Quality Control for Financial Risk

Management: Monitoring Disruptions

in the Distribution of Risk Exposure∗

Elena Andreou

Department of Economics,

University of Cyprus, and

CentER, Tilburg University.

Eric Ghysels

Department of Economics,

University of North Carolina.

September 24, 2002

First Draft: September 2, 2002

∗We would like to thank Bas Werker for insightful comments. The first author wouldlike to acknowledge the financial support of a Marie Curie Individual Fellowship (MCFI-2001-01645). This paper subsumes part of the material presented in a 1999 working papertitled “Testing for Disorders in Asset Returns and Volatility.”

ABSTRACT

Extreme tail observations and structural breaks are two types of rare events. The

former is one of the main focuses of financial risk management, the latter represents a

fundamental shift in the distribution of risky outcomes. Structural changes may be due

to deregulation (e.g. the electricity market) or other policy shifts (e.g. introduction of

the Euro currency). They may also be due to mergers of firms or other events. The pre-

vailing underlying assumption for the majority of risk management statistical tools, such

as VaR and other procedures, is the distributional time-homogeneity of the underlying

financial returns process. Failure to recognize the presence of structural breaks, confound

them with tail observations and facing heterogenous distributions as a result of breaks,

can have devastating and costly effects on financial risk management. The paper proposes

several procedures, known as change-point tests, that have their roots in the quality con-

trol literature. The change-point tests we consider apply to the conditional distribution

of return and pertain to shifts in conditional distribution functions or quantiles, i.e. the

objects that are important in financial risk management. We discuss the theory providing

certain extensions that are of practical use for monitoring the distribution of financial risk

exposure and show via simulation the performance of the procedures in standard sample

sizes encountered in the practice of financial risk management. The paper concludes with

empirical illustrations.

Key words: Change-point tests, empirical process, expected loss, financial market volatil-

ity, high-frequency data, GARCH, jump process, Levy process, location-scale distribution

family, quantile process, rank tests, Monte Carlo simulations, sequential EDF tests, sto-

chastic volatility, volatility estimators, Value at Risk.

1 Introduction

Extreme tail observations and structural breaks are two types of rare events.The former is one of the main focuses of financial risk management, thelatter represents a fundamental shift in the distribution of risky outcomes.The prevailing underlying assumption of the majority of risk managementstatistical tools is the distributional time-homogeneity of the underlying fi-nancial returns process. Prominent examples are Expected Shortfall (hence-forth ES) and Value-at-Risk (henceforth VaR), the latter being the mostwidely applied tool used by risk managers and financial institution regula-tors in dealing with market risk (see, for instance, Dowd (1998) and Jorion(1995) interalia). VaR attempts to forecast likely future losses based on thequantiles of the portfolio returns distribution evaluated using historical dataand ES (or tail conditional expectation) measures the expected loss givenit exceeds some threshold. The underlying assumption is that the mecha-nism generating portfolio returns remains time invariant, which is crucial forthe estimation, simulation and prediction of the tails of a distribution thatmay belong to a large class of leptokurtic and/or asymmetric distributions(ranging from Stable Paretian, Extreme Value, Elliptical, Beta and PowerExponential distributions, as suggested by the empirical literature). The as-sumption of homogeneity is, however, challenged by the presence of economicand institutional structural changes For instance, institutional changes suchas industry deregulations (e.g. the energy sector in the US and other coun-tries), mergers and acquisitions of companies or economic policy changes suchas monetary and exchange rate regime changes (e.g. the recent Euro regime),are events that may cause permanent change-points in the market or portfo-lio returns processes. Failure to recognize the presence of structural breaks,confound them with tail observations and facing heterogenous distributionsas a result of breaks, can have devastating and costly effects on financial riskmanagement with VaR, ES or any other tools.

The paper proposes several procedures, known as change-point tests, thathave their roots in the quality control literature. The change-point testswe consider pertain to shifts in distribution functions or quantiles, i. e. theobjects that are important in financial risk management. The literatureon change-point tests is considerable and goes back to at least Shiryaev(1963) who also proposes the disorders for change-points in technologicalprocesses. Hence a similar term namely disruptions is also adopted here toreflect change-points in financial asset returns processes. The application of

1

distributional change-point tests in financial risk applications is not straight-forward since the usual setup assumes an independent process whereas finan-cial returns are modelled by strongly nonlinear processes.

The main thrust of our procedures is to examine distributional shifts inde-volatilized returns defined as: Xt ≡ (rt − µ

t)/σt, where rt is the return

on an asset (or a portfolio of assets) and σt, is some estimator of condi-tional volatility and µ

ta conditional mean estimator.1 At this point, we

deliberately keep vague the specifics of the estimators except to say thatwe consider various data-driven and model-based estimators for conditionalvolatility and conditional means. Examining de-volatilized returns has theadvantage that we examine the conditional distribution of returns, which istypically the object of interest in risk management. Our work is in this re-spect very distinct from recent work, notably by Quintos et al. (2001), whoexamine the asset returns process rt directly. Testing for breaks in the tailbehavior of actual returns precludes one from analyzing the conditional dis-tribution directly and may confound changes in volatility dynamics and thetail behavior of ˆXt. We apply Kolmogorov-Smirnov and Cramer-von Misestype tests (e.g. Picard, 1985, Carlstein, 1988, Bai, 1994, Koul, 1996, In-oue, 2001, Horvath et al, 2001) applied to ˆXt. The finite sample propertiesof these asymptotic tests are evaluated via a Monte Carlo analysis (whichare also useful for the small sample requirements of the 1998 Basle Accordregulators). Nonparametric methods and in particular rank-based statisticsare also proposed. Rank-based methods are known to be robust to outliersthat have a transitory effect on the tails of the distribution. This feature isparticularly attractive since we would expect the source of rejections of thehomogeneity hypothesis to be fundamental change-points in the probabilitylaw and not isolated jumps (or extreme events). We adapt the Bhattacharyaand Frierson (1981) and Gordon and Pollak (1994, 1995) tests developedfor independent processes to transformations of the returns series that yieldindependence. The rank-based test do not apply under all circumstances,however, as will be explained.

The methods proposed are based on both a posteriori and sequential,parametric and nonparametric change-point procedures as well as dynamicsampling methods. A posteriori or ‘off-line’ procedures are useful for the de-cision about stochastic homogeneity of the random sequence of returns (i.e.

1The term de-volatilization is due to Zhou (1996). Henceforth we will interchangeablythe terms de-volatilized returns or standardized returns when refering to the ˆXt process.

