sampling frequency

increases the realised volatility approaches integrated volatility. Integrated volatility is

a theoretical representation of ex-post (actual) volatility and can be expressed as

follows:

∫ +−≡1

0

21

2int, τσσ τ dtt (3.8)

where 2int,tσ is integrated volatility over a period t of length [ ]1,0 , and 2

1 τσ +−t is

instantaneous volatility measured over intervals 0→τ (equivalently, number of

observations per period ∞→k ).

To illustrate the asymptotic properties of the realised volatility with respect to

the actual volatility, the following simulation experiment is designed. Let

)',...,( 1 Tyyy = , 1000=T , be a vector of zero mean independently normally

distributed (i.n.d.) variables with arbitrarily selected actual variance 25:

)25,0(~ Nyt . To see that 2ty is an unbiased estimator of variance, 10000=M

vectors with properties similar to y are generated. Then the expectations of squared

variables are computed for each vector: ∑=

−=T

titi yTv

1

2,

11, , Mi ,...,1= . As a result,

10000=M estimates are obtained. The histogram of estimates 1,iv is shown in Figure

3.1.

24

Figure 3.1 Histogram of 1,iv

Note: ∑=

−=T

titi yTv

1

2,

11, , Mi ,...,1=

The estimates are centred at 25 indicating that the square of a random normal variable

is the unbiased variance estimator: 22 )( ttyE σ= . The dispersion of estimates around

the true variance is an indication of the presence of the measurement or sampling

error. The simple standard deviation of the obtained series 1,iv is 1.1201.

To see how the realised volatility estimator can reduce the sampling error, let

each variable ty be decomposed into 5=k variables )',...,( 5,1, tt xx where

)5,0(~, Nx nt . It should be noted that the variance of i.n.d. variables is additive and

therefore

= ∑=

5

1,)(

nntt xVyV . Next accumulate squared variables ntx , as follows:

∑=

=5

1

2,5,

nntt xv . As in the previous stage of the experiment, the procedure is repeated

25

10000=M times and expectations for each of resulted vectors are computed:

∑=

−=T

titi vTv

1,

15, , Mi ,...,1= . The histogram of estimates 5,iv is shown in Figure 3.2.

Figure 3.2 Histogram of 5,iv

Note: ∑=

−=T

titi vTv

1,

15, , Mi ,...,1=

The increase in efficiency is obvious. The simple standard deviation of 5,iv is now

substantially smaller at 0.5035. When 25=k the estimator becomes even more

efficient. The histogram of 25,iv is shown in Figure 3.3.

26

Figure 3.3 Histogram of 25,iv

Note: ∑=

−=T

titi vTv

1,

125, , Mi ,...,1=

The simple standard deviation of 25,iv in this instance is 0.2227. Barndorff-Nielsen

and Shephard (2002a) demonstrated that realised volatility converges to the true a

priori selected volatility at the rate k when ∞→k . Indeed,

252227.055035.01201.1 ⋅≈⋅≈ .

Thus, the above simulation experiment has demonstrated that the realised

volatility estimator is both unbiased [ 2

1

2, )( t

k

nntxE σ=∑

=

] and consistent (it

asymptotically converges to the actual variance as ∞→k ). However, in practice

because of the physical nature of financial markets, the sampling frequency k cannot

approach infinity. The highest available frequency of return sampling is tick-by-tick.

Furthermore, even the use of tick-by-tick data does not guarantee that the best

available approximation of the integrated volatility can be obtained. The irregular

spacing of tick-by-tick returns is just one of the reasons for this. However, even when

tick-by-tick returns are aggregated up to fixed equally spaced intervals, microstructure

27

noise still creates problems for realised volatility estimation. Although irregular

spacing and market microstructure can be modelled explicitly, Andersen et al. (2001c)

noted that this approach is excessively complicated and subject to numerous

drawbacks. The authors suggested using an alternative procedure for selecting the

optimal sampling frequency for realised volatility estimation.

Andersen et al. (1999c) noted that for the estimation of realised volatility series,

one should find a sampling frequency that keeps a reasonable balance between market

microstructure on the one hand and measurement error on the other. Andersen et al.

(1999c) argued that “…the optimal sampling frequency will likely not be the highest

available, but rather some intermediate value, ideally high enough to produce a

volatility estimate with negligible sampling variation, yet low enough to avoid

microstructure bias” and suggested a method for selecting the optimal sampling

frequency. Authors constructed several volatility series by sampling prices over

different equally spaced intervals, calculated simple averages of realised volatility

over the whole sample and then plotted them against the intra-period sampling

interval values. This approach, which is called a volatility signature plot, is based on

the idea that the microstructure bias should reveal itself as the sampling frequency

gets higher. Although, at least at this stage, this method is subjective, it is the only one

currently available for selecting the optimal sampling frequency for realised volatility

estimation. It should be noted here that for return series with distinct trading

intensities, the optimal sampling frequency is more often different. For example,

Andersen et al. (1999c) noted that highly liquid assets are likely to suffer from the

bid-ask bounce effect as sampling interval approaches zero. However, for less

actively traded assets, a non-trading effect is likely to be of a main concern. This

follows directly from the nature of bid-ask bounce and non-trading effects that have

been reviewed earlier in this study (see Section 2.3).

In practice, several different studies selected different sampling intervals to

construct realised volatility series. For example, Andersen et al. (2001b) used 5

minute sampling intervals to estimate the realised volatility of foreign exchange rates.

However, Andersen et al. (2001c) argued that 30 minute sampling intervals are a

reasonable balance between a sampling variation on the one hand and microstructure

noise on the other. Bai, Rassell and Tiao (2001) and Andersen et al. (2001a) also used

28

30 minute sampling intervals when working with foreign exchange rates and equity

returns respectively. The following numerical example shows how the volatility

signature plot can be applied to the selection of the optimal sampling frequency.

Numerical Example 3.1: Volatility Signature Plot

A volatility signature plot has been suggested in Andersen et al. (1999c) and is a

graphical diagnostic, designed to provide some guidance in selecting the optimal

return sampling frequency for realised volatility series computation. AUD/USD and

JPY/USD futures exchange rates series, containing tick-by-tick data, are resampled

with different frequencies. Eight sampling frequency intervals have been used: 5, 10,

20, 40, 50, 100, 200 and 400 minutes. The corresponding numbers of intraday prices

are =k 80, 40, 20, 10, 8, 4, 2 and 1. For return calculation, the last prices that were

observed over a sampling period have been used. If no transactions occurred over a

sampling period, then the latest recorded prices were taken.

Realised volatility in day t based on k intraday price observations is calculated

as follows:

∑=

=k

nntkt xv

1

2,, (3.9)

As a result, sixteen realised volatility series, eight for AUD/USD AUDT

AUDAUDT

AUD vvvv 1,1,180,80,1 ,...,;...;,..., and eight for JPY/USD JPYT

JPYJPYT

JPY vvvv 1,1,180,80,1 ,...,;...;,..., ,

were obtained. To define a volatility signature plot, the average values of each of

sixteen series were calculated:

∑=

−=T

tktk vTv

1,

1 (3.10)

29

The average realised volatilities kv , =k 80,40,…,1 were plotted against the

lengths of sampling intervals in minutes 400,...,10,5 . Figures 3.4 and 3.5 show the

results.

Figure 3.4 Realised volatility signature plot for AUD/USD series

0.00003

0.000032

0.000034

0.000036

0.000038

0.00004

0.000042

0.000044

0 100 200 300 400 500

Sampling interval (min)

Ave

rage

rea

lised

vol

atili

ty

Figure 3.5 Realised volatility signature plot for JPY/USD series

0.0000360.0000370.0000380.0000390.00004

0.0000410.0000420.0000430.0000440.000045

0 100 200 300 400 500

Sampling interval (min)

Ave

rage

rea

lised

vol

atili

ty

Two volatility signature plots in Figures 3.4 and 3.5 notably differ from each other.

The potential reason for this is different trading intensities for AUD/USD and

JPY/USD futures contracts. JPY/USD futures contracts during the period 2585=T

days have been traded on average at 1351.27 times per day while AUD/USD contracts

Optimal: 50 min

Optimal: 40 min

30

only 90.84 times per day. The sharp increase in the AUD/USD series average realised

volatility for the highest sampling frequencies (5, 10 and 20 minutes) is most likely

due to negative serial correlation induced by the non-trading effect. A similar pattern

has been documented in Andersen et al. (1999c) for the asset with high trading

intensity, in contrast to the low intensity of AUD/USD, and attributed by authors to

the bid-ask bounce. However, non-trading effect and bid-ask bounce both induce

negative serial correlation (see Section 2.3) and therefore may affect the average

realised volatility in a similar manner. The difference between the shapes of the

volatility signature plots should be attributed to the different strengths of serial

correlation induced by non-trading effect and bid-ask bounce.

Following Andersen et al. (1999c) the microstructure bias “manifests itself”

around sampling intervals at 40-50 minutes (around 10v and 8v ). Remembering that

microstructure effects are of the main concern in series with the highest sampling

frequency, one should follow the volatility signature plot from the left to the right.

Andersen et al. (1999c) argued that the point where the volatility signature plot is

“stabilised” should be selected as optimal. In the AUD/USD series plot, this point

corresponds to the 50 minute sampling interval ( AUDv8 ) and for the JPY/USD series

plot to the 40 minute sampling interval ( JPYv10 ). This means that for obtaining the

optimal realised volatility estimates, returns should be sampled 8 and 10 times per

trading day, for AUD/USD and JPY/USD series respectively. In addition to these,

interesting information can be obtained from the right tails of the volatility signature

plots. The points on the far right of the plots correspond to 400 minute sampling

intervals, which are equivalent to daily squared returns and often used as a benchmark

for volatility forecast evaluation [see Cumby et al. (1993) and Jorion (1995)].

Assuming that the chosen optimal average realised volatility values are not

substantially contaminated by microstructure effects, but at the same time contain the

minimal measurement error, the difference between average daily squared returns and

the optimally estimated average realised volatility is a measure of the sampling error

in daily series. Without calculating the exact values of the measurement bias, it can be

seen that the bias is negative for the weakly traded AUD/USD contracts and positive

for the heavily traded JPY/USD contracts. In other words, remembering that realised

volatility is a highly efficient approximation of integrated volatility, daily squared

31

returns on average underestimate and overestimate volatility for low and high trading

intensity assets respectively. However, it should be noted that the observed bias can

be unique for considered assets and for interpreting the difference in signs of the

measurement bias the out of sample testing is necessary.

3.3 Modelling Realised Volatility

For modelling the realised volatility, the stochastic properties of realised

volatility series should be studied first. Andersen et al. (2001b) explored the realised

volatility of foreign exchange rates and found that the unconditional distribution of

realised volatility series is highly right skewed. However, the square root of realised

volatility, or realised standard deviation series, appeared to be much less skewed.

Furthermore, the logarithmic standard deviation series is symmetrically bell shaped.

Andersen et al. (2001c) argued that since the realised volatility is a highly efficient

volatility estimator, the realised volatility series can be modelled as an ordinary time

series. They used a vector autoregression model for forecasting volatility and found

that this approach had “superior” performance compared to alternative volatility

forecasting methods, such as forecasting with GARCH models.

In this study, the alternative approach to modelling realised volatility is

proposed. ARMA and GARCH models are not used for modelling returns, but applied

directly to the realised logarithmic standard deviations for explaining serial

correlations in levels and squared variables. If series are modelled successfully, then

the resulting residuals must be uncorrelated and free from the volatility clustering

effect. The realised volatility is modelled as follows. First, the chosen optimal realised

volatility series are transformed into the realised standard deviations:

ktkt v ,, =σ (3.11)

Next, the realised standard deviations are transformed into the logarithmic realised

standard deviation:

)ln( ,, ktktL σσ = (3.12)

32

As has been mentioned, according to Andersen et al. (2001b), the ktL ,σ series should

have a distribution that is reasonably close to normal. As a next step, linear serial

dependencies in ktL ,σ series are explained by:

ttkt cxL εσ += ', (3.13)

where tx' is a vector of explanatory variables that explain serial correlation in ktL ,σ

series, c is a vector of parameters, and residuals tε are serially uncorrelated. The

uncorrelated residuals are modelled as an ARCH process for explaining serial

dependence in squared variables:

ttt hz=ε )1,0(~ Nzt (3.14a)

∑∑=

−=

− ++=q

jjtj

p

iitit hbawh

11

2ε (3.14b)

where th is a conditional second moment of log realised standard deviation series.

The procedure has allowed arriving at uncorrelated series tz that is free from

volatility clustering.

Numerical Example 3.2: Modelling Optimal Realised Volatility Series

In Numerical Example 3.1, two realised volatility series AUDtv 8, and JPY

tv 10, were

found to be optimal for AUD/USD and JPY/USD returns respectively. These series

are modelled here. Relying on the argument of Andersen et al. (2001b) that realised

volatility can be treated as observable, traditional time series modelling techniques are

used. The main objective of the proposed modelling approach is to arrive at serially

uncorrelated error term that is free from the volatility clustering effect. Histograms of AUDtv 8, and JPY

tv 10, series are shown in Figures 3.6 and 3.7.

33

Figure 3.6 Histogram of AUD/USD realised volatility series AUDtv 8,

0 0.5 1 1.5 2 2.5

x 10-3

0

500

1000

1500

2000

2500

Figure 3.7 Histogram of JPY/USD realised volatility series JPYtv 10,

0 0.5 1 1.5 2 2.5

x 10-3

0

500

1000

1500

2000

2500

As can be seen, the series are highly right skewed. The next step is to calculate the

realised standard deviations ktkt v ,, =σ . Histograms of AUDt 8,σ and JPY

t 10,σ series are

presented in Figures 3.8 and 3.9 respectively.

34

Figure 3.8 Histogram of AUD/USD realised standard deviation series AUDt 8,σ

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

50

100

150

200

250

300

350

400

450

Figure 3.9 Histogram of JPY/USD realised standard deviation series JPYt 10,σ

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.050

50

100

150

200

250

300

350

400

450

500

The distributions of realised standard deviation series become notably less skewed

respective to the realised volatility distributions. Finally, the logarithmic standard

deviation series AUDtL 8,σ and JPY

tL 10,σ are computed as )ln( ,, ktktL σσ = . The histograms

of AUDtL 8,σ and JPY

tL 10,σ series are presented in Figures 3.10 and 3.11 respectively.

35

Figure 3.10 Histogram of AUD/USD log realised standard deviation series AUDtL 8,σ

Figure 3.11 Histogram of AUD/USD log realised standard deviation series JPYtL 10,σ

The distributions of AUDtL 8,σ and JPY

tL 10,σ series are apparently much closer to Gaussian

than the distributions of ktv , and kt ,σ series. The descriptive statistics of ktv , , kt ,σ and

ktL ,σ series are presented in Table 3.1.

36

Table 3.1 Descriptive statistics of ktv , , kt ,σ and ktL ,σ series

Realised volatility

( ktv , ) Realised standard deviation ( kt ,σ )

Log realised standard deviation ( ktL ,σ )

AUD/USD JPY/USD AUD/USD JPY/USD AUD/USD JPY/USD Mean 0.0000 0.0000 0.0052 0.0051 -5.4354 -5.4823Median 0.0000 0.0000 0.0044 0.0041 -5.4292 -5.5023Maximum 0.0023 0.0025 0.0474 0.0497 -3.0484 -3.0011Minimum 0.0000 0.0000 0.0003 0.0004 -8.1671 -7.7960Std. Dev. 0.0001 0.0001 0.0034 0.0039 0.5937 0.6193Skewness 14.2086 11.7883 2.9048 3.4040 -0.0676 0.1849Kurtosis 316.0385 210.2765 22.6198 24.2367 3.3398 3.3088JB statistics* 10637701 4685590 45079 53547 14.3966 24.9876

Note: * - Jarque-Bera test statistics; the critical value for 95% confidence level is 5.99

Although results of the Jarque-Bera test, reported in the last row of Table 3.1,

indicate that 0H of a normal can be rejected for all series, test statistics show that the

ktL ,σ series are reasonable close to normal. Additionally, the parameters of kurtosis

for the ktL ,σ series are only slightly above 3. This suggests that if serial dependencies

in log realised standard deviation series ( ktL ,σ ) can be modelled, then, knowing the

models’ parameters, normal pseudorandom innovations can be used for simulation of

realised volatility series with stochastic characteristics close to empirically observed.

The examination of a serial structure of AUDtL 8,σ and JPY

tL 10,σ series has indicated that

variables are serially correlated. The serial correlations in both series have been

explained by ARMA (1,1) models. The coefficient estimates with associated standard

errors are shown in Table 3.2.

Table 3.2 Estimates from ARMA (1,1) applied to AUDtL 8,σ and JPY

tL 10,σ series

Series Constant AR(1) MA(1) AUDtL 8,σ

-5.44 (0.07)

0.99 (0.00)

-0.91 (0.01)

JPYtL 10,σ

-5.49 (0.07)

0.98 (0.01)

-0.86 (0.01)

Note: Standard errors in parentheses

37

The residuals AUDt ,ε and JPYt ,ε obtained from ARMA (1,1) models presented above

are found to be serially uncorrelated. Although residuals from both models are serially

uncorrelated, squared residuals are serially correlated. This situation is similar to

volatility clustering in return series and therefore the ARCH modelling technique can

be used for explaining serial correlation in squared variables. The GARCH (1,1)

specification has been found the best for both series. The coefficient estimates with

associated standard errors of the GARCH (1,1) variance equations are shown below.

Table 3.3 Estimates from GARCH (1,1) applied to AUDt ,ε and JPYt ,ε series

Series Constant ARCH(1) GARCH(1)AUDt ,ε

0.14

(0.05) 0.07

(0.02) 0.45

(0.20) JPYt ,ε

0.13

(0.08) 0.04

(0.02) 0.53

(0.27) Note: Standard errors in parentheses

Notably, the kurtosis parameters of the resulting uncorrelated and free from volatility

clustering residuals are 3.26 and 3.32 for the AUD/USD and JPY/USD series

respectively, which are very close to 3 of the normal distribution.

3.4 Concluding Remarks

In this chapter the realised volatility estimator was introduced. In Section 3.2,

the main assumptions underlying realised volatility were presented. It was

demonstrated that realised volatility is an unbiased and consistent estimator of

integrated volatility. In Numerical Example 3.1, the volatility signature plot method

was applied to selection of the sampling frequency for intraday returns, which is

“…high enough to produce a volatility estimate with negligible sampling variation,

yet low enough to avoid microstructure bias” [Andersen et al. (1999c)]. Section 3.3

demonstrated how traditional time series modelling tools can be applied directly to the

realised volatility series.

38

Chapter 4

The Two-Component Effect

4.1 Introduction

A distribution of financial returns plays a central role in financial econometrics.

It is important to distinguish between unconditional and conditional return

distributions. The unconditional distribution reflects stochastic characteristics of

returns observed in the markets. Since unconditional returns can be directly observed,

they have been widely studied. The main stochastic properties of unconditional

returns, such as leptokurtosis, have been described in Section 2.3. The situation is

more complicated in the case of conditional returns. The term conditional generally

refers to conditioning returns on underlying volatility. Since financial volatility is not

directly observable, volatility estimates are used for conditioning. For example,

Bollerslev (1987) and Andersen (1992) studied returns conditional on ARCH and

stochastic volatility estimates respectively and found that standardised returns are

leptokurtic, although less than unconditional. Koedijk et al. (1990) applied the

Extreme Value Theory (EVT) to study tail probabilities of financial return distribution

and also found that conditional returns are leptokurtic. Authors noted that conditional

returns can follow Student t distribution. However, it is important to note that only

conditioning returns on actual volatility values could provide direct evidence in

favour of one or another distribution.

It has been shown in Chapter 2 that from the theoretical point of view financial

returns should be conditionally normally distributed conditional on underlying

volatility. A normal variable normalised by its standard deviation is expected to be

standard normally distributed [ )1,0(~ N ]. Since realised volatility is an unbiased,

consistent and highly efficient estimator of ex-post (actual) volatility the distribution

of standardised by realised standard deviations returns is expected to be

approximately standard normal. Indeed, Andersen et al. (2000a) normalised exchange

rate returns by realised standard deviations and obtained innovations that are only

slightly platykurtic or “nearly” standard normally distributed. Andersen et al. (2000a)

noted that the distribution of standardised returns is “remarkably close” to a standard

39

normal but did not attempt to find the reason for platykurtosis in empirical

standardised return series. In this study, a return standardisation identical to Andersen

et al. (2000a) has been performed. Table 4.1 below presents descriptive statistics of

returns standardised by optimal realised standard deviations ( =k 8 and 10 for

AUD/USD and JPY/USD series respectively).

Table 4.1 Descriptive statistics of standardised returns Standardised returns AUD/USD JPY/USD Mean -0.0441 0.0381Median -0.0651 0.0534Maximum 2.3129 2.3503Minimum -2.3285 -2.7689Std. Dev. 0.9236 0.9963Skewness 0.0165 -0.0753Kurtosis 1.9940 2.0128

The descriptive statistics of obtained standardised returns are similar to ones reported

in Andersen et al. (2000a). However, the visual examination of the empirical

distributions of standardised returns obtained in this study has revealed the highly

unusual two-peaked shape. Two histograms demonstrate that:

Figure 4.1 Standardised by 8,tσ AUD/USD returns

40

Figure 4.2 Standardised by 10,tσ JPY/USD returns

This effect is entitled the two-component effect and examined in this chapter. In

Section 4.2, the likelihood ratio test is applied for testing 0H of a single normal

against AH of a mixture of two normals with unequal mean parameters. Section 4.3

presents the alternative non-parametric test 0H of a mixture of two normals against

AH of no mixture of two normals. In Section 4.4, the parameter-free transformation

of a set of random variables to a smaller set of random variables distributed on the

interval from zero to one is introduced. Based on this transformation, the simple (as

opposed to composite) test for normality is presented and applied. In Section 4.5, the

simulation experiment is performed in order to find the cause of the two-component

effect. The final section briefly summarises the chapter.

4.2 Likelihood Ratio Test

The first hypothesis put forward in order to explain the discovered two-

component effect states that standardised returns can be distributed as a mixture of

two normals with unequal mean parameters. Noting that a mixture of two normals can

cause platykurtosis of the standardised return series distributions, it was decided to

41

test 0H of a single normal against AH of a mixture of two normals. The test aims to

indicate objectively the presence of the two-component effect in standardised return

series. Standardised return series have been calculated as follows:

kt

tkt

rz,

, σ= (4.1)

where tr is a daily log return in day t and kt ,σ is realised volatility estimated from

returns sampled =k 80, 40, 20, 10, and 8 times per trading day.

Suppose the variables )',...,,( 21 NN XXXX = with unknown distribution are

given. It is required to test the null hypothesis that iX is normally distributed with

single mean and variance ),;(~ 2σµii xfX against the alternative that iX is a

mixture of two normals with distinct mean and variance parameters and unknown

mixing proportion ),;()1(),;(~ 222

211 σµασµα iii xfxfX ⋅−+⋅ . A mixing proportion

in the mixture of two normals is denoted by α and indicates the probability that

observation is drawn from ),;(~ 211 σµixf . This hypothesis could be tested by

applying a likelihood ratio test, which requires finding the maximum log likelihood of

iX for the following functional specifications:

2,0 max:

σµH ∑

=

−−⋅=N

i

iR xLL1

2

2

2)(exp

21ln

σµ

πσ (4.2)

...max:222

211 ,,,,

=URA LLH

σµσµα

∑=

−−⋅⋅−+

−−⋅⋅=N

i

ii xx1

22

22

221

21

1 2)(exp

21)1(

2)(exp

21ln...

σµ

πσα

σµ

πσα (4.3)

where RLL and URLL are restricted and unrestricted log likelihood respectively and

10 ≤≤α . This test is associated with the important problem that is known as

parameters’ non-identifiability [see Ripley (1996)]. The problem arises because

42

),( 211 σµ and ),( 2

22 σµ can be exchanged by replacing α for α−1 with obtaining the

same density. Therefore the additional constraint such as 5.0≤α or 21 µµ ≥ must be

imposed. The latter constraint is used in this study. Although the additional constraint

allows obtaining unambiguous parameter estimates, the distribution of the likelihood

ratio statistics is different from the case with fully identifiable parameters.

Some researchers have explored the distribution of the likelihood ratio statistics

for a likelihood ratio test applied to testing the presence of a mixture of two normals.

For example Mendell et al. (1991, 1993) and Thode et al. (1988) performed

simulations for finding the distribution of likelihood ratio statistics of a mixture of

two normals with unknown and unequal means, and unknown but equal variances.

Goffinet et al. (1992) also studied the likelihood ratio test statistics for a mixture of

two normals though with a known mixing proportion. Garnel (2001) gave some

theoretical results of the likelihood ratio statistics for seven distinct cases of a mixture

of two normals. However, the case considered in this study is probably the most

general out of a mixture of two normals cases, since in addition to unequal mean and

variance parameters and unknown mixing proportion, the mean of a single normal

does not coincide with any of two means in a mixture of two normals. Distribution of

resultant statistics for this case is unknown. In this study series have been tested using

the following log likelihood statistic [see Press (1992)]:

( ) 23~2 χURR LLLLLR −−= (4.4)

Tables 4.2 and 4.3 below show the results of the test. The last line of each table

reports the likelihood ratio test statistics (with critical value for 95% confidence level

81.7205.0,;3 ==αχ crit ).

43

Table 4.2 Parameters of a mixture of two normals model fitted to AUD/USD series AUD/USD AUD/USD AUD/USD AUD/USD AUD/USD k = 80 k = 40 k = 20 k = 10 k = 8 Mean 1 0.69 0.70 0.73 0.81 0.77 Mean 2 -0.67 -0.68 -0.70 -0.70 -0.76 Variance 1 0.26 0.25 0.26 0.22 0.24 (Standard deviation 1) 0.51 0.50 0.51 0.47 0.49 Variance 2 0.30 0.29 0.29 0.32 0.28 (Standard deviation 2) 0.55 0.54 0.54 0.56 0.53 Alfa 0.46 0.46 0.46 0.44 0.47 Log likelihood two normal -3196.04 -3187.51 -3237.94 -3287.77 -3295.94 Log likelihood normal -3279.45 -3283.66 -3345.88 -3440.97 -3463.34 Likelihood ratio test 166.82 192.31 215.88 306.4 334.79

Table 4.3 Parameters of a mixture of two normals model fitted to JPY/USD series JPY/USD JPY/USD JPY/USD JPY/USD JPY/USD k = 80 k = 40 k = 20 k = 10 k = 8

Mean 1 0.86 0.88 0.89 0.96 0.92 Mean 2 -0.65 -0.65 -0.68 -0.69 -0.76 Variance 1 0.27 0.26 0.26 0.23 0.25 (Standard deviation 1) 0.52 0.51 0.51 0.48 0.50 Variance 2 0.40 0.39 0.37 0.40 0.35 (Standard deviation 2) 0.63 0.63 0.61 0.63 0.59 Alfa 0.45 0.45 0.46 0.44 0.47 Log likelihood two normal -3433.93 -3433.23 -3455.82 -3489.86 -3476.01 Log likelihood normal -3530.87 -3540.93 -3586.83 -3659.35 -3669.47 Likelihood ratio test 193.87 215.41 262.02 338.98 386.92

If the test’s statistics were distributed as 23χ , then 0H of a single normal could be

easily rejected for all series. However, the result of the test is not conclusive because

the actual distribution of the likelihood ratio test statistics is unknown. Therefore an

alternative test has been developed and applied. In the next section the Pearson

goodness of fit (chi-squared) test is applied as an alternative to the likelihood ratio

test.

44

4.3 Pearson Goodness of Fit Test

Suppose that innovations in return kttkt rz ,, σ= are sorted in ascending order

and subdivided into M non-overlapping cells. Each actual observation ktz , falls to

one of M cells. As a result, the empirical frequencies EmY of ktz , being in each cell

can be recorded. Recall that the likelihood ratio test has provided the estimates of

parameters for the hypothesised mixture of two normals specification (see Tables 4.2

and 4.3). Knowing 1µ , 2µ , 21σ , 2

2σ and α of a mixture of two normals

),;()1(),;(~ 222

211, σµασµα iikt xfxfz ⋅−+⋅ , theoretical frequencies T

mY can be

calculated. The objective of the test is to compare theoretical and empirical

frequencies in order to find whether they differ significantly. The test statistic follows 2χ distribution and can be calculated as shown below:

( )∑=

−=M

iE

i

Ei

Ti

YYY

1

22χ (4.5)

The null hypothesis of insignificant deviations from theoretical outcomes is tested

against AH which states that empirical frequencies differ significantly from

theoretically expected values. If the total number of observations T is large

( 2586=T can certainly be considered as large), then the distribution of the statistics

is approximately 21−−rMχ , where M is a specified number of cells and r is a number

of unknown parameters [see Conover (1999)].

The actual test has been implemented for arbitrarily selected 51=M . The

number of degrees of freedom with the number of unknown parameters 5=r is

4515511.. =−−=−−= rMfd . Table 4.4 presents results of the Pearson goodness of

fit test applied to the innovations ktz , given by (4.1) with different sampling

frequencies k (with critical value for 95% confidence level 66.61205.0,;45 ==αχ crit ).

45

Table 4.4 Pearson goodness-of-fit statistics for a mixture of two normals Sampling frequency k AUD/USD stats JPY/USD stats

80 76.13* 46.96 40 43.17 58.94 20 51.93 59.13 10 63.74* 53.47 8 48.75 70.84*

Note: * - 0H of a mixture of two normals can be rejected for 95% confidence level

As can be seen, 0H of a mixture of two normals can be rejected for three series

only. The results are graphically presented in Figures 4.3 and 4.4 showing two

examples – AUDtz 40, (the best fit series) and AUD

tz 80, (the worst fit series):

Figure 4.3 AUDtz 40, : Theoretical against empirical frequencies

0

20

40

60

80

100

120

140

Empirical frequencies Theoretical frequencies

46

Figure 4.4 AUDtz 80, : Theoretical against empirical frequencies

0

20

40

60

80

100

120

140

Empirical frequencies Theoretical frequencies

The same test for a single normal strongly rejects 0H for all series:

Table 4.5 Pearson goodness-of-fit statistics for a single normal Sampling frequency k AUD/USD stats JPY/USD stats

80 195.57* 223.36* 40 220.35* 249.34* 20 258.51* 304.29* 10 363.89* 372.05* 8 354.08* 453.89*

Note: * - 0H of a single normal can be rejected for 95% confidence level

Importantly, the results of the presented test should not be accepted as

conclusive. The reason for that is the result reported by Becker and Hurn (2002). B

and H (2002) demonstrated that the distinction between the simple hypothesis tests,

where distributional parameters are known a priori, and composite hypothesis tests,

where distributional parameters are estimates, must be made. In the simulation

experiment, it has been shown that the true 0H can almost never be rejected for the

composite case, which indicates the problem with the size of statistics. Since the

presented test is based on the composite hypothesis, the result cannot be accepted

without further investigation. Therefore in the next section the simple test 0H of a

normal against AH of no normal is presented.

47

4.4 Simple Test for Normality

The question of whether or not a set of random variables follows a specific

theoretical distribution often arises in many theoretical and applied disciplines. When

true population parameters of hypothesised distribution are known the test is known

as simple. However, more often one does not have the luxury to deal with the whole

population because of reasons such as the absence of access to a whole dataset or

unacceptable cost of information collection. In this case, parameters have to be

estimated from a sample and the test that is based on parameter estimates is known as

composite. Inferences that are based on a composite test are in general less reliable

than in the case of a simple test. To understand the reason for that, it should be noted

that estimates obtained from a sample will often differ from parameters of a

population. The sample estimates are the ‘best fit’ parameters, or parameters that fit

especially well to the sample that they are estimated from. We can say that a sample

and estimates are ‘relatives’ and therefore composite tests are too liberal, failing to

reject 0H of a hypothesised distribution reliably enough. In this section, a simple test

0H of a normal against AH of no normal is introduced. The test is based on the

method first suggested by Nechval, Nechval and Vasermanis (2001) and extended in

Moldovan et al. (2002). This method is originally designed for change-point detection

in regression relationship and especially efficient in small samples of data. The idea

behind the method can be stated as follows. A stochastic process observed up to time t

can be described by an invariant statistic that fully defines the character of the process

with respect to known assumptions. This statistic follows a known theoretical

distribution. At time t+1 the next outcome of the statistic becomes available. Since

distribution of the statistic is known, the confidence interval α−1 can be specified. If

a new outcome of the statistic falls outside the confidence interval it can be concluded

that the process is broken with confidence α−1 . The analytical derivation of the

method is presented below.