2

the absence of change) made after observing a sample of a fixed length. Thishelps establishing a long historical sample necessary for the estimation of thedistribution and its tails. Once an invariant historical sample is established,sequential change-points are applied according to which the decision is made‘on line’ with the arrival of each observation. The sequential character ofthe procedures is particularly useful for real time monitoring of the statisticthat assesses the homogeneity hypothesis of VaR estimations on a daily basisas reported by financial institutions internally to the risk manager and ex-ternally to financial industry regulators. Moreover, the sequential statisticsare useful for the quality control of VaR forecasts (e.g. Kupiec, 1995, asrecommended by the Basle, 1998, among others).

The paper is organized as follows: Section 2 discusses the scope of testingfor breaks in the context of financial risk management and reviews the variouschange-point in distribution tests. The next section 3 covers both data-drivenand model-based de-volatilization. The next two sections, 4 and 5, deal withsequential empirical density function based, denoted EDF-based, and rank-based change-point tests, respectively. So far the test being discussed pertainto tests for breaks covering the entire distribution function (henceforth DF).The next section 6 suggests procedures that focus specifically on the quantilesand on the tails, both being objects of particular interest in a risk manage-ment context. The Monte Carlo design and results are presented in section 7followed by the empirical illustration in section 8. The last section concludesthe paper.

2 Financial Risk Management and Breaks

Has the introduction of a single European currency changed the exchangerate risk of small EU countries or more generally the risk of European finan-cial assets? A priori the answer may not be so obvious, since some smallEU countries historically tied their currency to say the German DM. Henceshould a foreign stockholder worry about fundamental shifts in exchange raterisk when holding European stocks? A similar example for corporate finan-cial assets arises when a manager holding Chrysler and Daimler stocks needsto assess the relevant risks before and after the merger of the two companies.In such a case, the manager is faced with the question: Has the cross-Atlanticmerger fundamentally altered the riskiness of the assets? Another example isthe deregulation of the electricity supply markets in the US and other coun-

3

tries, including the spectacular price hikes in California and other states.Most likely we do not need any statistical procedure to come to the conclu-sion that risks in energy markets changed completely. More often than not,however, as the first two examples illustrate, it is not clear whether thereare fundamental shifts and the need for change-point analysis, or testing forstructural breaks, in financial risk management are certainly clear.

From the statistical point of view one needs a large historical sample ofasset returns for the estimation of the distribution and its moments, particu-larly its conditional variance and tails. A large historical sample may provideestimation and testing precision gains, yet it also possesses the danger of thepresence of fundamental structural changes during that period (such as eco-nomic, policy or institutional structural breaks mentioned above). Indeed thefinancial industry regulators (in the 1998 Basle Accord) are particularly sen-sitive to market ‘disruptions’ and suggest the use of only the most recent 250days historical sample for variance-covariance methods of VaR calculation.Although this is an attempt by regulators to avoid contaminating sampleswith rare or extreme events, this does guarantee the absence of structuralbreaks in the returns process. Moreover, discarding the long sample historywill not only typically yield less precision, especially for tail estimation, butalso defy the purpose of an institutions’ prudent VaR and capital allocationduring periods of market ‘disruptions’. The large sample is also necessary forthe asymptotic approximation of most methods. It is interesting that in ananalogous manner a large historical sample is required for the estimation ofthe equity premium (one of the most important quantities in finance) recentlyfound to be subject to multiple structural breaks (Pastor and Stambaugh,2001) which are advocated as an explanation for the high equity premiumpuzzle (see for instance, Mehra and Prescott, 1985). Other examples arethe existence of volatility persistence and long memory that is sometimesspuriously overstated if structural breaks are ignored (e.g. Lamoureux andLastrapes, 1990, Diebold and Inoue, 2001). Similarly for the estimation ofVaR we first need to establish homogeneity of a long historical sample.

The main thrust of our procedures is to examine distributional shiftsin de-volatilized returns defined as: (rt − µ

t)/σt, where rt is the return on

an asset (or a portfolio of assets) and σt, is some estimator of conditionalvolatility and µ

tthe conditional mean estimator. In most of our analysis we

will almost always suppress the conditional mean and simply define Xt ≡

rt/σt. However, when parametric models are used it is As we noted before,the Xt process pertains to the conditional distribution of returns, which

4

is typically the object of interest in risk management. In this respect ourwork is very distinct from recent work, notably by Quintos et al. (2001)who examine the asset returns process rt directly. Testing for changes inbreaks in the tail behavior of actual returns precludes one from analyzingthe conditional distribution directly and may confound changes in volatilitydynamics and the tail behavior of ˆXt.

The choice of de-volatilized or standardized returns as an object of interestis motivated by both finance and statistics arguments. From a finance pointof view there are several arguments. First VaR and other tools typically applyto the location-scale family of distributions, for which the de-volatilizationtransformation is entirely natural (more on this in section 6). Second, the ˆXt

process represents the fundamental measure of reward-to-risk consistent withthe Mean-Variance theorem (Markowitz, 1952) which can be considered thebackbone of many asset pricing models. At long and intermediate horizons(say monthly and beyond) the risk adjusted returns are known as the marketprice of risk, when it involves returns on the market portfolio, or as the Sharperatio (Sharpe, 1975, 1994) when it involves excess returns. Finally, by directlyfocusing on the ratio of two components we also immunize our procedures toa large extend to changes in the volatility dynamics that may not affect theconditional distribution relevant for risk management purposes.2

We consider ˆXt as a time series process sampled at daily, monthly or anyother frequency. In the context of risk management this typically involvesdaily observations, in portfolio applications the data are usually sampledmonthly. Alternatively, we can also examine the standardized returns froma cross-section point of view (keeping subscripts t for convenience). In eitherthe time series or cross-section context we assume that ˆXt is drawn from adistribution Ft for t = 1, ..., T . The null hypothesis of interest is F1 = F2 =

... = FT . When Xt is fat-tailed i.i.d. we can apply for instance nonparametric

rank-based tests. In a time series context we apply sequential EDF- or rank-

based tests for the distributional invariance of risk adjusted returns, discussed

later. Rejecting the null hypothesis implies that there was a change in the

distribution of risk adjusted returns at data-determined points of the sample.3

2From the statistics point of view testing for breaks in conditional volatility models hasnot been an easy task because it is a highly persistent, possibly non-ergodic process andnon-Markovian. See Lee and Hansen (1994), Chu (1995), de Lima (1998) and Andreouand Ghysels (2002a) for further discussion. The latter show that the standardized returnprocess is largely immune to changes in the volatility dynamics.