Suppose the following basic regression model is given:

Ttwy tttt 1,2,..., , =+′= ax (4.6)

48

where at time t, yt is the observation on the dependent variable and xt is the column

vector of observations on p regressors. The first regressor, xt,1, will be taken equal to

unity for all values of t if the model contains a constant. The other regressors are

assumed to be non-stochastic and therefore auto-regressive models are excluded from

consideration. The column vector of parameters, at=(at,1,…,at,p)′, is written with the

subscript t to indicate that it may vary with time. It is assumed that the error terms, wt,

are independently and normally distributed with zero mean and variances 2tσ ,

Tt ,...,1= . The problem is to construct a test for constancy of regression relationships

over time, which consists of testing the null hypothesis:

TtNyH tttttt ,...,1 ),0( ),0( ),,(~: 2220 ===′ σσσ aaax (4.7)

against the alternative

;,...,1 ),0( ),0( ),,(~: 222 τσσσ ===′ tNyH ttttttA aaax

Tttt ,...,1 ),1( ),1( 22 +=== τσσaa (4.8)

where the parameters a(0)=[a1(0),…,ap(0)]′, a(1)=[a1(1),…,ap(1)]′, σ2(0), σ2(1), and τ,

the point after which the change occurs, are unknown. The problem as stated is

meaningful only if 1≤τ≤T–1.

Assuming that H0 is true, let ta) be the least-squares maximum likelihood

estimate of at for the regression model (4.6), if we set at= a(0), 2tσ =σ2(0), ∀ Tt ,...,1= ,

based on the first t observations, i.e.,

Tptttttt , ... 1, ,)( 1 +=′′= − yXXXa) (4.9)

49

where

)y, ... ,( ),, ... ,( t1tt1 ′==′ yt yxxX (4.10)

and the matrix tt XX′ is assumed to be non-singular. The unbiased estimator of 2tσ is

given by

)/(22 ptstt −=σ) (4.11)

where 2ts is the residual sum of squares after fitting the model to the first t

observations, i.e.,

)()( tt2 aXyaXy ))

ttttts −′−= (4.12)

The estimates ta) and 2tσ) are independently distributed as follows:

))( ,(~ˆ 12 −′ ttttpt N XXaa σ (4.13)

22

2

~)(pt

tpt−

− χσ

σ) (4.14)

with tp − degrees of freedom.

50

Lemma 4.1. (Recurrence Relations).

111

1 )()( −−−

− ′=′ tttt XXXXtttt

tttttt

xXXxXXxxXX

111

111

111

)(1)()(

−−−

−−−

−−−

′′+′′′

− (4.15)

)()( 11

1 −−

− ′−′+′=′ tttttttt y axxXXaa ) (4.16)

Tptvss ttt , ... 1, ,221

2 +=+= − (4.17)

where

2/11111 ])(1[)( tttttttt yv xXXxax −

−−− ′′+′−= ) (4.18)

Proof. The relation (4.15) was given by Plackett (1950) and Bartlett (1951). It

was used in order to avoid having to invert the matrix (X′ tXt) directly at each stage of

the calculations. It is proved by multiplying the left-hand side by X′ tXt and the right-

hand side by X′ tXt = X′ t-1Xt-1+x′ txt. Since ta) is the least-squares estimate it satisfies

ttttttttt yxyXyXaXX +′=′=′ −− 11)

ttttt yxaXX +′= −−− 111) )( 11 −− ′−+′= ttttttt y axxaXX )) (4.19)

This implies (4.16).

)()(2ttttttts aXyaXy )) −′−= )()( 11 −− −′−= tttttt aXyaXy )) )()( 11 −− −′′−− tttttt aaXXaa ))))

21

21 )( −− ′−+= tttt ys ax ) )()( 1

1−

− ′−′′− ttttttt y axxXXx ) (4.20)

which gives (4.17) on substituting for (X′ tXt)-1 from (4.15).

51

Lemma 4.2 (Normality and Independence of Prediction Errors). Under H0

[assuming that at=a(0), 2tσ =σ2(0), ∀ t=1,…,T, for the regression model (4.6)],

vp+1,…,vT are independent, N[0,σ2(0)], where vt [t=p+1,…,T, see (4.18)] is the

standardised prediction error of yt when predicted from y1,…,yt-1.

Proof. The unbiasedness of vt is obvious and the assertion Var{vt}= 2tσ follows

immediately from the independence of yt and 1−ta) . Also,

2/1111

1

1

111 ])(1[ )( −−

−−

−

=

−−− ′′+

′′−= ∑ tttt

t

iiittttt wwv xXXxxXXx (4.21)

Since each vt is a linear combination of the normal variates wj, the vt-s are jointly

normally distributed. Now

′′− ∑−

=

−−−

1

1

111 )(

t

iiitttt wwE xXXx

′′− ∑−

=

−−−

1

1

111 )(

j

iiijjjj ww xXXx

tjjj xXXx )(00)[0( 112

−−′′−−=σ ]))(()( 11111

111 jjjttttt xXXXXXXx ′′′′′+ −

−−−−−

−− )( 0 jt <= (4.22)

It follows that vp+1,…,vT are uncorrelated and therefore independent in terms of their

joint normality.

The transformation from the wt’s to the vt’s is a generalised form of the Helmert

transformation [see Kendall and Stuart (1969)].

Lemma 4.3 (First Characterisation of Prediction Errors). Let yt, t=1,…,T, be T

real independent random variables, with means x′ ta(0), t=1,…,T, respectively, and

common variance σ2(0) (σ(0)>0). Then vt, t=p+1,…,T, given by (4.18) are

52

independently and identically normally distributed with mean zero and variance σ2(0)

if and only if the yt (t=1,…,T) are normal with means x′ ta(0) and variance σ2(0).

Proof. This rests on Lemma 4.2 and a result of Cramer [see Lukacs (1960)].

Lemma 4.4 (Student’s Random Variables). Suppose that vp+1,…,vT are T-p real

independent random variables, T-p>1. If vt (t=p+1,…,T) are independently and

identically normally distributed variates with zero mean and common standard

deviation σ(0) (σ(0) > 0), then zp+2,…,zT given by

Tptsvptz ttt , ... 2, ,/)]1([ 12/1 +=+−= − (4.23)

are random variables independently distributed according to Student’s law with

1,2,…,T-(p+1) degrees of freedom respectively.

Proof. This follows by using the fact that ta) and 2ts , ∀ t=p+1,…,T, are

independently distributed and the result of Basu’s lemma [see Basu (1955)].

Lemma 4.5 (Second Characterisation of Prediction Errors). The necessary and

sufficient condition for vt, t=p+1,…,T, to be independently and identically normally

distributed with zero mean and common standard deviation σ(0) is that zp+2,…,zT

given by (4.23) are independently distributed according to Student’s law with

1,2,…,T-(p+1) degrees of freedom respectively, and T≥p+3.

Proof. It can be shown, after some algebra (see Lemma 4.1), that

Tptvvptzt

pjjtt , ... 2, ,)]1([

2/11

1

22/1 +=

+−=

−−

+=∑ (4.24)

53

We can therefore apply Kotlarski’s result [see Kotlarski (1966)] to the random

variables vp+1,…,vT and obtain zp+2,…,zT. Then the proof follows by using this result

and Lemma 4.4.

zt is a statistic that defines a character of a process observed up to time t. How

knowledge about distribution of zt can be applied to detecting brakes in the process

and simple hypothesis testing is shown next.

Theorem 4.1 (Characterisation of a Normal Regression via the Student

Distribution). Let yt, t=1,…,T, be T real independent random variables (T≥p+3) with

means x′ ta(0), t=1,…,T, respectively, and common variance σ2(0) (σ(0)>0). Let vt,

t=p+1,…,T, be defined by (4.18) and let zt, t=p+2,…,T, be defined by (4.23), then the

yt (t=1,…,T) are N[x′ ta(0),σ2(0)] if and only if zp+2,…,zT are independently distributed

according to Student’s law with 1,2,…,T-(p+1) degrees of freedom respectively.

Proof. This follows immediately by applying Lemma 4.3 and Lemma 4.5.

Thus, we can already replace the composite null hypothesis

3)( 1,..., )],0(),0([~: 20 +≥=′ pTTtNyH tt σax (4.25)

with the simple equivalent null hypothesis

T20 , ... ,: zzH p+• (4.26)

are independently distributed according to Student’s law with 1,2,…,T-(p+1) degrees

of freedom respectively. It should be also appreciated here that the number of the

original variables has only been decreased by the number of eliminated unknown

parameters through these transformations.

54

Theorem 4.2a (Characterisation of a Normal Regression via the F

Distribution). Let yt, t=1,…,T, be T real independent random variables (T≥p+3) with

means x′ ta(0), t=1,…,T respectively, and common variance σ2(0) (σ(0)>0). Let 2tz ,

t=p+2,…,T, be defined by

Tptssptzt

tt , ... 2, ,1)]1([ 2

1

22 +=

−+−=

−

(4.27)

then the yt (t=1,…,T) are N[x′ ta(0),σ2(0)] if and only if 22pz + ,…, 2

Tz are independently

distributed according to the central F distribution with 1 and 1,2,…,T-(p+1) degrees

of freedom respectively ( )1(,12 ~ +− ptt Fz ).

The univariate characterisation can be extended to the multivariate process.

Theorem 4.2b (Characterisation of a Multivariate Normal Distribution via the

F Distribution). Let yt, t=1,…,T, be T independent ν-variate random variables

(T≥ν+2) with common mean a and covariance matrix (positive definite) G. Let 2tz& ,

t=ν+2,…,T, be defined by

( ) ( ) ...1)1(1

111

2 =−′−−+−= −−

−− kkkkkt kkkz yyGyy

νν

&

Tt

t

t , ... ,2 t,1)1(...1

+=

−+−=

−

ννν

GG

(4.28)

where

∑−

=− −=

1

11 )1/(

t

iit tyy (4.29)

55

∑−

=−−− ′−−=

1

1111 ))(

t

ititit yyy(yG (4.30)

then the yt (t=1,…,T) are Nν(a,G) if and only if 222 ,..., Tv zz && + are independently

distributed according to the central F distribution with ν and 1,2,…,t-(ν+1) degrees of

freedom respectively ( )1(,2 ~ +− vtvt Fz& ).

Proof. Theorems 4.2a and 4.2b can be proved by using the results of Lemma 4.4

and Theorem 4.1. The proof being straightforward is omitted.

Here the following theorem clearly holds.

Theorem 4.3 (Characterisation of a Normal Regression via the Uniform

Distribution). The yt (t=1,…,T) are N[x′ ta(0),σ2(0)] if and only if the random

variables up+2,…,uT are independently and uniformly distributed on the interval from

zero to one [i.i.d. U(0,1)], where

TptzFu tptt 2,..., ),(1 2)1(,1 +=−= +− (4.31)

F1,t-(p+1) )(⋅ is the cumulative distribution function of the central F distribution with 1

and T-(p+1) degrees of freedom respectively.

Proof. This follows immediately by applying Theorem 4.2a and the probability

integral transformation theorem.

To illustrate the suggested technique, Moldovan et al. (2002) gave the example

based on data presented by Beckman and Cook (1979):

56

Table 4.6 Yield of hydrogen sulphide from gamma radiolysis as a function of krypton pressure

Krypton pressure (xt) 1.00 1.50 1.75 2.10 2.35 2.65 3.00 Yield (yt) 6.40 7.00 7.40 8.80 9.00 6.40 6.60

It is necessary to test the null hypothesis

1,...,7 ,)0()0( : 210 =++= twxaayH ttt (4.32)

against the alternative

;1,..., ,)0()0( : 21 τ=++= twxaayH tttA

71,..., ,)1()1( 21 +=++= τtwxaay ttt (4.33)

where wt, t=1,2…,7 is a sequence of independent random variables assumed to be

N(0,σ2). The parameters a1(0), a2(0), a1(1), a2(1), σ2 and τ , the point after which the

change occurs, are unknown.

In order to detect a possible change in the regression coefficients, we used the

technique proposed above. This technique gave 5=τ) and 61.49 26 =z . Since

,1.3401.0,;3,126 => =αcritFz the above null hypothesis is rejected using type I error

α=0.01. This result is the same as reported in Beckman and Cook (1979) but it is

obtained without resorting to computationally intensive simulation methods. In this

study, we use Theorem 4.3 and replace the composite null hypothesis

)ˆ,ˆ(~: 2,0 σµNzH kt , t=1,2,…,T (4.34)

with the simple equivalent null hypothesis

57

1),i.i.d.U(0,~:0 tuH • t=3,4,…,T (4.35)

where ktz , is given by (4.1) and tu is given by (4.31). Note that the length of tu is

two variables less than the number of the initial standardised return variables since

two parameters µ̂ and 2σ̂ in (4.34) have been eliminated from the test.

Next, the Kolmogorov-Smirnov (KS) test [see Conover (1999)] is applied to the

set of variables tu in order to find if they are independently uniformly distributed on

the interval from zero to one [i.i.d.U(0,1)]. According to Theorem 4.3, ktz , is

normally distributed with unknown parameters µ and 2σ [ ),(~ 2, σµNz kt ] if and

only if tu is independently uniformly distributed with the known support region )1,0( .

The KS test applied to AUD/USD and JPY/USD standardised return series has

strongly rejected 0H of 1)i.i.d.U(0,~tu for all series. The associated p-values are

almost indistinguishable from zero and therefore are not reported. Thus, with

application of Theorem 4.3, the hypothesis of normally distributed standardised

returns ktz , can be rejected for all standardised return series. Notably, no knowledge

about distributional parameters of ktz , series has been used to arrive at this inference.

It should be noted here that none of the three tests presented in Sections 4.2-4.4

has reliably indicated that ktz , is most likely to follow one of the known theoretical

distributions. Therefore it has been decided to go another way and attempt to simulate

the density that ktz , belongs to. The results are reported in the next section.

4.5 Simulation-Based Test

The careful examination of intraday return series has revealed that simple

standard deviations of intraday returns that correspond to different times of a day have

one particularity. Specifically, simple standard deviations of first intraday returns are

substantially higher than the standard deviations of the rest of intraday returns. Simple

58

standard deviations of intraday returns ntx , , kn ,...,2,1= with k that have been used

for estimation of the optimal realised volatility series are shown in Table 4.7.

Table 4.7 Simple standard deviations of intraday returns

n = AUD/USD JPY/USD 1 0.0051 0.0053 2 0.0015 0.0013 3 0.0015 0.0014 4 0.0015 0.0013 5 0.0014 0.0013 6 0.0012 0.0012 7 0.0011 0.0011 8 0.0012 0.0010 9 0.0010

10 0.0010

As can be seen from the table, the standard deviations of the first intraday returns

( 1=n ) are more than three times greater than standard deviations of the rest of the

intraday returns. This means that the variability of returns that are sampled first in a

day is substantially higher than the variability of the rest of the intraday returns. This

observation is consistent with Madhavan et al. (1997), Hasbrouck (1993), Lam and

Tong (1999) and Bildik (2000), amongst others, who also documented that volatility

of first intraday returns is generally higher than volatility of intraday returns on

average. Some potential reasons for this effect that have been listed by authors include

the release of information accumulated over non-trading hours, strategic trading and

increase in liquidity risk in non-trading periods. We see dynamics of financial returns

as a reflection of underlying informational shocks that continuously affect our

existence. Even when a market is close and prices cannot be observed, information

continues to arrive with associated effect on dynamics of financial returns. According

to this view the release of information accumulated over non-trading hours is the most

plausible reason for jumps in volatility of intraday returns. Noting that heterogenous

intraday volatility violates the assumption of the continuous time process behind

estimation of realised volatility (see Section 3.2), it has been decided to simulate the

stochastic process that is similar to empirically observed. The settings of the

stochastic experiment are presented next.

59

Suppose the sequence of intraday log returns is given by:

nttntx ,, εσ ⋅= )/1,0(~, kNntε , kn ,...,1= (4.36)

where ntx , is the n-th intraday log return, tσ is a prior (given) volatility over a day t

and k is the number of intraday returns recorded within a day t . Thus, the intraday

returns are distributed as )/,0(~ 2, kNx tnt σ . Note that when ∞→k , this process is

equivalent to the process underlying realised volatility estimation given by (3.2). The

log return in day t is computed as a sum of intraday log returns:

∑=

=k

nntt xr

1, (4.37)

The variance of the sum of uncorrelated normal random variables is equal to the sum

of variances of these variables. Since ∑=

=k

nnttz

1,ε is distributed as )1,0(~ Nzt and

intraday returns ntx , are serially uncorrelated by construction, daily returns are

distributed as ),0(~ 2tt Nr σ . Realised volatility and corresponding realised standard

deviation over the day t are computed as follows:

∑=

=k

nntkt xv

1

2,, (4.38)

∑=

=k

nntkt x

1

2,,σ (4.39)

As has been demonstrated in Chapter 3, the realised standard deviation kt ,σ converges

to a prior volatility tσ as ∞→k . Daily standardised returns are computed as follows:

kt

ktt

kt

tt

xxrz,

,1,

,

...σσ

++== (4.40)

60

Since a random normal zero mean variable standardised by its standard deviation is

standard normally distributed and recognising that kt ,σ is an unbiased and consistent

estimator of tσ , tz is expected to be standard normally distributed )1,0(~ Nzt .

As a next step, the heterogeneity in the variance of intraday return process given

by (4.34) is introduced. The normal random innovation ),0(~ 2tt Nj σ is added to the

first intraday return within each day

ttj

t jxx += 1,1, 0),( 1, =tt jxCov (4.41)

The daily log return in the presence of a jump in variance becomes

kttttj

t xxjxr ,2,1, ...)( ++++= (4.42)

The realised standard deviation in the presence of a jump becomes

2,

22,

21,, ...)( ktttt

jkt xxjx ++++=σ (4.43)

The corresponding standardised return series is given by

2,

22,

21,

,2,1,

...)(

...

ktttt

kttttjt

xxjx

xxjxz

++++

++++= (4.44)

In this study, it has been found that if the variance of tj is greater than zero,

then standardised returns jtz follow a specific two-component distribution with the

shape that is visually identical to the shape of the empirically observed distributions

of AUD/USD and JPY/USD standardised returns (see Figures 4.1 and 4.2). The

Figure 4.5 demonstrates this.

61

Figure 4.5 Histogram of simulated variables jtz

Note: jtz is given by (4.41), )/1,0(~, kNx nt , )3,0(~ Njt , 10=k and 2585=T

This finding allows the conclusion that the special case of heterogeneity in

variances of intraday returns is most likely to be responsible for the two-component

effect that is observed in standardised by realised standard deviations returns.

Remembering that the release of information accumulated over night has been noted

as the most plausible reason for jumps in intraday volatility, the absence of the ability

to observe price dynamics continuously should be accepted as the direct cause of the

two-component effect.

There is at least one important consequence of overlooking the two-component

effect. Andersen et al. (2001c) suggested the model that heavily relies on the standard

normal distribution of conditional on realised volatility returns. Authors demonstrated

how their approach can produce “well-calibrated density forecasts of future returns”.

The presence of the two-component effect documented in this study means that the

model suggested by Andersen et al. (2001c) is misspecified and one must be cautious

with using it in “asset pricing, asset allocation and financial risk management

applications”.

62

4.6 Concluding Remarks

In this chapter, the unusual shape of empirical distributions of standardised by

realised standard deviations returns was documented. This finding has been entitled

the two-component effect. In Section 4.2, the likelihood ratio test was applied for

testing 0H of a single normal against AH of a mixture of two normals. Although 0H

was rejected for all examined series, the results were not accepted as conclusive due

to the parameters’ non-identifiability problem. In Section 4.3, the nonparametric

Pearson goodness of fit test was performed as an alternative to the likelihood ratio

test. The test confirmed that most standardised return series are likely to follow a

mixture of two normals. Since all the parameters in this test are maximum likelihood

estimates, the test is composite. Becker and Hurn (2002) demonstrated that composite

tests are often too liberal and therefore the results of the Pearson goodness of fit test

also were not accepted as conclusive. In Section 4.4, the method for detection of

change-points in regression relationship was introduced. Based on this method, the

simple test 0H of a normal against AH of no normal was suggested and applied.

Although the test strongly rejected 0H for all series, it was not indicated which

theoretical distribution the series were most likely to belong to. In Section 4.5, it was

noted that the result of intraday return series examination revealed the special case of

heteroscedasticity. Specifically, the simple standard deviations of the first intraday

returns are more than three times higher than the simple standard deviations of the rest

of intraday returns. This finding was applied for simulation of the stochastic process

with characteristics that are similar to empirically observed. The shape of the obtained

distribution was found to be visually identical to the shapes of the empirical

distributions of AUD/USD and JPY/USD standardised returns. This allowed the

conclusion that the two-component effect is most likely to be caused by the special

case of heteroscedasticity in intraday returns.

63

Chapter 5

Modelling of Random Variables

5.1 Introduction

A combination of independent random variables sometimes produces a random

variable with new distinct stochastic characteristics. Random variables with new

stochastic characteristics can often be applied for approximation of empirically

observed phenomena. For example, the sum of T squared i.n.d. variables is distributed

as chi-squared random variable with T degrees of freedom. The chi-squared

distribution is widely used for approximating distributions of test statistics, such as

statistics of the Pearson goodness of fit test applied earlier in the study (see Section

4.3). The following chapter illustrates how modelling techniques can be applied not

only to empirically observable variables, such as financial returns, but also to purely

random variables. In the previous chapter it has been shown that a certain

combination of independent normal random variables given by (4.44) produces the

probability density with an unusual two-peaked shape. This chapter demonstrates how

the combination (4.44) can be generalised with application of modelling techniques.

The result of generalisation is applied for validation of the model for forecasting

density of future asset prices.

In Section 5.2, the derivation and some characteristics of the new probability

density, which has been entitled J for expositional purposes, are presented. Section

5.3 is demonstrates how ),( αγJ can be applied to forecasting probability quintiles of

future price distributions via Monte Carlo simulation. Section 5.4 summarises the

chapter.

5.2 Derivation and Characterisation of ),( αγJ

In the previous chapter it was found that a certain combination of normal

random variables given by (4.44) produces a new random variable with the two-

peaked probability density. Recognising that if ∞→k then 0, →ntx and relating the

64

homogenous variance 2tσ to the variance of a jump as 22

, ttj σασ ⋅= , the function

(4.44) can be presented as follows:

22tt

ttt

jjyz

+

+=σ

(5.1)

where ),0(~ 2tt Ny σ , ),0(~ 2

tt Nj σα ⋅ , 0≥α reflects the portion of 2tσ in 2

,tjσ and

0),( =tt jyCov .

Since the denominator of (5.1) is an unbiased and consistent estimator of the standard

deviation of the numerator (see Section 3.2) and noting that the variance of ty is

directly related to the variance of tj through the parameter α , it can be demonstrated

that the distribution of tz does not depend on the variance of ty . Therefore (5.1) can

be generalised as follows:

21 t

ttt

jjyz

++= (5.2)

where )1,0(~ Nyt , ),(~ αγNjt , 0≥α and 0),( =tt jyCov .

The new function is a combination of two i.n.d. random variables one of which is

standard normally distributed and the other is normally distributed with arbitrary

parameters. Since the distribution of tz given by (5.2) depends only on two

parameters γ and α , this probability density will be referred to as ),( αγJ . In this

section some characteristics (properties) of ),( αγJ are given. The set of analytical

proofs is not yet completed and therefore is not reported here. However, all

characteristics have been proven experimentally with application of Monte Carlo

simulation techniques.

),(~ αγJ given by (5.2) has the following characteristics:

65

Property 5.1. If 0=α and 0=γ , then tz is standard normally distributed

)1,0(~ Nzt .

Comments: This property follows directly from equation (5.2).

Property 5.2. If 0=γ , then the mean and variance parameters of tz are

expected to be 0 and 1 respectively.

Comments: This property follows from the fact that the denominator of (5.2) is

the unbiased and consistent estimator of the standard deviation of the numerator.

Property 5.3. If 0>α , then the kurtosis of tz is expected to be less than 3.

Comments: At this stage, this is a simulation-based property.

Property 5.4. If 0>α , then the distribution of tz contains two peaks (means)

−µ and +µ that are expected to be symmetrical around zero.

Comments: Consider the extreme case when ∞→α . When this is the case, the

magnitude of tj is generally much larger than the magnitude of ty ( tt yj >> ). As a

result, tz in (5.2) can be approximated as follows:

t

t

t

tt j

jjjz =≈

2 (5.3)

tz takes the value close to 1 if 0>tj and the value close to –1 if 0<tj . Therefore

for ∞→α two expectations are 1−→−µ and 1→+µ . When α becomes smaller

the magnitude of tj respectively to ty decreases reducing the dominance of these

variables. Finally, if 0=α , then both −µ and +µ are expected to be zero. Figures

5.1-5.4 below illustrate this property.

66

Figure 5.1 )1,0(~ J

Figure 5.2 )2,0(~ J

67

Figure 5.3 )20,0(~ J

Figure 5.4 )200,0(~ J

68

Intuitively, this result follows from the fact that as α is getting large, the probability

of the magnitude of tj being close to the magnitude of ty decreases. As a result, tj

dominates increasingly more often, making −µ and +µ close to –1 and 1

respectively.

In presenting the following characteristics suppose that the distance between

expectations −µ and +µ of ),( αγJ is given by:

−+ −= µµd (5.4)

Property 5.5. d increases with α and goes to 2 as α goes to infinity.

Comments: This property follows directly from the intuition behind Property

5.4. The simulation-based plot of parameters α against distances between peaks d is

shown in Figure 5.5.

Figure 5.5 Relationship between d and α

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 5 10 15 20 25

alfa

d

The relationship is shown up to 20=α . As α further increases, d approaches 2.

69

Property 5.6. If 0<tj , then tz belongs to the left-side component ( −tz ) with

the expectation −µ . If 0>tj , then tz belongs to the right-side component ( +tz ) with

the expectation +µ .

Comments: This follows directly from the Property 5.4.

Property 5.7. If 0<γ or 0>γ , then more random innovations tz are expected

to belong to left and right components respectively.

Comments: This property follows from Property 5.6. γ is somewhat similar to

the mixing proportion in a mixture of two normals specification (see Section 4.2).

Figures 5.6 and 5.7 below demonstrate this property.

Figure 5.6 )5,1(~ −J

70

Figure 5.7 )5,1(~ J

Property 5.8. If 0=γ and 0>α , then left ( −tz ) and right ( +

tz ) components are

right and left skewed respectively with the parameter of skewness that increases with

α .

Comments: This property follows from the fact that if tj is close to zero, then

tz is approximately standard normally distributed )1,0(~ Nzt . Thus, the expectation

of tz is located on the right and on the left of −µ and +µ respectively inducing right

and left skewness. This property is demonstrated graphically in Figures 5.8 and 5.9

below.

71

Figure 5.8 The left component of )5,0(~ J : Right skewness

Figure 5.9 The right component of )5,0(~ J : Left skewness

A more complete list of characteristics together with analytical proofs will be

presented in a subsequent study. Here we would like to note that although J has not

72

been sufficiently explored yet, it appears to be very flexible. Firstly, multiplying

innovations drawn from ),( αγJ by a positive constant δ and adding a real number

m , a variable with characteristics similar to ),( αγJ but with desired unconditional

mean m and variance 2δ can be obtained. Secondly, the sum of m ),0( αJ variables

( 0>α ) divided by m produces a new variable distributed with (m+1)-peaked

density and unconditional mean and variance 0 and 1 respectively:

mzzm

iitt ∑

=

=1

,& ),0(~, αJz it , 0>α (5.5)

Thirdly, varying γ and α can produce stochastic processes with outliers that lie up to

several hundred standard deviations from the mean. It is also important to note that at

the present moment a probability density function (p.d.f.) for ),( αγJ is not available

and this probability distribution is simulated only. The absence of a p.d.f. is very

restrictive in terms of a practical application. For example, maximum likelihood

estimation (MLE) cannot be applied in the search for optimal parameters γ and α .

However, since the random number generator for ),( αγJ is available, this

distribution can already be applied. In the next section it is demonstrated how ),( αγJ

can be applied for improving the forecast of probability quintiles via Monte Carlo

simulation.

Section 5.3 Application of ),( αγJ : Forecasting Probability Quintiles of Future

Price Distributions

Since its acceptance by the Basle Committee [Basel Committee on Bancing

supervision (1996)] Value at Risk (VaR) has become the most widely used risk

measurement tool in the banking sector. The VaR of a portfolio is defined as the

maximum loss that can be expected with a certain level of confidence over a

particular interval of time [Guermat et al. (2002)]. Since the potential lose is

expressed by the amount of money, it is an unambiguous and easily interpretable

measure. VaR is presented graphically in Figure 5.10.

73

Figure 5.10 Illustration of VaR

Figure 5.10 shows the distribution of gains/losses on the hypothetical portfolio. VaR

indicates the loss that is likely to be exceeded with probability α percent. In practice

VaR is used for many purposes. For example, financial institutions use VaR for

evaluation of the overall riskiness of a business activity and as indicator of the amount

of resources that should be put aside for compensation of unexpected losses.

Prudential authorities use VaR for calculating the amount of required regulatory

capital. In this section the Monte Carlo simulation method is applied for forecasting

distributions of future prices on AUD/USD and JPY/USD futures contracts. Since

prudential authorities require banks and other financial organisation to report 95% and

99% VaR for 1 to 10 days ahead [see Jorion (2001)], we are interested in forecasting

distributions of future prices for 1 to 10 days ahead and selecting quintiles that

correspond to 95% and 99% VaR. VaR is a stochastic measure and therefore

deviations from theoretically expected values that can be attributed to a chance should

be anticipated. Kupiec (1995) gave the following confidence intervals for VaR

backtesting that is based on 1000 periods:

95% confidence level: 6537 << T (5.6)

99% confidence level: 174 << T (5.7)

For 95% and 99% confidence levels 50 and 10 observations respectively from 1000

are expected to lie below the VaR estimates. Too few exceptions (say, 35 for the 95%

VaR (quintile)

Confidence level )1( α− Significance level

α

Losses Gains

74

confidence level) indicate that the model is overconservative and that can lead to

suboptimal allocation of resources. In contrast, underestimation of the risk (say, 20

exceptions for the 99% confidence level) can result in a shortage of resources for

covering unexpected losses and/or cause sanctions from prudential authorities. In this

study the VaR backtesting serves as objective criterion for evaluation of the

forecasting performance of the following financial model:

)exp(1 ttt rPP ⋅= − (5.8a)

tktt zr ,σ= )1,0(...~ Ndiizt (5.8b)

)ln( ,kttq σ= (5.8c)

tttt cqccq ηη +++= −− 13121 (5.8d)

ttt hξη = )1,0(...~ Ndiitξ (5.8e)

12

1 −− ++= ttt bhawh η (5.8f)

where tP and tr are the price and log return at time t respectively, kt ,σ is the realised

standard deviation estimated from returns sampled k times per time interval, 1c , 2c

and 3c are an unconditional mean, AR and MA are coefficients in (5.8d) respectively,

tη is the serially uncorrelated innovation, th is the conditional second moment of tη .

w , a and b are the unconditional mean, ARCH and GARCH coefficients in (5.8f)

respectively. The presented specification was found to be optimal for both AUD/USD

and JPY/USD series.