3When ˆXt represents a cross-section of T portfolios, then other tests for distributional

5

The methods proposed are based on both a posteriori and sequential,parametric and nonparametric change-point procedures as well as dynamicsampling methods. A posteriori or ‘off-line’ procedures are useful for thedecision about stochastic homogeneity of the random sequence of returns (i.e.the absence of change) made after observing a sample of a fixed length. Thisis useful for establishing a long historical sample necessary for the estimationof the distribution and its tails for VaR. Once an invariant historical sampleis established, sequential change-points are applied according to which thedecision is made ‘on line’ with the arrival of each observation. The sequentialcharacter of the procedures is particularly useful for real time monitoring ofthe statistic that assesses the homogeneity hypothesis of VaR estimations on adaily basis as reported by financial institutions internally to the risk managerand externally to financial industry regulators. Moreover, the sequentialstatistics are useful for the quality control of VaR forecasts (e.g. Kupiec, 1995,as recommended by the Basle, 1998, among others). The use of warning linesin statistics (see for instance, Lai (1995)) is also a useful tool for practitionersin order to trace the trend of the risk exposure statistic and to timely adjustportfolios and capital according to the institution’s internal risk attitudes.Examples of these methods are provided and interpreted in terms of differentmonetary losses and risk exposures with the corresponding graphical toolsdepicting control limits and warning lines that can be easily adopted byrisk managers. A complementary useful strategy is the dynamic samplingscheme (e.g. Assaf et al., (1992)) according to which once a change-pointis detected the sampling frequency rate increases in an attempt to enlargethe post-break sample size. This approach is well suited in financial marketsgiven the high frequency data arrival (with negligible measurement error)and the need to re-estimate the distribution or the tail index that requirea relative large sample size after a change-point. Hence we suggest optimalintraday sampling frequencies for monitoring risk statistics when these passthe warning lines and control limit.

homogeneity, such as the Wilcoxon rank test, can be considered. They are a nonparametricversion of the standard Sharpe ratio tests (see Jobson and Korkie, 1981). In the remainderof the paper we will focus exclusively on time series applications.

6

3 Data-driven and Model-based

De-volatilization

Financial markets volatility is time-varying and has predictable patternsacross time and different assets due to persistence and co-movements. Thesubject is an extremely active area of research and there are a staggeringnumber of papers written on volatility modeling. The most commonly usedmodels fall either in one of two classes: the ARCH class or the SV class.The best references one can provide are surveys such as Bollerslev, Chou andKroner (1992), Bollerslev, Engle and Nelson (1994), for ARCH models andGhysels, Harvey and Renault (1996), Shephard (1996), for SV models. An-dersen, Bollerslev and Diebold (2002) discuss data-driven estimators whichhave roots in the SV as well as ARCH literature. Finally, surveys on optionpricing, Bates (1996b) and Garcia, Ghysels and Renault (2002) also deal withseveral aspects of volatility modeling.

Given an asset return process rt we describe several ways to obtain thede-volatilized returns defined as: ˆXt ≡ rt/σt, where σt, is some estimatorof conditional volatility (ignoring the presence of a conditional mean). Thedistinction between Xt ≡ rt/σt, and ˆXt ≡ rt/σt, is obviously important.For the parametric class of ARCH models σt(b) is parameterized by someparameter vector b and it is common to assume that Xt ≡ rt/σt(b) is i.i.d.However, since we have to estimate b we need to examine the estimatedresiduals of the ARCH process, i. e. we need to examine Xt where b is replacedby b to compute σt. Likewise, with data-driven estimators we will rely onincrements of quadratic variation (discussed in detail later). Here again, weface the fact that the quadratic variation needs to be estimated since it isnot directly observed.

In this section we will seek to establish that Xt satisfies either one of thefollowing two assumptions:

Assumption 3.1 The process ˆXt is weakly dependent.

This assumption will be the most commonly used in our analysis and wewill show that for a large class of continuous time and discrete time volatil-ity models we can construct a process ˆXt that will satisfy this assumption.Broadly speaking we divide the subject into two main categories: (1) purelydata-driven measures of volatility and (2) parametric model-based volatilityestimators. A subsection is devoted to each of the two categories. Before we

7

do, it should also noted that sometimes a more restrictive assumption willbe made, namely:

Assumption 3.2 The process ˆXt is i.i.d.

Assumption 3.2 is necessary when we use rank-based statistics. The

sequential EDF-based change-point tests, apply to processes satisfying As-

sumptions 3.1. Rank-based change-point tests, appearing in section 5, apply

to processes satisfying the more stringent second assumption. In some special

cases we can establish that the process is i.i.d.

3.1 Data-Driven De-volatilization

It will be convenient to start with a continuous time setting to discuss data-

driven de-volatilization. Consider the logarithmic price process defined on

a complete probability space (Ω,F , P ) over some finite interval [0, T ]. Thelogarithmic price process is assumed to be a semi-martingale and let rt be the

continuously compounded return over the interval [0, t] defined as the càdlàg

process rt ≡ pt+ - p0.4 Moreover, consider the associated natural filtration

(Fτ τ∈[0,T ]) and the so called Doob-Meyer decomposition of returns rt :

rt = µt +Mt ≡ µt+M c

t+M J

t(3.1)

where µtis a predictable and finite variation process, M c

tis a continuous sam-

ple path infinite variation local martingale and MJ

tis a compensated jump

process, both add up infinite variation local martingale Mt. The processesthat can be represented as in (3.1) include continuous time stochastic volatil-ity processes with or without a jump component. Such type of processes arecommonly used for asset returns, see e.g. considered recently by

A process central to many recent developments in data-driven volatilityestimation is the quadratic variation defined as [M,M ]t ≡M 2

t−2

∫t

0Ms−dMs

for 0 < t < T.5 Recently, various authors, including Andersen, Bollerslev,

Diebold, and Labys (2001) Andersen, Bollerslev and Diebold (2002) andBarndorff-Nielsen and Shephard (2001, 2002a,b), have advocated to use in-crements in the quadratic variation over discrete intervals as a measure of

4Henceforth we will simplify the notation t+ to t.5The quadratic variation is directly defined on Mt since the predictible component with

finite variation has the property [µ,µ]t = 0 ∀t.