The model (5.8) is used for forecasting VaR via Monte Carlo simulation. To

understand how this model works, it is convenient to follow from the bottom to the

top. To start with, it should be noted that estimates of the parameters 1c , 2c , 3c , w , a

and b are known since they can be estimated by fitting the model to tq , Tt ,...,1=

series. First, starting values 1−tη and 1−th must be supplied to the model. Starting

values can be obtained from the same model fitted up to time 1−T . Having the

parameter estimates and starting values, th in (5.8f) can be computed. Second,

conditional second moment th is multiplied by the standard normal innovation

75

)1,0(...~ Ndiitξ in (5.8e). As a result of this operation, the linearly independent

innovation, tη with volatility dependent on the previous realisations is obtained.

Third, the ARMA (1,1) structure in (5.8d) makes tη linearly dependent on the

previous realisations arriving to the value tq with stochastic characteristics close to

ones observed in the empirical series. Fourth, the exponent of tq should be taken for

obtaining the value similar to the empirical realised standard deviation. Fifth,

recalling the return decomposition (3.5), a realised standard deviation is multiplied by

the standard normal innovation with obtaining a value with stochastic characteristics

similar to empirically observed log returns. Finally, the price tP is a product of the

previous price 1−tP and exponent of a computed return tr . The price is calculated in

this way because log (continuously compounded) returns have been used. Repeating

the whole procedure M times allows obtaining M possible outcomes of the price at

time t . If M is large (say, 10000=M ), then resulting price outcomes will represent

the distribution of possible future prices. The computation of a probability quintile of

this distribution is straightforward. The outcomes of prices are sorted and the price

corresponding to a required quintile is selected. For the n-step-ahead ( 1>n ) forecast

of probability quintiles the procedure is repeated recursively.

The Monte Carlo simulation experiment is specified as follows. Suppose that at

date T the realised standard deviation series )',...,,( 21 Tσσσσ = are available. All

parameters in the model (5.8) are estimated from these series. Distributions of prices

for 1+T to 10+T days are simulated by drawing random innovations from desired

distributions and feeding them to the models. The number of trials 10000=M is

selected arbitrarily. Next, from simulated distributions 5 and 1 percent quintiles,

which correspond to 95 and 99 percent VaR, are recorded. As a result, in day T a set

of 20 prices corresponding to 5 and 1 percent quintiles are recorded: 9510

951,..., ++ TT PP and

9910

991,..., ++ TT PP . The whole procedure is repeated adding one observation 1+T to the

previous information set: )',...,,( 121 += Tσσσσ . Note that only the information up to

date lT + ( 999,...,1,0=l ) that corresponds to one of the trading days within the 1000-

days backtesting interval is used for forecasting. The actual forecasting is performed

for 1000 trading days: 2500,...,1501=t . As a result, for each series 20000 forecasted

76

prices are recorded: 95102500

9512500

95101501

9511501 ,...,;...;,..., ++++ PPPP and

99102500

9912500

99101501

9911501 ,...,;...;,..., ++++ PPPP . Next, the VaR backtestng method is applied for

evaluation of the forecasting performance of the model. If the model is specified

correctly, then 50 and 10 actual rates for 95 and 99 percent confidence levels

respectively are expected to fall below estimates obtained from simulations. The

acceptable deviations from the expectations are given by (5.6) and (5.7). Prices that

fall below VaR estimates are called exceptions:

dPP cnT

actualnT =− ++ , 0<d (5.9)

where actualnTP + corresponds to the actually observed price at time nT + , c

nTP + is the

estimate obtained from the model with parameters estimated from the information set

up to time T with a confidence level c , 10,...,2,1=n and 0<d indicates the

exception.

First, the model (5.8) with the standard normal innovations in (5.8b) is used. The

results of backtesting are presented in Tables 5.1 and 5.2.

Table 5.1 Numbers of exceptions of (5.8) with )1,0(~ Nzt in (5.8b) applied to AUD/USD series

Days ahead 1 2 3 4 5 6 7 8 9 10 Var95 54 47 47 44 42 41 33* 35* 36* 34*Var99 4* 7 9 8 5 7 9 7 6 7

Note: * indicates the numbers of exceptions 0<d that lie outside the non-rejection regions given by (5.6) and (5.7)

Table 5.2 Numbers of exceptions of (5.8) with )1,0(~ Nzt in (5.8b) applied to JPY/USD series

Days ahead 1 2 3 4 5 6 7 8 9 10 Var95 64 60 58 52 48 53 58 50 47 55 Var99 5 10 5 6 6 7 8 2* 3* 0*

Note: * indicates the numbers of exceptions 0<d that lie outside the non-rejection regions given by (5.6) and (5.7)

77

As can be seen, this model is too conservative. This misspecification is attributed to

the platykurtic empirical distributions of tz (see Table 4.1) as opposed to the standard

normal that has been used in (5.8b) for forecasting. To correct the model, we replace

)1,0(~ Nzt in (5.8b) for ),( αγJ . The parameters γ and α are selected arbitrarily

with respect to the mixing proportions and the distances between mean parameters

that have been estimated for a mixture of two normals specification and, are given in

Tables 4.2 and 4.3. The results of backtesting of the model with ),( αγJ innovations

in (5.8b) are presented in Tables 5.3 and 5.4.

Table 5.3 Numbers of exceptions of (5.8) with )8.14,08.0(~ J in (5.8b) applied to AUD/USD series

Days ahead 1 2 3 4 5 6 7 8 9 10 Var95 57 51 47 47 42 43 38 38 38 36* Var99 9 10 11 12 8 7 10 9 8 8

Note: * indicates the numbers of exceptions 0<d that lie outside the non-rejection regions given by (5.6) and (5.7)

Table 5.4 Numbers of exceptions of (5.8) with )5.15,15.0(~ J in (5.8b) applied to JPY/USD series

Days ahead 1 2 3 4 5 6 7 8 9 10 Var95 74* 71* 64 59 59 60 69* 68* 71* 69* Var99 11 11 9 12 10 11 9 7 7 5

Note: * indicates the numbers of exceptions 0<d that lie outside the non-rejection regions given by (5.6) and (5.7)

While the model applied to the AUD/USD series becomes almost correctly specified,

the same model applied to the JPY/USD series has become too liberal

underestimating the risk. This misspecification is attributed to the lack of the

objective method for selection parameters in ),( αγJ . Indeed, the mean parameters of

left and right components of a mixture of two normals specification are not

symmetrical with respect to zero (see Table 4.3), and this feature has been ignored

completely. When simulated density )5.15,15.0(J was shifted to the left by 0.03, the

following backtesting results were obtained:

78

Table 5.5 Numbers of exceptions of (5.8) with [ )5.15,15.0(~ J – 0.03] in (5.8b) applied to JPY/USD series

Days ahead 1 2 3 4 5 6 7 8 9 10 Var95 63 63 60 52 50 56 62 50 47 56 Var99 9 10 9 10 10 6 8 6 5 1*

Note: * indicates the numbers of exceptions 0<d that lie outside the non-rejection regions given by (5.6) and (5.7)

It can be seen from the table that the model becomes almost correctly specified.

Section 5.4 Concluding Remarks

In this chapter, the technique of modelling of random variables was suggested

and illustrated on the example of derivation of the new probability density. In Section

5.2, the general form of ),( αγJ was presented. ),( αγJ is a combination of two i.i.d.

random variables, one of which is standard normally distributed and the other is

normally distributed with arbitrary parameters. It was shown how parameters γ and

α influence the shape of ),( αγJ . Particularly, the intuition behind the nature of two

components of ),( αγJ was given in Property 5.4. Although ),( αγJ at the present

moment has not a p.d.f., the random number generator is already available. In Section

5.3, it was demonstrated how ),( αγJ can be used in the model (5.8) for forecasting

probably quintiles via Monte Carlo simulation. The standard normal innovations

)1,0(~ Nzt in (5.8b) were replaced for ),(~ αγJzt with the theoretically expected

shift in forecasting performance of the model. However, application of ),( αγJ

indicated the weakness of the arbitrary parameter selection procedure that was used.

After appropriate adjustments, the model became almost correctly specified for both

AUD/USD and JPY/USD series.

79

Chapter 6

Conclusion

6.1 General Overview

This study has been devoted to the problem of stochastic model validation. In

particular, the modelling of financial prices has been considered. It has been

demonstrated that due to the profit maximising actions of market participants,

financial price time series are generally non-stationary (or have no memory). It has

been shown how non-stationary prices can be transformed to stationary return time

series. The Efficient Market Hypothesis has been presented which requires returns to

be independent and therefore, according to the Central Limit Theorem, normally

distributed. The ‘fat’ tails of empirical distributions of financial returns have been

attributed to non-constant volatility (the Mixture of Distributions Hypothesis). It has

then been suggested that the returns conditional on underlying volatility should be

approximately standard normally distributed. To test if this is true, the realised

volatility estimator has been selected as the only one that asymptotically converges to

integrated volatility and does not require information outside the estimation interval.

This choice has been motivated by Andersen et al. (2000a) work where authors

studied standardised by realised standard deviations return series.

Andersen et al. (2000a) found that according to the theoretical expectations,

conditional on realised volatility returns are “nearly Gaussian” or “remarkably close

to a standard normal”. In this study, a return standardisation similar to that of

Andersen et al. (2000a) has been performed. Although the descriptive statistics of

resulted standardised return series are similar to those reported in Andersen et al.

(2000a), the visual examination of histograms revealed the highly unusual two-peaked

shape that has been entitled the two-component effect. In our opinion, this effect has

not been documented in Andersen et al. (2000a) because the authors worked with

financial instruments that are trading 24 hours per day. The only non-trading periods

in the datasets used in Andersen et al. (2000a) are weekends and holidays. However,

it has been demonstrated in this thesis that the two-component effect is most likely to

arise due to ‘overnight’ jumps in the volatility of returns, and probably the specifics of

80

the data used by Andersen et al. (2000a) made this effect less evident. Indeed,

Andersen et al. (2000a) interpreted the “nearly Gaussian” distribution of standardised

returns as “…consistent with the distributional assumptions underlying the Mixture-

of-Distributions-Hypothesis” and “…providing indirect support for the assertion of a

jumpless diffusion, because the presence of jumps is likely to result in a violation of

the normality”. Surprisingly, Andersen et al. (2000a) completely ignored the

platykurtosis of standardised return series, which clearly violates normality and

indicates the potential presence of the two-component effect. However, it is important

to note that even the actual presence of the two-component effect in conditional on

realised volatility return series does not contradict “…support for the assertion of a

jumpless diffusion” reported in Andersen et al. (2000a), because this effect can be

theoretically expected under the assumption of a jumpless diffusion with the presence

of non-trading periods.

The results of the analysis that underlies the two-component effect have been

applied for the financial model validation. Specifically, it is demonstrated that the

precision of forecasts of probability quintiles (VaR) can be improved by taking into

account the specifics of the return dynamics.

6.2 Potential Applications and Limitations

The area of applications of the results obtained in this study includes, in

particular, financial risk analysis and management, asset pricing, asset allocation,

regulatory impact analysis, forecasting, modelling, optimisation, simulation, logic

programming, pattern recognition, automatic control, signal processing, operations

research, climate issues, ecology, medical and biological sciences amongst others. In

particular, the following aspects should be highlighted. The first aspect is related to

the specifics of conditional on realised volatility returns documented in this study. It

has been demonstrated in Section 5.2 that taking these specifics into account can help

to forecast probability density of future returns more precisely. The suggested

forecasting technique can be easily extended to multivariate cases [see Moldovan

(2002a)] and used in asset pricing, asset allocation and financial risk management

applications. The high computational intensity is the obvious but not serious

limitation of this technique.

81

The second aspect is related to the parameter-free transformation technique that

has been introduced and applied in Section 4.4. The simple test for normality that has

been performed in this section allowed to test 0H of a normal against AH of no

normal without knowing the parameters of the initial samples. Although 0H has been

strongly rejected for all samples, the test has not given any information about a

theoretical distribution that the initial samples could be drawn from. This is obviously

the limitation of the suggested test. However, it should be noted that the proposed

technique has been originally designed for detection of change-points in regression

relationship and is especially efficient in small samples of data. The practical potential

of the suggested method is difficult to overestimate, since in everyday life,

practitioners and expert systems are forced to draw inferences relying on the limited

amount of information. The shortage of data arises due to several reasons and here we

give some examples of situations where our method can be applied. First, the time for

data collection and processing can be limited. An example of such a time-critical

situation is the Digital Scene Matching Area Correlation (DSMAC) system that is

used in so-called smart weapons, such as Tomahawk missiles. Due to the short time

intervals between obtaining data and implementation of decisions, the system cannot

perform the full-scale analysis and has to rely on a limited set of parameters that

defines a landscape. The improvement in reliability of decisions produced by the

DSMAC system means the increase in the military strikes precision and thus the

reduction of casualties among civilians. See Waldemark et al. (2000) for an

introduction to the problem. Second, the exceptional nature of events can reduce the

length of a sample to just several observations though demanding regulatory actions

after each event. This situation is known as the end-of-sample problem [Andrews

(2002)] and can be of a particular concern to insurance companies that have to adjust

their policies with respect to extreme events, such as failures of space rocket launches

or earthquakes. See Chongfu (1996) for an introduction to the problem. Third,

frequently data is collected experimentally and the cost of one experiment can be very

high. Industrial car crash tests can be viewed as an example of the situation in which

the car manufacturers often look for a compromise between the increase in safety of a

new model and the cost of an additional crash test. Fourth, there are areas where the

performance of an additional experiment can cause unacceptable consequences. This

situation is common for clinical dose-response trials where side effects of new drugs

82

are often unknown. See Friede et al. (2002) for the example. Finally, sometimes the

collection of new information is an ethical issue. For example, according to the ethical

norms of the modern society we should try to reduce the number of animals that are

used in scientific experiments.

The third result of this study that should be indicated is the technique of

modelling of random variables suggested in Chapter 5. Specifically, it has been shown

how normal variables can be modelled for obtaining the new probability density that

has been entitled J . The obvious limitation of J is the absence of a p.d.f. that would

allow considering this density as a theoretical distribution. However, we have

demonstrated that since the random number generator is already available, J can be

studied and applied even without a p.d.f. In particular, we have applied platykurtosis

of ),( αγJ for improving the forecasts of probability quintiles via Monte Carlo

simulation. We expect that because of the bimodal structure, ),( αγJ will find

applications in naturally dichotomised disciplines, such as biological and medical

sciences. For example, weights of a group of people that includes both genders are

expected to have a bimodal distribution and ),( αγJ can be more appropriate for

describing this empirical distribution than a traditional mixture of two normals. We

are confident that time will indicate other applications as well as limitations of J and

illustrated technique of modelling of random variables.

6.3 Avenues for Further Research

The results of this study have indicated some avenues for future research. The

first and probably the most interesting avenue is related to the realised volatility

estimator, which in our view will replace most of the traditional volatility estimation

techniques, including the most popular at the moment, ARCH estimators. Recall that

in Numerical Example 3.1 (Section 3.2), for reduction of the measurement error of

realised volatility estimates we applied the volatility signature plot approach proposed

by Andersen et al. (1999c). However, it should be noted that the volatility signature

plot approach is clearly subjective and therefore has to be replaced by an alternative

objective method. Recognising that for realised volatility estimation over a time

interval no information outside this interval is necessary, different sampling

83

frequencies can be optimal for different intervals. Furthermore, the presence of the

special case of heteroscedasticity in intraday return series, which manifests itself as a

documented two-component effect, must be accounted for. There are some potential

ways to deal with heteroscedasticity of intraday returns. For example, Andersen and

Bollerslev (1997c) and Müller (1990, 1993) applied modelling procedures to intraday

return series in order to smooth the process. Alternatively, since it has been concluded

that the two-component effect is a result of a drawback in the data collection, an

indirect inference about variability of returns from information flows in markets

located in different time zones can help to replace missing data. Ito and Lin (1993)

provided the evidence that such an inference is possible. Finally, the two-component

effect can be eliminated by a mathematical transformation of J distributed

standardised returns. In order to understand why the special case of heteroscedasticity

should be eliminated from intraday return series it is important to note that the two-

component effect provides the strong indirect evidence that prices follow jumpless

diffusion and, therefore conditional on the actual (integrated) volatility returns should

be approximately standard normally distributed. Realised volatility is a highly

efficient volatility estimator and therefore returns standardised by properly measured

realised standard deviations are expected to be approximately standard normally

distributed. Having such a series will enable the following research. Recall Theorem

4.2b presented in Section 4.4. This theorem allows a simple test for detection of a

change-point in a multivariate normal process:

H0: zt~ Nν(at,Gt), at = a(0), Gt = G(0), t = 1,…,T (6.1)

against

HA: zt~ Nν(at,Gt), at = a(0), Gt = G(0), t = 1,…,τ

at = a(1), Gt = G(1), t = τ+1,…,T (6.2)

where zt is the multivariate (ν-variate) series under the null distributed with the mean

at = a(0) and covariance matrix Gt = G(0). Since conditional on volatility returns are

84

expected to be approximately standard normally distributed )1,0(~ Nzt , the test can

be simplified to the test for a change-point detection in a covariance structure:

H0: zt~ Nν(0,Gt), Gt = G(0), t = 1,…,T (6.3)

against

HA: zt~ Nν(0,Gt), Gt = G(0), t = 1,…,τ

Gt = G(1), t = τ+1,…,T (6.4)

In words, we test if the covariance structure of financial assets is stable over

time (H0) against a break (say, increase) in covariance at time τ (HA). Since

1)max( −=Tτ , the test is end-of-sample and a covariance structure of multivariate

asset return series can be observed in the nearly real time. The power of this test

increases with the number of assets ν presented in a multivariate series z. The test can

be applied for empirical investigation as well as used practically in asset pricing, asset

allocation and financial risk management applications. For example, it would be

interesting to see if the covariance structure of multivariate return series was stable

just before the market crash in October 1987. The detection of abnormal changes

would mean that the crash could be predicted.

The second question that has been generated by this study is related to the

simple test 0H of a normal against AH of no normal proposed in Section 4.4. Often it

is required not only to reject 0H of a specified distribution but also to find which

particular theoretical distribution a sample has been most likely drawn from. The

same parameter-free technique that has been applied for testing 0H of a normal

against AH of no normal can be used for attribution of a sample to one specified

theoretical distribution. In some applied disciplines the existence of such a test could

eliminate less reliable composite tests completely.

The final research avenue that we would like to note is indirectly related to the

new probability density that has been introduced in this study. This density is a result

85

of the series of stochastic experiments that allowed reproducing the empirically

observed phenomenon. The practical implementation of a stochastic experiment can

be subdivided into the following three stages. First, the empirical phenomenon should

be observed and analysed. As it has been demonstrated in Sections 4.2, 4.3 and 4.4,

the traditional analytical tools not always lead to better understanding of the nature of

a phenomenon. Second, the stochastic experiment has to be designed and stated in

mathematical terms. For designing the stochastic experiment it is important to

recognise that almost any empirical phenomenon can be defined in probabilistic terms

and then presented as a combination of related stochastic phenomena or building

blocks. For example, a conditional on realised volatility daily return can be presented

as a sum of conditional intraday returns and it has been stated by equation (4.44) in

mathematical terms. Third, the experiment must be conducted and consequences

observed. If the experiment does not lead to the result similar to empirically observed

phenomenon then building blocks and relations between them must be examined once

again in order to redesign the experiment. It has been demonstrated in this study that

the suggested approach can be successfully applied and we strongly believe that

systematisation of experimental techniques with respect to stochastic methods can

provide an additional tool for the inductive scientific investigation.

86

Bibliography: Alexander, S., 1961, Price movements in speculative markets: Trends or Random

walks, Industrial Management Review 2, 7-26. Alexander, S., 1964, Price movements in speculative markets: Trends or Random

walks, in P. Cootner (ed.), The Random Character of Stock Market Prices, MIT Press, Cambridge, MA.

Alizadeh, S., M. Brandt and F. Diebold, 2001, High- and low-frequency exchange rate

volatility dynamics: Range-based estimation of stochastic volatility models, National Bureau of Economic Research, Working Paper No. 8162.

Alspach, D.L., 1972, Dual control based on approximate a posteriori density

functions, IEEE Transactions on Automatic Control AC-17, 689-702. Alster, J., and P.R. Belanger, 1974, A technique for dual adaptive control, Automatica

10, 627-630. Amihud, Y. and G. Mendelson, 1991, Trading mechanisms and stock returns: An

empirical investigation, Journal of Finance 42, 533-553. Andersen, T., 1992, Volatility, Working Paper No. 144, Kellogg Graduate School,

Northwestern University. Andersen, T., 1996, Return volatility and trading volume: An information flow

interpretation of stochastic volatility, Journal of Finance 51, 169-204. Andersen, T., 2000, Some reflections on analysis of high-frequency data, Journal of

Business and Economic Statistics 18, 146-153. Andersen, T. and T. Bollerslev, 1997a, Answering the critics: yes, ARCH models do

provide good volatility forecasts, National Bureau of Economic Research, Working Paper No. 6023.

Andersen, T. and T. Bollerslev, 1997b, Heterogeneous information arrivals and return

volatility dynamics: Uncovering the long-run in high frequency returns, Journal of Finance 52, 975-1005.

Andersen, T. and T. Bollerslev, 1997c, Intraday periodicity and volatility persistence

in financial markets, Journal of Empirical Finance 4, 115-158. Andersen, T., T. Bollerslev and F. Diebold, 2002, Parametric and nonparametric

volatility measurement, Handbook of Financial Econometrics, forthcoming. Andersen, T., T. Bollerslev and S. Lange, 1999, Forecasting financial market

volatility: Sample frequency vis-à-vis forecast horizon, Journal of Empirical Finance 6, 457-477.

87

Andersen, T., T. Bollerslev, F. Diebold and H. Ebens, 2001a, The distribution of realised stock return volatility, Journal of Financial Economics 61, 43-76.

Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 1999a, Microstructure bias and

volatility signatures, Manuscript in progress. Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 2001b, The distribution of

realised exchange rate volatility, Journal of the American Statistical Association 96, 42-55.

Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 1999c, Realised volatility and

correlation, Working Paper, University of Pennsylvania. Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 2000a, Exchange rates returns

standardised by realised volatility are (nearly) Gaussian, Multivariate Finance Journal 4, 159-179 (National Bureau of Economic Research, Working Paper No. 7488.).

Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 2000b, Great realisations, Risk,

105-108. Andersen, T., T. Bollerslev, F. Diebold and P. Labys, 2001c, Modelling and

forecasting realised volatility, Kellogg School of Management, Working Paper 276, www.kellogg.nwu.edu/faculty/andersen/htm/abdl4.pdf.

Andrews, D.W.K., 1993, Tests for parameter instability and structural change with

unknown change-point, Econometrica 61, 821-856. Andrews, D.W.K., 2002, End-of-sample instability test, Working Paper, Yale

University. Andrews, D.W.K. and R.C. Fair, 1988, Inference on nonlinear econometric models

with structural change, Review of Economic Studies, 615-640. Atkinson, A.C., 1980, A note on the generalized information criterion for a choice of

a model, Biometrika 67, 413-418. Baba, Y., R. Engle, D. Kraft and K. Kroner, 1991, Multivariate simultaneous

generalised ARCH, Working Paper, University of California, San Diego. Bachelier, L., 1900, Theory of speculation, in Cootner P. edition, The random

character of stock market prices, Massachusetts Institute of Technology Press, Cambridge.

Backus, D., and A. Gregory, 1993, Theoretical relations between risk premiums and

conditional variances, Journal of Business and Economic Statistics 11, 177-185. Bai, X., J. Russell and G. Tiao, 2001, Beyond Merton’s utopia: Effect of non-

normality and dependence on precision of variance estimates using high-frequency financial data, Working Paper, University of Chicago.

88

Bali, T. G., H. Genberg and S. N. Neftci, 2002, Excessive variation in risk-factor correlations and volatilities, Journal of Futures Markets 22, 1119-1146.

Barndorff-Nielsen, O. and N. Shephard, 2002a, Econometric analysis of realised

volatility and its use in estimating stochastic volatility models, Journal of the Royal Statistical Society, Series B, 64, 253-280.

Barndorff-Nielsen, O. and N. Shephard, 2002b, Estimating quadratic variation using

realised volatility, Journal of Applied Econometrics, forthcoming. Barndorff-Nielsen, O. and N. Shephard, 2002c, How accurate is the asymptotic

approximation to the distribution of realised volatility? In D.W.F. Andrews, J.L. Powell, P.A. Ruud and J.H. Stock (eds.), Identification and Inference in Econometric Models, forthcoming.

Barndorff-Nielsen, O. and N. Shephard, 2002d, Econometric analysis of realised

covariation: High frequency covariance, regression and correlation in financial economics, Nuffield Economics Research Working Paper, Oxford University.

Barndorff-Nielsen, O., N. Shephard and C. Ysusi, 2002, Measuring and forecasting

financial variability using realised variance with and without a model, Working Paper, University of Aarhus.

Bartlett, M.S., 1951, An inverse matrix adjustment arising in discriminant analysis,

Annals of the Mathematical Statistics 22, 107-111. Bar-Shalom, Y. and E., Tse, 1974, Dual effect, certainty equivalence and separation in

stochastic control, IEEE Transactions on Automatic Control AC-19, 494-503. Basel Committee on Banking Supervision, 1996, Overview of the amendment to the

capital accord to incorporate market risks, January. Basu, D., 1955, On statistics independent of a complete sufficient statistics, Sankhyā

15, 377-380. Baxter, M. and N. Gale, 1998, Testing for multivariate normality via univariate tests:

A case study using isotope ratio data, Journal of Applied Statistics 25, 671-683. Becker, R. and S. Hurn, 2002, A bootstrap test for uniformity using the Gini

coefficient, Working Paper, Queensland University of Technology. Beckman, R.J. and R.D. Cook, 1979, Testing for two-phase regressions,

Technometrics 21, 65-69. Beirlant, J., D Mason and C. Vynckier, 1999, Goodnes-of-fit analysis for multivariate

normality based on generalized quintiles, Computational Statistics and Data Analysis 30, 119-142.

Bertsekas, D., 1987, Dynamic Programming: Deterministic and Stochastic Models,

Prentice Hall, Upper Saddle River, N.J.

89

Bhattacharya, P.K., 1987, Maximum likelihood estimation of a change-point in the distribution of independent random variables: general multi-parameter case, Journal of Multivariate Analysis 23, 183-208.

Bhatti, M.I. and J. Wang, 1998, Power comparison of some tests for testing change-

point, Statistica, Anno. LVIII, n.1, 127-137. Bierens, H.J., 1984, Model specification testing of time series regressions, Journal of

Econometrics 26, 323-353. Bildik, R., 2000, Intra-day seasonalities on stock returns: Evidence from the Turkish

stock market, Working Paper, http://papers.ssrn.com/ Black, F., 1976, Studies of stock price volatility changes, Proceedings of the 1976

Meetings of the Business and Economics Statistics Section, 177-181. Black, F., and M. Scholes, 1972, The valuation of option contracts and a test of

market efficiency, Journal of Finance 27, 399-418. Black, F., and M. Scholes, 1973, The pricing of options and corporate liabilities,

Journal of Political Economy 81, 637-654. Blume, M. and R. Stambaugh, 1983, Biases in computed returns: An application to

the size effect, Journal of Financial Economics 12, 387-404. Bollerslev, T., 1986, Generalized autoregressive conditional heteroscedastisity,

Journal of Econometrics, 307-327. Bollerslev, T., 1987, A conditional heteroscedastic time series model for speculative

prices and rates of return, Review of Economic Studies 69, 542-547. Bollerslev, T., 1990, Modelling the coherence in short-run nominal exchange rates: A

multivariate generalised ARCH approach, Review of Economics and Statistics 72, 498-505.

Bollerslev, T. and R. Engle, 1993, Common persistence in conditional variances,

Econometrica 61, 167-186. Bollerslev, T., R. Engle, and J. Wooldridge, 1988, A Capital Asset Pricing Model

with time-varying covariances, Journal of Political Economy 96, 116-131. Bollerslev, T. and J. Wooldridge, 1992, Quasi-Maximum Likelihood Estimation and

Inference in Dynamic Models with Time Varying Covariances, Econometric Reviews 11, 143-172.

Brailsford, T. and R. Faff, 1996, An evaluation of volatility forecasting techniques,

Journal of Banking and Finance 20, 419-438.

90

Brown, R.L., J. Durbin and J.M. Evans, 1975, Techniques for testing the constancy of regression relationships over time, Journal of the Royal Statistical Society 37, 149-192 (with discussion).

Brown, S., J.Karpoff and P. Malatesta, 1999, Of Smiles and Smirks: A Term-

Structure Perspective, Working Paper, Stern School of Business, New York University.

Busse, J., 1999, Volatility timing in mutual funds: Evidence from daily returns,

Review of Financial Studies 12, 1009-1041. Box, G. and Jenkins, 1970, Time series analysis, Forecasting and control, San

Francisco: Holden-Day. Box, G. and D. Pierce, 1970, Distribution of residual autocorrelations in

autoregressive-integrated moving average time series models, Journal of the American Statistical Association, vol. 65, December 1970.

Campbell, J., 1987, Stock returns and term structure, Journal of Financial Economics

18, 373-399. Campbell, J., A. Lo and C. MacKinlay, 1997, The econometrics of financial markets,

Princeton University Press, New Jersey. Campbell, J. and P. Perron, 1991, Pitfalls and opportunities: What macroeconomists

should know about unit roots, National Bureau of Economic Research Macroeconomics Annual 6, 141-201.

Chang, E., M. Pinegar and R. Ravichandron, 1993, International evidence on the

robustness of the day-of-the-week effect, Journal of Financial and Quantitative Analysis 28, 497-513.

Chen, J. and A.K Gupta, 1997, Testing and locating variance change-points with

applications to stock prices, Journal of the American Statistical Association 92, 739-747.

Chen, W. and I.S. Reed, 1991, A new CFAR detection test for radar, Digital Signal

Processing, Vol.1, no.4, 198-214. Chinchilli, V., B Schwab and P. Sen, 1989, Inference based on ranks for the multiple-

decision multivariate linear model, Journal of the American Statistical Association 84, 517-524.

Chongfu., H., 1996, Using fuzzy method to study small sample problem of natural

disaster risk assessment, In Proceedings of the 1996 Asian Fuzzy Systems Symposium, 435-440.

Chou, R., R. Engle and A. Kane, 1992, Measuring risk aversion from excess returns

on a stock index, Journal of Econometrics 52, 201-224.

91

Chow, G.C., 1960, Tests of equality between sets of coefficients in two linear regressions, Econometrica 28, 591-605.

Chriss, N. and W. Morokoff, 1999, Volatility and variance swaps, Risk 12, 55-59. Christie, A., 1982, The stochastic behaviour of common stock variances, Journal of

Financial Economics 10, 406-432. Clark, P., 1973, A subordinated stochastic process model with finite variance for

speculative prices, Econometrica 41, 135-156. Christoffersen, P. and F. Diebold, 1998, How relevant is volatility forecasting for

financial risk management?, National Bureau of Economic Research, Working Paper No. 6844.

Christoffersen, P., J. Hahn and A. Inoue, 2001, Testing and comparing Values-at-Risk

measures, Journal of Empirical Finance 8, 325-342. Conover, W. J., 1999, Practical nonparametric statistics, John Wiley & Sons Inc. Cootner, P. (ed.), 1964, The Random Character of Stock Market Prices, MIT Press,

Cambridge, MA. Copeland, T. and D. Galai, 1983, Information effects and the bid-ask spread, Journal

of Finance 38, 1457-1469. Corrado, C. J, and T. W. Miller Jr., 1996, Efficient option-implied volatility

estimators, Journal of Futures Markets 16, 247-272. Cowles, A., 1960, A revision of previous conclusions regarding stock price behaviour,

Econometrica 28, 909-915. Cumby, R., S. Figlewski and J. Hasbrouck, 1993, Forecasting volatility and

correlations with EGARCH models, Journal of Derivatives, Winter, 51-63. Curry, R.E., 1969, A new algorithm for suboptimal stochastic control, IEEE

Transactions on Automatic Control (Short Papers) AC-14, October 1969, 533-536.