8

asset return volatility. A natural estimator of the increments in the quadraticvariation is to compute sums of squared financial returns over some discreteintervals (to be made more precise shortly). Such estimators have been usedquite extensively since many years, see for example French, Schwert andStambaugh (1987), Poterba and Summers (1987), Schwert (1989) and therecent flurry of papers making use of high-frequency intra-daily data, includ-ing Andersen and Bollerslev (1998), Andersen et al. (2001a,b), Andersen etal. (2002), Andreou and Ghysels (2002b), Barndorff-Nielsen and Shephard(2001, 2002a,b, 2003), Taylor and Xu (1997), among others.

We will denote increments in the quadratic variation as:

Qt ≡ [M,M ]t − [M,M ]t−1 ≡M 2

t−M 2

t−1− 2

∫t

t−1

Ms−dMs

and consider r(m),t ≡ pt − pt−1/m, the discretely observed time series of con-

tinuously compounded returns with m observations per day. Hence, the

unit interval for r(1),t is assumed to yield the daily return. To estimate the

quadratic variation we sum intra-daily squared returns over a day, namely:

ˆQ(m),t =

m−1∑

j=0

r2(m),t−j/m (3.2)

where ˆQ(m),t is an estimator of Qt. Andersen and Bollerslev (1998) andBarndorff-Nielsen and Shephard (2001) noted that the theory of quadraticvariation (e.g. Jacod and Shiryaev (1987)) implies that ˆQ(m),t is a consistentestimator of Qt as m → ∞. Hence, the asymptotics pertains to incrementsof the quadratic variation over a fixed time interval, with increasingly finerpartitions to compute high frequency squared returns r2(m),t−j/m.

We are interested in the properties of ˆXt ≡ r(1),t/(Q(m),t)1/2 and function-

als thereoff such as r2(1),t/Q(m),t. So far, we refrained from writing down anexplicit law of motion for the continuous time process of prices or returns.To proceed let us therefore define:

drt = µ(t)dt+ σ(t)dWt + JtdNt (3.3)

where the function µ is the drift, σ2 is the diffusion coefficient and Wt isa standard Brownian motion whereas N is a Poisson process with intensityλ(t), and jump size J has distribution ν. The diffusion appearing in (3.3)

9

will henceforth be refered to as a stochastic volatility jump diffusion. Wewill also consider stochastic volatility processes without jumps, that is:

drt = µ(t)dt+ σ(t)dWt (3.4)

Barndorff-Nielsen and Shephard (2001, 2002a,b, 2003) develop an asymp-totic distribution theory for Q(m),t as an estimator of Qt for m → ∞ forstochastic volatility processes without jumps. The asymptotics involves tak-ing returns over fixed intervals, say daily, and computing ˆQ(m),t with intra-daily observations sampled with increasing frequency. In particular, theystudy the asymptotic properties of and log δ2t = log Q(m),t - logQt for datagenerated by (3.4) with the drift function µ(t) continuous and predictableand the volatility function σ(t) a càdlàg process on [0,∞). They show thatthe asymptotic distribution of log δ2t is mixed normal. Barndorff-Nielsen andShephard (2003) show through Monte Carlo simulations that even for smallm = 12, the approximation seems to be quite accurate. In general, comput-ing Xt ≡ r(1),t/(Q(m),t)1/2 results in heavy-tailed random sequences even ifthe underlying process Xt = r(1),t/(Q(m),t)

1/2 is Gaussian.

One implication of the Barndorff-Nielsen and Shephard asymptotic analy-

sis is that if∑m−1

j=0 r4(m),t−j/m/

ˆQ2(m),t is i.i.d. then log δ2t is a heavy tailed i.i.d.

process. Consequently, Xt is i.i.d. provided the process Xt is i.i.d. to beginwith. Andreou and Ghysels (2002c) show that Xt, using various volatilityestimators applied to representative financial market data, is indeed i.i.d.and heavy tailed or else approximately i.i.d. and heavy tailed. Hence, undersomewhat restrictive settings the use of Assumption 3.2 may be warranted.

The discussion so far indicates that typically ˆXt ≡ r(1),t/(Q(m),t)1/2 satis-fies Assumption 3.1, i. e. it is a weakly dependent process with heavy tails.This allows us to apply EDF-based tests. There are limitations, however,to the applicability of data-driven measures involving Q(m),t. Very little is

known about the asymptotic properties of Q(m),t when the return process isgenerated by a jump diffusion, such as (3.3). Moreover, typically it is alsoassumed that the diffusion does not feature leverage, i.e. correlation betweenWt and the innovations for the process for σ(t) in (3.3) which was deliber-ately left unspecified. For some empirical applications, assuming no leverageis warranted, in particular for FX markets. Applications involving equitymarkets would require leverage. Some of the parametric models discussed inthe next section can handle this.6

6Adjustment ˆQ(m),t to take into account the presence of leverage also have been dis-

10

Alternative data-driven de-volatilization approaches can be considered aswell. Alizadeh et al. (2002) suggest to use the daily range as a measure ofvolatility instead of the increments in quadratic variation. The appeal of thedaily range is that is measured without error, unlike the quadratic variationestimators discussed above. To proceed let us define the daily range, namely:

R(m),t = supt−1/m<τ≤t

pτ − inft−1/m<τ≤t

pτ (3.5)

and consider the stochastic process Xt ≡ r2(m),t/R(m),t. Note that unlike withthe quadratic variation, we no longer hold the time interval fixed, insteadwe consider returns over a ever shrinking time interval and the associatedrange. Note also that we do not denote the process as ˆXt since in prin-ciple no estimation is involved. Obviously we can view the range as anestimator of volatility, yet we do not directly link it to the parameters ofthe underlying process, nor to its quadratic variation. Such links can onlybe established in some special cases as discussed by Alizadeh et al. (2002).We view the ratio Xt ≡ r2(m),t/R(m),t, as a process not involving parametricor non-parametric estimation. In practice we will have to assume that wesample over sufficiently small intervals, instead of sampling squared returnsover some sufficiently small subintervals of [t − 1, t] to compute quadraticvariation estimators. Such asymptotic analysis is reminiscent of the continu-ous record asymptotic theory of Foster and Nelson (1996). Using argumentssimilar to those of Foster and Nelson, Andreou and Ghysels (2002d) showthat Xt ≡ r2(m),t/R(m),t becomes an i.i.d. random variable for m sufficientlylarge, or equivalently for sufficiently short sampling intervals. The basic intu-ition driving the results is that returns, r(m),t/(R(m),t)1/2, over short intervalsappear like approximately i.i.d. with zero conditional mean and finite condi-tional variance and have regular tail behavior. It should be noted that thisresult is weaker than that of Foster and Nelson who excluded the presence ofjumps, i.e. returns are driven by (3.4). Using the range, instead of a rollingsample estimator of instantaneous volatility, as advocated by Foster and Nel-son, removes the requirement of a smooth volatility process, so that jumpscan be accommodated. It should also be noted that we can accommodateleverage effects with the range.