Danielsson, J., 1994, Stochastic volatility in asset prices: estimation with simulated

maximum likelihood, Journal of Econometrics 61, 375-400. Danielsson, J. and J.-F. Richard, 1993, Accelerated Gaussian importance sampler with

application to dynamic latent variable models, Journal of Applied Econometrics 8, S153-S174.

Demeterfi, K., E. Derman, M. Kamal and J. Zou, 1999, A guide to volatility and

variance swaps, Journal of Derivatives 6, 9-32.

92

Dickey, D. and W. Fuller, 1979, Distribution of the estimators for autoregressive time series with a unit root, Journal of the American Statistical Association 74, 427-431.

Dickey, D. and W. Fuller, 1981, Likelihood ratio statistics for autoregressive time

series with a unit root, Econometrica 49, 1057-1072. Diebold, F. 1988, Empirical modelling of exchange rate dynamics, Springer Verlag,

New York. Diebold, F. and M. Nerlove, 1989, The dynamics of exchange rates volatility : A

multivariate latent factor ARCH model, Journal of Applied Econometrics 4, 1-21.

Ding, Z. and C. Granger, 1996, Modelling volatility persistence of speculative returns:

A new approach, Journal of Econometrics 73, 185-215. Drenik, R.F. and L. Shaw, 1964, Optimal control of linear plants with random

parameters, IEEE Transactions on Automatic Control AC-9, 236-244. Dreyfus, S., 1964, Some types of optimal control of stochastic systems, SIAM Journal

on Control 2, No. 1, 120-134. Drost, F. and T. Nijman, 1993, Temporal aggregation of GARCH processes,

Econometrica 61, 909-927. Drost, F. and B. Werker, 1996, Closing the GARCH gap: Continuous time GARCH

modelling, Journal of Econometrics 74, 31-57. Duffie, D. and K. Singleton, 1993, Simulated moments estimation of Markov models

of asset prices, Econometrica 61, 929-952. Dufour, J.M., 1982, Generalized Chow tests for structural change: a coordinate-free

approach, International Economic Review 23, 565-575. Dufour, J.-M., 1982, Recursive stability analysis of linear regression relationships: an

exploratory methodology, Journal of Econometrics 19, 31-76. Dufour, J.-M., A. Farhat, L. Gardiol and L. Khalaf, 1998, Simulation-based finite

sample normality test in linear regressions, Econometrics Journal 1, C154-C173.

Ederington, L. and J. Lee, 1993, How markets process information: News realises and

volatility, Journal of Finance 48, 1161-1191. Embrechts, P., S. Klüppelberg and G. Micosch, 1997, Modelling extremal events for

insurance and finance, Springer, Berlin. Engle, R., 1982, Autoregressive conditional heteroscedastisity with estimates of the

variance of U.K. inflation, Econometrica 50, 987-1008.

93

Engle, R., 2000, The econometrics of ultra-high frequency data, Econometrica 68, 1-22.

Engle, R., T. Ito, and W. Lin, 1990, Meteor shower or heat waves? Heterascedastic

intra daily volatility in the foreign exchange market, Econometrica 58, 525-542. Engle, R. and Gonzalez-Rivera, 1991, Semiparametric ARCH models, Journal of

Business and Economic Statistics 9, 345-359. Engle, R. and C. Granger, 1987, Co-integration and error correction: Representation,

estimation, and testing, Econometrica 55, 251-276. Engle, R., and K. Kroner, 1993, Multivariate simultaneous generalised ARCH,

Econometric Theory 11, 122-150. Engle, R., D. Lilien and R. Robins, 1987, Estimating time varying risk premia in the

term structure: The ARCH-M model, Econometrica 55, 391-407. Engle, R and V. Ng, 1993, Measuring and testing the impact of news on volatility,

Journal of Finance 48, 1749-1778. Engle, R. and J. Russell, 1994, Forecasting transaction rates: The autoregressive

conditional duration model, National Bureau of Economic Research, Working Paper No. 4966.

Epps, T. and M. Epps, 1976, The stochastic dependence of security price changes and

transaction volume: Implications for the mixture-of-distribution hypothesis, Econometrica 44, 305-321.

Fama, E., 1963, Mandelbrot and the stable Paretian distribution, Journal of Business

36, 420-429. Fama, E., 1965, The behaviour of stock market prices, Journal of Business 38, 34-

105. Fama, E., 1970, Efficient capital markets: A review of theory and empirical work,

Journal of Finance 25, 383-417. Fama, E., 1997, Market efficiency, long-term returns, and behavioural finance, Social

Science Research Network, papers.ssrn.com/ Farison, J.B., R.E. Graham, and R.C. Shelton, 1967, Identification and control of

linear discrete systems, IEEE Transactions on Automatic Control AC-12, 438-442.

Figlewski, S., 1997, Forecasting volatility, Financial markets, institutions and

instruments, 6, 1-88. Fischer, L., 1966, Some new stock market indices, Journal of Business 39, 191-225.

94

Fischer, N. and P. Hall, 1990, On bootstrap hypothesis testing, Australian Journal of Statistics 32, 177-190.

Fleming, J., Kirby C. and B. Ostdiek, 2001, The economic value of volatility timing

using realised volatility, Journal of Financial Economics, forthcoming. Follmann, D., 1996, A simple multivariate test for one-sided alternatives, Journal of

the American Statistical Association 91, 854-861. Franses, P. and D. van Dijk, 2000, Non-linear time series models in empirical finance,

Cambridge University Press. French, K. and R. Roll, 1986, Stock return variances: The arrival of information and

the reaction of traders, Journal of Financial Economics 17, 5-26. French, K., G.Schwert and R. Stambaugh, 1987, Expected stock returns and volatility,

Journal of Financial Economics 19, 3-29. French, S., 1986, Decision theory: an introduction to the mathematics of rationality,

Chichester, England: Ellis Horwood; New York: Halsted Press. Friede, T., F. Miller, W. Bischoff and M. Kieser, 2001, A not on change-point

estimation in dose-response trials, Computational Statistics and Data Analysis 37, 219-232.

Friedman, M., 1969, The optimum quantity of money, Chicago: Aldine Publishing

Co. Fung, W. and D. Hsieh, 1991, Empirical analysis of implied volatility: Stocks, Bonds

and Currencies, Manuscript, Fuqua School of Business, Duke University. Ganesalingam, S., 1989, Classification and mixture approaches to clustering via

maximum likelihood, Applied Statistics 38 (No.3), 455-466. Garel, B., 2001, Likelihood ratio test for univariate Gaussian mixture, Journal of

Statistical Planing and Inference 96, 325-350. Garman, M. and M. Klass, 1980, On estimation of securities price volatilities, Journal

of Business 56, 67-78. Giannopoulos, K., 2000, Measuring volatility, in Professional’s Handbook of

Financial Risk Management, M. Lore and L. Borodovske eds., 42-74. Gibbons, J.D., 1985, Nonparametric methods for quantitative analysis, 2d. ed.,

American Scientific Press Inc., University of Alabama, Ohio. Ghonem, H. and S. Dore, 1987, Experimental study of the constant-probability crack

growth curves under constant amplitude loading, Engineering Fracture Mechanics 27, 1-25.

95

Glosten, L., R. Jagannathan, and D. Runkle, 1993, On the relation between the expected value and the volatility of the nominal excess return on stocks, Journal of Finance 48, 1779-1801.

Glosten, L. and P. Milgrom, 1985, Bid, ask, and transaction prices in a specialist

market with heterogeneously informed traders, Journal of Financial Economics 14, 71-100.

Goffinet, B., P. Loisel and B. Laurent, 1992, Testing in normal mixture models when

the proportions are known, Biometrika 79, 842-846. Gombay, E. and L. Horvath, 1994, An application of the maximum likelihood test to

the change-point problem, Stochastic Processes and Their Applications 50, 161-171.

Goodhart, C. and M. O’Hara, 1997, High frequency data in financial markets: Issues

and applications, Journal of Empirical Finance 4, 73-114. Gourieroux, C., A. Monfort and E. Renault, 1993, Indirect inference, Journal of

Applied Econometrics 8, S85-S118. Granger, C., 1981, Some properties of time series data and their use in econometric

model specification, Journal of Econometrics, 121-130. Granger, C. and O. Morgenstern, 1970, Predictability of stock market prices,

Lexington, Mass.: D. C. Health and Co. Granger, C. and A. Weiss, 1983, Time series analysis of error-correcting models, in

Studies of Econometrics, Time Series, and Multivariate Statistics, New York: Academic Press, 255-278.

Gray, P., E. Kalotay and J. McIvor, 1998, Testing for multivariate normality of

Australian stock returns, Australian Journal of Management 23, 135-150. Guermat, C. and R. Harris, 2002, Forecasting Value at Risk allowing for time

variations in variance and kurtosis of portfolio returns, International Journal of Forecasting 18, 409-419.

Hamao, Y., R. Masulis and V. Ng, 1990, Correlation in price changes and volatility

across international stock markets, Review of Financial Studies 3, 281-307. Hamilton, J., 1994, Time series analysis, Princeton University Press. Hansen, B., 1994, Autoregressive conditional density estimation, International

Economic Review 35, 705-730. Hansen, L., 1982, Large sample properties of generalised method of moments

estimators, Econometrica 50, 1029-1054.

96

Harris, L., 1986, Transaction data tests of the mixture of distributions hypothesis, Journal of Financial and Quantitative Analysis 21, 39-46.

Harris, L., 1987, Cross-security tests of the mixture of distributions hypothesis,

Journal of Financial and Quantitative Analysis 22, 127-141. Harvey, C., 1984, A unified view of statistical forecasting procedures, Journal of

Forecasting 3, 247-275. Harvey, C., 1989, Time-varying conditional covariances in tests of asset pricing

models, Journal of Financial Economics 24, 289-317. Harvey, A., E. Ruiz and N. Shephard, 1994, Multivariate stochastic variance models,

Review of Economic Studies 61, 247-264. Hasbrouck, J., 1993, Assessing the quality of a security market: A new approach to

transaction-cost measurement, The Review of Financial Studies 6, 191-212. Hawkins, D.M., 1980, A note on continuous and discontinuous segmented

regressions, Technometrics 22, 443-444. Hawkins, I., 2000, Choosing appropriate VaR model parameters and risk-

measurement methods, in Professional’s Handbook of Financial Risk Management, M. Lore and L. Borodovske eds., 111-151.

Heagerty, P. and B. Kurland, 2001, Misspecified maximum likelihood estimates and

generalised liner mixed models, Biometrika 88, 973-985. Henze, N. and T. Wagner, 1997, A new approach to the BHEP tests for multivariate

normality, Journal of Multivariate Analysis 62, 1-23. Hiemstra, C. and J. Jonathan, 1994, Testing for linear and nonlinear granger causality

in the stock price-volume relation, Journal of Financial 49, 1639-1664. Hong, P., 1991, The autocorrelation structure for the GARCH-M process, Economic

Letters 37, 129-132. Howison, S. D., A. Rafailidis and H. O. Rasmussen, 2000, A note on the pricing and

hedging of volatility derivatives, Unpublished paper: Mathematical Institute, University of Oxford.

Hsieh, D., 1989, Modelling heteroscedastisity in daily foreign exchange rates, Journal

of Business and Economic Statistics 7, 307-318. Hsieh, D., 1991, Chaos and nonlinear dynamics: Application to financial markets,

Journal of Finance 46, 1839-1877. Huffer, F. amd C. Park, 2000, A test for multivariate structure, Journal of Applied

Statistics 5, 633-650.

97

Hüsler, G., R. Lui and K. Singh, 2002, A formula for the tail probability of a multivariate normal distribution and its applications, Journal of Multivariate Analysis 82, 422-430.

Hwang, S. and S. Satchel, 2000, Market risk and the concept of fundamental volatility: Measuring volatility across asset and derivative markets and tasting for the impact of derivatives markets on financial markets, Journal of Banking and Finance 24, 759-785.

Jacquier, E., N. Polson and P. Rossi, 1994, Bayesian analysis of stochastic volatility

models, Journal of Business and Economic Statistics 12, 371-417. Jaffe, J. and R. Westerfield, 1985, Patterns in Japanese common stock returns: Day-

of-the-week and turn-of-the-year effects, Journal of Financial and Quantitative Analysis 20, 261-272.

Jorion, P., 1995, Predicting volatility in the foreign exchange market, Journal of

Finance 50, 507-528. Jorion, P., 2001, Value at Risk, McGraw Hill. Ito, T. and W. Lin, 1993, Price volatility and volume spillovers between the Tokyo

and New York stock markets, National Bureau of Economic Research, Working Paper No. 4592.

Kant, I., 1781, Critique of pure reason, Reprinted in 1978, London; New York:

Macmillan. Kato, K., S. Schwartz and W. Ziemba, 1990, Day of the week effects in Japanese

stocks, in Japanese Capital Markets, E. Elton and M. Gruber eds., New York: Ballinger, p. 253.

Keim, D., 1989, Trading patterns, bid-ask spreads, and estimated security returns: The

case of common stocks at calendar turning points, Journal of Financial Economics 25, 75-97.

Kendall, M.G. and A. Stuart, 1969, The Advanced Theory of Statistics, London:

Griffin. King, M.L., 1987, An alternative test for regression coefficient stability, The Review

of Economics and Statistics 69, 379-381. King, M.L., 1996, Hypothesis testing in the presence of nuisance parameters, Journal

of Statistical Planning and Inference 50, 103-120. Koedijk, K., M. Schafgan and C. de Vries, 1990, The tail index of exchange rate

returns, Journal of international Economics 29, 93-108. Kotlarski, I., 1966, On characterizing the normal distribution by Student’s law,

Biometrika 53, 603-606.

98

Kraut, S. and L. Scharf, 1999, The CFAR adaptive sub-space detector is a scale-invariant GLRT, IEEE Transactions on Signal Processing 47, 2538-2541.

Kraut, S., L.L. Scharf and L.T. McWhorter, 2001, Adaptive subspace detectors, IEEE

Transactions on Signal Processing 49, no.1, 1-16. Krishnaian, P.R. and B.Q. Miao, 1988, Review about estimation of change-points,

Handbook of Statistics, vol. 7. (P.R. Krishnaian and C.R. Rao, eds) Elsevier, Amsterdam, 375-402.

Kupiec, P., 1995, Techniques for verifying the accuracy of risk management models,

Derivatives 2, 73-84. Kuruc, A., 2000, Implementation of a Value-at-Risk system, Measuring volatility, in

Professional’s Handbook of Financial Risk Management, M. Lore and L. Borodovske eds., 185-214.

Lam, P. H. and W. H. Tong, 1999, Intra-day volatility in continuos order-driven

markets, Journal of Business Finance and Accounting, September/October, 1013-1036.

Lamoureux, C. and W. Lastrapes, 1990, Heteroscedastisity in stock return data:

Volume versus GARCH effects, Journal of Financial 49, 221-229. Lamoureux, C. and W. Lastrapes, 1994, Endogenous trading volume and momentum

in stock-return volatility, Journal of Business and Economic Statistics 12, 253-260.

Lévy, P., 1925, Calcul des Probabilités, Paris: Gauthier-Villars. Leybourne, S.L. and B.P.M. McCabe, 1989, On the distribution of some test statistics

for coefficient constancy, Biometrika 76, 169-177. Lin, W., R. Engle and T. Ito, 1994, Do bulls and bears move across borders?

International transition of stock returns and volatility as the world turns, National Bureau of Economic Research, Working Paper 3911.

Lintner, J., 1965, The valuation of risk assets and selection of risky investments in

stock portfolios and capital budgets, Review of Economics and Statistics 47, 13-37.

Lo, A. and A. MacKinlay, 1988, Stock market prices do not follow random walk:

Evidence from a simple specification test, Review of Financial Studies 1, 41-66. Lo, A. and A. MacKinlay, 1990, An econometric analysis of nonsynchronous trading,

Journal of Econometrics 45, 181-212. Login, F., 1997a, Beyond the VaR, Technical Report 97-011, CERESSEC.