We can use the range-based de-volatilization and assume the resultingprocess satisfies Assumption 3.1. However, it is also clear from the above

cussed, see Meddahi (2002).

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discussion that for sufficiently small intervals Assumption 3.2 is warranted

and rank-based tests for testing change-points in distributions can be used.

3.2 Model-based De-volatilization and Empirical Processes

Instead of using data-driven volatility estimators to construct ˆXt ≡ (rt −

µ)/σt, we now resort to parametric models. It will be assumed that σt(b) isparameterized by some parameter vector b. One of the most popular para-metric families for financial asset returns is the class of ARCH models (seefor instance the reviews by Bollerslev et al., 1994). The residuals sequencethat arises from parametric and semiparametric estimators of ARCH mod-els will be the process monitored for change-points. Horvath, Kokoszka andTeyssière (2001a), Koul (2002) and Koul and Moukherjee (2002) providerecent contributions on the empirical process properties of ˆXt for differentestimators of b which can be used for change-point analysis.

Horvath, Kokoszka and Teyssière (2001a) (henceforth HKT) and Koul(2002) show that unlike the residuals of ARMA processes (e.g. Bai, 1994),the residuals of the ARCH models yield sequential empirical processes thatdo not behave like asymptotically independent random variables. In particu-lar they show that the asymptotic distribution involves a term depending onthe parameters of the model. In certain interesting cases, including the de-tection of changes in the distribution function of unobserved innovations, thesequential Empirical Distribution Function (EDF) tests yield asymptoticallydistribution free statistics.

The discussion below holds for the Generalized ARCH (GARCH) familyof models and for simplicity purposes we focus on the ARCH model (Engle,1982) for the returns process rt given by:7

rt = σtεt (3.6)

σ2

t= b0 +

∑p

j=1bjr

2

t−j (3.7)

where the volatility σt is assumed to be driven by an i.i.d. innovationprocess εt −∞ < t < ∞ with mean zero and unit variance and bj > 0.We emphasize that our analysis is based on the normalized returns process

7In the discussion below we present more general nonlinear dynamic structures thatincorporate various GARCH specifications and nonlinear AR models. We also suppressfor simplicity the presence of a mean component in (3.6).

12

Xt ≡ rt/σt that is equivalent to εt in (3.6), the latter being the representativespecification of most (G)ARCH type processes. Hence we adopt the Xt

notation since it represents the process of interest here and establishes thecoherency with the data-driven de-volatilitization methods in the previoussection. HKT assume that the unknown distribution function F (X2

0) needs

to satisfy the general conditions so that the density function f exists and is

continuous on (0,∞). The continuity assumption implies the derivation of

the conditional quantiles and the relevant empirical process which are the

focus of VaR and tail change-point tests. An additional crucial condition in

HKT is that the innovation process has a finite fourth moment, E(X4

0) <∞,

and the ARCH process satisfies the stationarity condition:8

E(X4

0)1/2∑

1≤j≤pbj < 1. (3.8)

Hence the ARCH equations have a unique strictly stationary solution suchthat E(r4

t) <∞ and the squares r

2

thave a Volterra representation

r2

t=

∑∞

l=0

∑p

j1,...,jl=1bj1...bjlX

2

t X2

t−j1...X

2

t−j1−...−jl.

Thus r2tis a function of Xt,Xt−1, ..., and so it follows that r2

t is ergodic.

It is also well-known that ARCH-type sequences are not only ergodic butmixing with geometric rate (Chen and Carrasco, 2001).

The properties of the normalized returns (or residual) empirical processintimately depend on the family of estimators of the parameter vector b of

the ARCH given by ˆbp = (b0, b1, ...bp) which are assumed to satisfy certain

conditions such as asymptotic linearity and√T -consistency given by:

ˆbi − bi =1

T

∑1≤t≤T

li(X2

t )fi(Xt−1,Xt−2, ...) + o(T−1/2), 0 ≤ i ≤ p (3.9)

The functions li and fi above are regular in the sense that:

E(li(X2

0)) = 0, E

[li(X

2

0)]2

<∞, E [fi(X0,X1, ...)]2<∞, 0 ≤ i ≤ p

(3.10)

8It is worth noting that Inoue (2001), Giraitis et al. (1997), Mikosch and Starica(1999), Quintos et al. (2001) also require existence of the fourth moment of the return

process itself, rather then the residuals which is directly monitored for change-points. Ourapproach requires moment restrictions on Xt which can be easier to evaluate if such anassumption is verifiable as opposed to the analogous condition on the strongly dependentprocess rt.

13

Berkes et al. (2002) show that the above conditions hold for GARCH(p, q)processes. Commonly used estimators for GARCH models admit the repre-sentation (3.9) e.g. conditional likelihood, pseudo maximum likelihood, con-ditional least squares estimators (see for instance, Gourieroux, 1997, Ch.4).Moreover, in general (3.10) can be considered as score functions which areassumed to be bounded, nondecreasing, and real-valued, and can be relatedto robust estimators, as shown in Koul (2002) and discussed below.

The sequential (or two-time parameter) empirical process results in HKTare valid for ARCH type estimators that satisfy conditions (3.9)-(3.10) andyield a squared residual process X2

t= r2

t/σ2

tfor which the sequential empir-

ical process is defined as:

eT (x, s) = T 1/2s(ˆFT (x, s)− F (x)

)

where

FT (x, s) =

1

T s

∑p<t≤T s 1

ˆX2

t≤ x if p/T < s ≤ 1

0 if 0 ≤ s ≤ p/T.

Theorem 1.1 in Horvath et al. (2001a, p.3) shows that under the aboveconditions this converges to a Kiefer process:

eT (x, s)→ Γ(x, s) (3.11)

where Γ(x, s) is the limiting Gaussian process with zero mean and covari-

ance function Γ. This is equivalent to the convergenence eT (F−1(x), s) →Γ(F−1(x), s), 0 ≤ x, s ≤ 1, in D([0, 1]× [0, 1]).

The covariance depends on several unknown parameters and functions

involving the innovation process, its density function, the volatility process

and their expected values. The proof of the structure of eT (x, s) and the

formula for its covariance function relies on the properties of the sequence

δ2t = σ2

t/σ2

t . It is observed by HKT that:

sup0≤x<∞

sup0≤s≤1

|eT (x, s)− (eT ,1(x, s) + eT,2(x, s))| = Op(T−1/2) (3.12)

whereeT,1(x, s) = T−1/2

∑p<t≤T s

(1 ˆX

2

t≤ xδ

2

t − F (xδ2

t))

14

and

eT,2(x, s) = T−1/2∑

p<t≤T s

(F (xδ2t )− F (x)

).

The proofs in HKT show that δ2tis so close to one that the difference between

eT,1(x, s) and

eT (x, s) = T−1/2∑

p<t≤ns

(1 ˆ

X2

t≤ x − F (x)

)(3.13)

is negligible. The third term in (3.12) is approximated by

hT (x, s) = xF (x)T−1/2∑

p<t≤ns(δ2t − 1) (3.14)

where

∑p<t≤Ts

(δ2t− 1) =

∑p<t≤T s

( σ2t−σ2

t

σ2t

) = (b0 − b0)∑

p<t≤T s

1

σ2t

+(b1 − b1)∑

p<t≤T s

y2t−1

σ2t

+ ...+ (bp − bp)∑

p<t≤T s

y2t−p

σ2t

= Ts(b0 − b0)β0+ (b1 − b1)β1

+

...+ (bp − bp)βp+ op(T−1/2)

Hence

sup0≤x<∞

sup0≤s≤1

∣∣∣∣∣eT (x, s)−

(eT (x, s) + xF (x)s

∑1<i≤p

T 1/2(bi − bi)βi

)∣∣∣∣∣ = op(1)

(3.15)and therefore the joint convergence of eT (x, s) and

√T (bT − b) imply the

result in (3.11) (Theorem 1.1 in Horvath et al., 2001a).The above results show that Empirical Distribution Function (EDF) of

the squared returns normalized by the ARCH variance (X2

tor ε2

t) can be

used to study the distribution change-point problem just like the EDF testsfor an i.i.d. process, the latter being widely used and studied statisticalproblem (e.g. Csörgö and Horvath, 1997, section 2.6, Szyszkowicz, 1998).The sequential two-parameter EDF process, wT (x, s) for X2

t is:

wT (x, s) =

0 0 ≤ s ≤ p/T[T s](T−[Ts])

T 3/2

(ˆFT (x, s)− F ∗

T(x, s)

)p/T < s ≤ (T − 1)/T

0 (T − 1)/T < s ≤ 1,(3.16)

15

where ˆF ∗

T(x, s) = 1

T−T s

∑T s<t≤T

1 ˆX2

t≤ x so that it compares the EDF

of ˆX2

p+1, ...,ˆX2[Ts] to that of ˆX2

[Ts]+1, ...,ˆX2T . What is interesting about (3.16)

is that, given the results in the above theorem, the terms hT (x, s) in (3.14)from the two-parameter process cancel out so that the process wT (x, s)converges to a tied-down Kiefer process. Hence the following well knownstatistics can be used to examine the distributional homogeneity hypothesis.

The supremum statistic:

sup

0≤x<∞,0≤s≤1

|wT (x, s)|d→ sup

0≤u,s≤1

|K∗(u, s)| (3.17)

has an asymptotic distribution equivalent to the Kolmogorov-Smirnov statis-tic and has been studied for time series models originally by Picard (1985).Similarly since F is continuous the quadratic statistic:

∫1

0

∫∞

0

w2

T(x, s)dFT (x, 1)ds

d

→

∫1

0

∫1

0

[K∗(u, s)]2duds (3.18)

has an asymptotic distribution equivalent to the Cramer-von Mises repre-sentation studied for instance in Blum et al. (1961). The properties ofthese statistics (3.17) and (3.18) are examined for the squared normalizedreturns of a general family of ARCH processes and estimators that satisfythe above conditions and for members of the location-scale family of distri-butions that are fat tailed and asymmetric and exhibit the stylized facts ofjumps or extreme values. From the continuous mapping theorem the quan-tile empirical process of X2

tfor the aformentioned estimators is also used to

focus on change-points in the tails of the distribution that represent the coreof risk management measures. The alternative versions of sequential EDFtype change-points tests evaluated in the paper are revisited in section 5 inthe context of monitoring the distribution of risk exposure.

Koul (2002) also establishes the asymptotic uniform linearity of weigthedand sequential residual empirical processes for ARCH models using a class ofnonparametric estimators, namely M- and minimum distance (m.d.) estima-tors. The sequential results are used to provide asymptotically distributionfree tests for examining the hypothesis of a change in the error distributiongenerating this class of nonlinear time series. Koul (2002) considers a generaltime series model for the returns process rt :

rt = µ(r0t−1

,α)+σ(r0t−1

,β)εt, t ≥ 1 (3.19)

16

where the errors εt, t ≥ 1 are independent of r0t−1

:= (rt−1, rt−2, ..., rt−p)′

and i.i.d. r.v.’s with F . The known functions µ→ R and σ → R+ have un-

known parameters α and β, respectively. This model nests the ARCHmodel(3.6)-(3.7) as well as other dynamic (non)linear conditional mean and vari-ance models. For example, the dynamic model in (3.19) can capture the lever-age effect in rt which can be modelled by σ

2

t= b0+b1r

2

t−1+δ0dt−1r

2

t−1where

dt−1 = 1 if rt−1 ≥ 0 and zero otherwise, such that if there is a leverage effectδ0 < 0 (Glosten, Jagannathan and Runkle, 1993). Model smoothness as-sumptions about the functions µ and σ and moment restrictions are requiredin order to show the weak convergence of certain basic randomly weightedempirical processes for model (3.19) which are general enough and can befound in Koul (2002, p.830-1 and p.383-4). The M- and m.d.-estimators forα and β are based on:

εt(k) ≡rt − µ(r0

t−1,k1)

σ(r0t−1

,k2)

where k := (k′

1,k

′

2)′denote the scores of the estimators and εt is by definition

the normalized returns process Xt. For the sake of simplicity and coherence

with the the previous section as well as the stylized fact of zero serial cor-

relation in stock returns we will assume that µ(r0t−1

,α) =0. Moreover, Koul(2002) shows that the estimators of α and β can be obtained by a two-stepprocedure such that in the first step a

√T -consistent estimator for α can be

used to capture only the nonlinear AR structure in (3.19) and then α is usedto construct an estimator for β which takes into account the heteroskedas-ticity in the model. The asymptotic distribution of ˆβ does not depend onthe preliminary estimator α used in defining ˆβ. Hence there is no loss ofgenerality in assuming µ(r0

t−1,α) =0 which nests Engle’s ARCH process.