99

Login, F., 1997b, From value at risk to stress testing, the extreme value approach, Technical Report 97-004, CERESSEC.

Looney, S., 1995, How to use tests for univariate normality to access multivariate

normality, The American Statistician 49, 64-70. Lu, C.J. and W.Q. Meeker, 1993, Using degradation measures to estimate a time-to-

failure distribution, Technometrics 35, 161-174. Lukacs, E., 1960, Characteristic Functions, London: Griffin. Luu, J. and M. Martens, 2001, Testing the mixture of distributions hypothesis using

realised volatility, Working Paper, University of New South Wales. MacKinnon, J., 1991, Critical Values for Cointegration Tests, Chapter 13 in R. F.

Engle and C. W. J. Granger (eds.), Long-run Economic Relationships: Readings in Cointegration, Oxford University Press.

Madhavan, A., M. Richardson and M. Roomans, 1997, Why do securities prices

change? A transaction-level analysis of NYSE stocks, The Review of Financial Studies 10, 1035-1064.

Mallows, C.L., 1973, Some comments on Cp, Technometrics 15, 661-676. Mandelbrot, B., 1963, The variation of certain speculative prices, Journal of Business

36, 394-416. Markowitz, H., 1952, Portfolio selection, Journal of Finance 7, 77-91. Marquardt, D.W., 1963, An algorithm for least squared estimation of non-linear

parameters, Journal of the Society for Industrial and Applied Mathematics 11, 431-441.

Meier, L., J. Perschon, and R.M. Dressler, 1967, Optimal control of measurement

subsystems, IEEE Transactions on Automatic Control AC-12, 528-536. Melino, A. and S. Turnbull, 1990, Pricing foreign currency options with stochastic

volatility, Journal of Econometrics 45, 239-265. Mendelbort, B., 1963a, New methods in statistical economics, Journal of Political

Economy 71, 421-440. Mendelbort, B., 1963b, The variation of certain speculative prices, Journal of

Business 39, 394-419. Mendell, N., H. Thode and S. Finch, 1991, The likelihood ratio test for the two-

component normal mixture problem: Power and sample size analysis, Biometrics 47, 1143-1148.

100

Mendell, N., S. Finch and H. Thode, 1993, Where is the likelihood test powerful for detecting two component normal mixtures? Biometrics 49, 907-915.

Menegetti, A., 1997, A course of ontopsychology (Учебник по онтопсихологии),

Slovenian Association of Ontopsychology, Moscow. Merton, R., 1973, An intertemporal asset pricing model, Econometrica 41, 867-887. Merton, R., 1980, On estimating the expected return on the market: An explanatory

investigation, Journal of Financial Economics 8, 323-361. Miller, L., 1956, Table of percentage points of Kolmogorov statistics, Journal of the

American Statistical Association 51, 111-121. Moldovan, M., 2001, Conditional against realised volatility: Which ARCH model fits

better?, Economic and Financial Modelling Term Paper, Queensland University of Technology.

Moldovan, M., 2002a, Forecasting portfolio Value at Risk via Monte Carlo

simulation, Risk Management Project, Queensland University of Technology. Moldovan, M., 2002b, Test for double normal mixture with unequal unknown

parameters, Econometric Project, Queensland University of Technology. Moldovan, M., N. Nechval, E. Vasermanis, U. Rozevskis & K. Rozite, 2002, Testing

for two-phase multiple regressions, in Proceedings of the International Conference on Reliability and Statistics in Transportation and Communication (RelStat’02).

Moldovan, M., K. Nechval, N. Nechval, E. Vasermanis and V. Strelchonok, 2003,

UMPI Test for Identifying an Observable Process with One of Several Simulation Models, Join International Meeting “New Opportunities For Operations Research”, Istanbul, Turkey, July 6-10.

Moran, P.A.P., 1951, A characteristic property of the Poisson distribution,

Proceedings of the Cambridge Philosophical Society 48, 206-207. Morgan, I., 1976, Stock prices and heteroscedasticity, Journal of Business 49, 496-

508. Müller, U., M. Dacorogna, R. Davé, O. Pictet, R. Olsen, R. Ward, 1993, Fractal and

intrinsic time: A challenge to econometricians, Working Paper, Olsen and Associates, Research Institute for Applied Economics, Zürich, Switzerland.

Müller, U., M. Dacorogna, R. Olsen, O. Pictet, M. Schwarz and C. Morgenegg, 1990,

Statistical study of foreign exchange rates, empirical evidence of a price change scaling law, and intraday analysis, Journal of Banking and Finance 14, 1189-1208.

101

Naiman, D., 1987, Characterization of risk sets for simple versus simple hypothesis testing, The American Statistician 41, 218-220.

Nechval, N., K. Nechval and E. Vesermanis, 2001, Change point detection in

regression relationship, Managemnt of Organizations: Systematic Research 19, 81-88.

Nechval, N., M. Moldovan, K. Nechval and E. Vesermanis, 2003, Airline Seat

Inventory Control Under Uncertainty, Join International Meeting “New Opportunities For Operations Research”, Istanbul, Turkey, July 6-10.

Nechval, N., K. Nechval, E. Vesermanis and M. Moldovan, 2002, Optimization of

Airline Seat Inventory Control System for a Single Leg Route, in Proceedings of The 6th World Conference on Systematics, Cybernetics and Informatics, Orlando, Florida, USA.

Nelson, D., 1990a, ARCH models as diffusion approximation, Journal of

Econometrics 45, 7-38. Nelson, D., 1990b, Stationarity and persistence in the GARCH(1,1) model,

Econometric Theory 6, 318-334. Nelson, D., 1991, Conditional heteroscedastisity in asset returns: A new approach,

Econometrica 59, 347-370. Nyblom, J., 1989, Testing for constancy of parameter over time, Journal of the

American Statistical Association 84, 223-230. O’Hara, M., 1995, Market microstructure theory, Blackwell Publisher, Cambridge,

MA. Officer, R., 1973, The variability of the market factor of the New York Stock

Exchange, Journal of Business 46, 434-453. Orme, C., 1990, The small sample performance of the information matrix test,

Journal of Econometrics 46, 309-331. Osborne, M., 1959, Brownian motion in the stock market, Operations Research 7,

145-173. Osborne, M., 1962, Periodic structure of the Brownian motion of stock structure,

Operations Research 10, 345-379. Pagan, A., 1996, The econometrics of financial markets, Journal of Empirical

Finance 3, 15-102. Pagan, A. and G. Schwert, 1990, Alternative models for conditional stock volatility,

Journal of Econometrics 45, 267-290.

102

Parkinson, M., 1980, The extreme value method for estimating the variance of the rate of return, Journal of Business 53, 61-65.

Pascual, F.G. and W.Q. Meeker, 1999, Estimating Fatigue Curves With the Random

Fatigue-Limit Model, Technometrics 41, 277-302. Pasquariello, P., 2000, The microstructure of currency markets: an empirical model of

intra-day return and bid-ask spread behaviour, Working Paper, New York University.

Patil, G.P. and V. Seshadri, 1964, Characterization theorems for some univariate

probability distributions, Journal of the Royal Statistical Society B26, 2, 286-292.

Pearson, K., 1900, On the criterion that a given system of deviations from the

probable in the case of correlated system of variables in such that it can reasonably be supposed to have arisen from random sampling, Philosophical Magazine 50 (5), 157-175 (reprinted 1948 in Karl Pearson’s Early Statistical Papers, ed. By E. S. Pearson, Cambridge: Cambridge University Press, 4.5, 4.7).

Phillips, P.C.B., 1983, ERA’s: a new approach to small sample theory, Econometrica

51, 1505-1527. Pidyck, R., 1984, Risk, inflation, and the stock market, American Economic Review

73, 335-351. Pilla, R. and B. Lindsay, 2001, Alternative EM methods for nonparametrc finite

mixture models, Biometrika 88, 525-550. Plackett, R.L., 1950, Some theorems in least squares, Biometrika 37, 149-157. Popper, K., 1972, The logic of scientific discovery, London: Hutchinson. Popper, K., 1982, The open universe, Hutchinson & Co. (Publishers) Ltd. Press, S.J., 1989, Bayesian statistics: Principles, models, and applications, John Wiley

& Sons. Press, W., S. Teukolsky, W. Vetterling and B. Flannery, 1992, Numerical recipes in

Fortran, Cambridge University Press. Prohorov, Yu.V., 1966, Some characterization problems in statistics, Proceedings of

the Fifth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1 (Berkeley: University of California Press), 341-349.

Raff, S.M., 1956, On approximating the point binomial, Journal of the American

Statistical Association 51, 293-303.

103

Reed, I.S. and X. Yu, 1990, Adaptive multiple-band CFAR detection of an optical pattern with unknown spectral distribution, IEEE Transactions on Acoustics, Speech, and Signal Processing 38, 1760-1770.

Reiss, R., and M. Thomas, 1997, Statistical analysis of extreme events, Birkhausel,

Basel. Richardson, M., and T. Smith,1993, Tests for multivariate normality of stock returns,

Journal of Business 66, 295-321. Ripley, B. D., 1996, Pattern recognition and neural networks, Cambridge University

Press. Robinson, M. E. and M.J. Crowder, 1999, Bayesian methods for a growth-curve

degradation model with repeated measures, Lifetime Data Analysis 6, 357-374. Rocha, M.M. and G.I. Schueller, 1996, A probabilistic criterion for evaluating the

goodness of fatigue crack growth models, Engineering Fracture Mechanics 53, 701-731.

Rogers, L. and S. Satchell, 1991, Estimating variance from high, low and closing

prices, Annals of Applied Probability 1, 504-512. Rogers, L., S. Satchell and Y. Yoon, 1994, Estimating the volatility of stock prices: A

comparison of methods that use high and low prices, Applied Financial Economics 4, 241-247.

Ross, S. M., 1996, Stochastic processes, 2nd ed., John Wiley & Sons. Rossi, P. E. (ed.), 1996, Modelling stock market volatility, Academic Press. Quandt, R.E., 1958, The estimation of the parameters of a linear regression system

obeying two separate regimes, Journal of the American Statistical Association 53, 873-880.

Quandt, R.E., 1960, Test of the hypothesis that a linear regression obeys two separate

regimes, Journal of the American Statistical Association 55, 324-330. Samuelson, P., 1965a, Proof that properly anticipated prices fluctuate randomly,

Industrial Management Review 6, 41-49. Samuelson, P., 1965b, Rational theory of warrant pricing, Industrial Management

Review 6, 13-31. Samuelson, P. 1973, Proof that properly discounted present values of assets vibrate

randomly, Bell Journal of Economics and Management Science 4, 369-374. Schilling, M., 1986, Multivariate two-sample tests based on nearest neighbours, The

Journal of the American Statistical Association 89, 799-806.

104

Scholes, M. and J. Williams, 1977, Estimating betas from non-synchronous data, Journal of Financial Economics 5, 309-327.

Schwert, G., 1989, Why Does Stock Market Volatility Change Over Time?, Journal

of Finance 44, 1115-1153. Schwert, G., 1990, Stock volatility and the crash of '87, The Review of Financial

Studies 3, 77-102. Schwert, G., 1998, Stock Market Volatility: Ten Years After the Crash, National

Bureau of Economic Research, Working Paper No. W6381. Scruggs, J., 1998, Resolving the puzzling intertemporal relation between the market

risk premium and conditional market variance: A two-factor approach, Journal of Finance, 576-603.

Shanmugam, R. and J. Singh, 1984, A characterization of the logarithmic series

distribution and its application, Communications in Statistics - Theory and Methods 13, 865-875.

Shannon, C., 1993, Claude Elwood Shannon: collected papers, edited by N.J.A.

Sloane and Aaron D. Wyner, Picataway, NJ: IEEE Press. Sharpe, W., 1964, Capital asset prices: A theory of market equilibrium under

conditions of risk, Journal of Finance, 425-442. Shiller, R., 1989, Market volatility, London: MIT Press. Skiadopoulos, G., S. Hodges and L. Clewlow, 1998, The dynamics of smiles,

Working Paper, University of Warwick. Smirnov, N., 1948, Table for estimating goodness of fit of empirical distributions,

Annals of Mathematical Statistics 19, 279-281. Sofianos, G., 1992, Expirations and stock price volatility, NYSE Working Paper 92-

04. Spang, H.A., 1965, Optimum control of an unknown linear plant using Bayesian

estimation of the error, IEEE Transactions on Automatic Control (Short Papers) AC-10, 80-83.

Springer, M. D., 1979, The algebra of random variables, John Wiley & Sons. Stephens, M., 1970, Use of the Kolmogorov-Smirnov, Kramer-von Mises and related

statistics without extensive tables, Journal of the Royal Statistical Society 32, 115-122.

Stock, J., 1988, Estimating continuous-time processes subject to time deformation,

Journal of the American Statistical Association 83, 77-85.

105

Streibel, C., 1965, Sufficient statistics in the optimum control of stochastic systems, Journal of Mathematical Analysis and Applications 12, No. 3, 576-592.

Tauchen, G. and M. Pitts, 1983, The price variability-volume relationship on

speculative markets, Econometrica 51,485-505. Taylor, H. and S. Karlin, 1998, An introduction to stochastic modelling, 3 ed.,

Academic Press. Taylor, S., 1986, Modelling financial time series, John Wiley: Chichester. Teräsvirta, T., 1970, On Stepwise Regression and Economic Forecasting (Economic

Studies XXXI, Helsinki: The Finnish Economic Association). Thode, H., S. Finch and N. Mendell, 1988, Simulated percentage points for the null

distribution of the likelihood ratio test for a mixture of two normals, Biometrics 44, 1195-1201.

Tijms, H.C., 1994, Stochastic models: An algorithmic approach, JOHN WILEY &

SONS. Tse, E. and M. Athans, 1972, Adaptive stochastic control for a class of linear systems,

IEEE Transactions on Automatic Control AC-17, 38-52. Waldemark, J., M. Millberg, T. Lindblad and K. Waldemark, 2000, Image analysis for

airborne reconnaissance and missile application, Pattern Recognition Letters 21, 239-251.

Wilson, A.L., 1978, When is the Chow test UMP?, The American Statistician 32, 60-

68. Wilson, S. P. and D. Taylor, 1997, Reliability assessment from fatigue microcrack

data, IEEE Transactions on Reliability 46, 165-172. Whitelaw, R., 1994, Time variations and covariations in the expectation and volatility

of stock market returns, Journal of Finance, 515-541. Woodward, W.A. and H.L. Gray, 1975, Minimum variance unbiased estimation in the

gamma distribution, Communications in Statistics 4, 907-922. Worsley, K.J., 1983, Testing for two phase multiple regression, Technometrics 25, 35-

42. Worsley, K.J., 1986, Confidence regions and tests for change-point in a sequence of

exponential family random variables, Biometrika 73, 91-104. Worsley, K.J., 1986, Confidence regions and tests for change-point in a sequence of

exponential family random variables, Biometrika 73, 91-104.

106

Xu, K., 2000, Inference for generalised Gini indices using the interated-bootstrap method, Journal of Business and Economics Statistics 18, 223-227.

Yang, D. and Q. Zhang, 2000, Drift independent volatility estimation based on high,

low, open and close prices, Journal of Business 73, 477-491. Yang, Z.-H., K.-T. Fang and J.-J. Liang, 1996, A characterisation of multivariate

normal distribution and its application, Statistics and Probability Letters 30, 347-352.

Yar, M. and C. Chatfield, 1990, Prediction intervals for the Holt-Winters forecasting

procedure, International Journal of Forecasting 6, 127-137. Yu, X. and I.S. Reed, 1993, Comparative performance analysis of adaptive multiband

detectors, IEEE Transactions on Signal Processing 41, 2639-2656. Yungtai, L., N. Mendell and D. Rubin, 2001, Testing the number of components in a

normal mixture, Biometrika 88, 767-778. Zadicario, J. and R. Sivan, 1966, The optimal control of linear systems with unknown

parameters, IEEE Transactions on Automatic Control AC-11, 423-426. Zou, F., J. Fine and B. Yandell, 2002, On empirical likelihood for a simiparametric

mixture model, Biometrika 89, 61-75.