To conclude we describe the specifics of an example involving both con-

ditional mean and volatility and a QMLE estimator. This setup is fairly

standard and therefore worth elaborating on more specifically. The ARCH

model now be re-written as:

Zt = b′Wt−1 + b

′Wt−1ηt

. (3.20)

where Zr = r2

tand Yt−1 = (Zt−1, ..., Zt−p)

′ = (r2t−1, ..., r2t−p)

′and Wt =

(1,Y ′

t−1), b = (b0, .., bp)′. This is an example of the general model (3.19)

with α = β, µ(r0t−1

,b) = b′w = σ(y,b),w′ = (1,y′). The distribution

function F refers to the error ηt≡ ε2

t− 1 which is assumed to satisfy the

17

general conditions in Koul (2002, Ch. 2 (2.2.49)-(2.2.55). Now considerε2t≡ r2

t/σt and we obtain the squared normalized returns process X2

t. Let

ηt≡ ε2

t− 1 ≡ X2

t− 1 and E(η2

t) = 1. Assume that rt is stationary

and ergodic and E(r4t) < ∞ and that this model satisfies all the smooth-

ness and moment conditions discussed in Koul (2002, p. 400-1).9 From

Corollary 8.3.4 in Koul (2002, p.839) T1/2(b − b)

d

→ N(0,Σ(b)) where

Σ(b) ≡ (E[W0W′

0/(b′W0)2])−1υ(ψ,F ). Given the stationarity of r2

t the

finite fourth moment assumption on the i.i.d. errors ηt, the asymptotic distri-

bution of the QMLE estimator is given in Weiss (1986): T 1/2(bQMLE−b)d→

N(0,ΣQMLE) where ΣQMLE ≡ (E[W0W′

0/(b′W0)

2])−1var(η).The empirical process for EDF-type tests based on the residuals of dy-

namic time series models is also studied in Koul (1996, 2002). Based on theabove model it is shown in Koul (2002, Corrolary 8.3.6) that the limitingbehavior of the one-sample EDF process is not asymptotically distributionfree since

supx∈R

∣∣∣∣∣T1/2[FT (x)− F (x)]− T

−1

T∑

t=1

W′

t−1

2(b′W′

t−1)1/2

T1/2(b − b)xf (x)

∣∣∣∣∣= o(1).

However, it is shown in Koul (1996, 2002) that the two-sample EDF supre-mum and quadratic statistics are asymptotically distribution free. The abovearguments not only apply to QMLE estimators but also M-estimator whichare asymptotically relatively more efficient than the widely used QMLE es-timator.

4 Sequential EDF based Change-point Tests

We are interested in testing for breaks in the distribution of Xt, constructed

via means described in the previous section, and assuming it is a weakly

dependent process, satisfying Assumption 3.1. The general idea of studying

change in distribution for a stochastic process can be expressed in terms of

comparing the empirical distribution function of the first Ts observations to

that of the last (T − Ts) observations. The distance between these EDFs is

9For estimation purposes in the ARCH reparameterized model (3.20) since α = β wemay use the two-step procedure discussed above by focusing on the first step and using aconsistent M- or m.d.-estimator αp instead of β to define the final b.

18

given by:∣∣∣∣∣

1

Ts

T s∑

t=1

1Xt ≤ x −1

T − Ts

T∑

t=T s+1

1Xt ≤ x

∣∣∣∣∣

where Ts denotes a canditate change 1 ≤ Ts ≤ T, T = 1, 2, ... for x ∈ R. The

asymptotic distribution of a sequence of such processes is used to construct

a nimber of statistics useful for examining distributional homogeneity.

4.1 Supremum statistics for change-points in the EDF

The following Kolmogorov-Smirnov statistics are examined:

KS1 =√T

∣∣∣∣1

T s

T s∑

t=1

1Xt ≤ x − 1

T−T s

T∑

t=Ts+1

1Xt ≤ x

∣∣∣∣=

∣∣∣∣

Ts∑

t=1

1Xt ≤ x − T s

T

T∑

t=1

1Xt ≤ x∣∣∣∣/√

T(Ts

T

(1−

T s

T

)) (4.1)

Considering the sup1≤T s≤T sup

x∈R functionals of (4.1), the resulting sequenceof random variables and even the one where Ts/T (1 − Ts/T ) is replaced by

(Ts/T (1 − Ts/T ))1/2, namely,

KS2 = sup

1≤T s≤T

sup

x∈R

∣∣∣∣∣T s∑t=1

1Xt ≤ x − Ts

T

T∑t=1

1Xt ≤ x∣∣∣∣∣/√

T

(Ts

T

(1−

Ts

T

))1/2

(4.2)

converges to ∞ in probability, as T → ∞, even if the null assumptionof no change in distribution were true (see for instance, Csörgö and Hor-vath, 1997, Szyszkowich, 1998). In order to have nondegenerate limits asT →∞, in supremum norm, one can consider weighting functions q(Ts/T ) =

((Ts/T ) (1 − Ts/T ))1/2 h(Ts/T ), where the function h(Ts/T ) necessarily goesto ∞ as Ts/T →∞ or as Ts/T → 1 (Picard, 1985, Szyszkowicz, 1994).

In the spirit of the classical Kolmogorov-Smirnov statistic, the ones basedon the the weighted difference between the sequential EDFs

sup

1≤T s≤T

sup

x∈R

√T

(Ts

T

(1−

Ts

T

)) ∣∣∣∣∣1

Ts

T s∑t=1

1Xt ≤ x −1

T − Ts

T∑

t=T s+1

1Xt ≤ x

∣∣∣∣∣

can detect change points in the distribution function of different types ofindependent and dependent stochastic processes (e.g. Picard, 1985, Carl-stein, 1988, Bai, 1994, Koul, 1996, Inoue, 2001, Horvath et al, 2001). This

19

weighted Kolmogorov-Smirnov expression can be written in a less computa-tionally intensive format as given by the second equality:

KS3 = sup

1≤T s≤T

sup

x∈R

√T(Ts

T

(1−

T s

T

))×

∣∣∣∣1

Ts

Ts∑

t=1

1Xt ≤ x − 1

T−Ts

T∑

i=T s+1

1Xt ≤ x

∣∣∣∣

= sup

1≤T s≤T

sup

x∈R1√T

∣∣∣∣

Ts∑

t=1

1Xt ≤ x − Ts

T

T∑

t=1

1Xt ≤ x

∣∣∣∣

(4.3)

The statistic in (4.3) should be more powerful in detecting changes that occurin the middle, namely near T/2, where T s

T

(1−

Ts

T

)has its maximum, than

for noticing the ones occuring near the endpoints 0 and T . Thus the weightedversion of (4.3) should emphasize changes that may have occurred near theendpoints while retaining sensitivity to possible changes in the middle aswell. Szyszkowicz (1994, 1998) suggests the following modification of (4.2)and (4.4)

KS4 =

sup1≤T s≤T supx∈R

∣∣∣∑

T s

t=11Xt ≤ x − T s

T

∑T

t=11Xt ≤ x

∣∣∣√T

(Ts

T(1− T s

T) log log 1

Ts

T(1−T s

T)

)1/2

(4.4)

which has a nondegenerate limiting distribution,

sup

0<s<1

sup

0≤x≤1

|K(x, s)− sK(x, s)|

/(s(1 − s) log log

1

s (1− s)

)1/2

where K(x, s), 0 ≤ x, s ≤ 1 is a Kiefer process.

4.2 Quadratic statistics for change-points in the EDF

A second wide measure of discrepancy between EDFs beyond the supremumis the family of quadratic statistics. In direct analogy with the weigthedKolmogorov-Smirnorv, the weighted Cramer-Von Mises stastistic is basedon:

CVM1 =1

T (T−1)1√T

(T s∑t=1

1Xt ≤ x − T s

T

T∑t=1

1Xt ≤ x

)2

=

T−1/2

T (T−1)

T−1∑k=1

T∑t=1

[(T s

T

(1−

Ts

T

))(1Ts

T s∑t=1

1Xt ≤ x − 1

T−Ts

T∑t=T s+1

1Xt ≤ x

)]2(4.5)

20

(see for instance, Picard, 1985) where the second equality yields a statisticsimilar to (4.3). Another way or writing the CVM statistic is also givenbelow:

CVM2 =1

T (T − 1)

(∑

Ts

t=11Xt ≤ x − Ts

T

∑T

t=11Xt ≤ x

)2[√T(Ts

T

(1−

Ts

T

))]2

(4.6)

The asymptotics is similar to the one for Kolmogorov-Smirnov statistics,except that the asymptotic distribution is a double integral of a squaredKiefer process instead of the double supremum, that is the norm is changedfrom to L2.

5 Rank based Change-point Tests

We are interested in testing for breaks in the distribution ofXt, assuming it isi.i.d., satisfying Assumption 3.2. This is a more restrictive setting that leadsto a different set of statistical tools based on ranks or on the signed ranks.These procedures are generally constructed with contiguous alternatives inmind; one expects to see many observations before the change, and expectsto see a small change if it occurs. Although there is an intimate relationbetween ranks and empirical processes, the former require an i.i.d. settingwhereas the latter apply to wide range of residual sequences of time seriesmodels. Our change-point analysis is based on two rank based tests: (1) theBhattacharya and Frierson (1981) test using sequential ranks and the Gordonand Pollak (1994,1995) test using the sequential likelihood-ratios for signedranks, an adaptation of the Shiryaev-Roberts procedures in a nonparametriccontext.

It should be noted that the application and extension of the above proce-dures is restricted to standardized returns Xt which is de-volatilized by theobserved range of the process and does not involve estimation. It is assumedthat the sampling of Xt is fine enough so it is i.i.d. Unlike in the previoussection, we work directly with Xt, not ˆXt which involved data-driven andmodel-based estimates of volatility (that is estimators unlike the range, thatinvolve estimation noise). One can of course apply the EDF-based tests toreturns normalized by the range. We suggest here procedures that are mostlikely more powerful and robust for this particular special case consideredhere.

21

Bhattacharaya and Frierson (1981) present a test for detecting changesin the unknown DF of an independent stochastic process, at an unknownchange-point and prove the asymptotic behavior of their test based on the fol-lowing assumptions: For each positive integer T (= 1, 2, ...), let Xt, 1 ≤ t ≤ Tbe a sequence of independent random variables. The first [sT ] members(0 < s < 1) of the random variables have a common continuous cumulativeDF F0 and the last T − [sT ] members are distributed as F1. The DF’s F0

and F1 are unknown but assumed continuous. The type of change in the DFconsidered is quite general. It is assumed that T is large and the change fromF0 to F1 is small as defined in condition (1) in Bhattacharaya and Frierson(1981, p.545).

The test statistic is constructed as follows: While observing the Xt’ssequentially, the objective is to stop soon after the time point [sT ]+ 1 wherethe process shifts from F0 to F1. The basic elements of a nonparametricdetection scheme are the sequential ranks R1, ..., RT where Ri denotes the(ordinary) rank of Xt when Xt is ranked among the subset X1, ...,Xt .Alternatively, we have:

Rt = 1 +∑t−1

k=1

Ukt (5.1)

where Ukt = 1 (0), if Xk < Xt (if Xk > Xt). The sequential ranks of i.i.d.random variables are themselves independent random variables (Barndoff-Nielsen, 1963) with Rt uniformly distributed on the integers 1, ..., t so thatE(Rt) = (t+1)/2 and V ar(Rt) = (t2−1)/12 (e.g. Bhattacharya and Frierson,1981, p.547). Let

Zt = i−1 (Rt − (t+ 1) /2) , St =∑t

i=1

Zi, (5.2)

The test procedure is based on (5.1)-(5.2) and the normalized statistic√

12/TSt

is used for detecting disruptions when it exceeds the nonparametric controllimit, c = Φ

−1 ((1 − a) /2) where Φ is the standard Normal DF.10 The para-

metric counterpart of the above control chart is based on the cumulative

sums T−1/2∑

t

i=1(Xi − µ

F) /σF where µ

Fand σF are the known mean and

standard deviation of F0, respectively. Simulation evidence in Bhattacharyaand Frierson (1981) shows that for distributions with heavy tails, the non-

10The behavior of the test statistic can also be represented on a nonparametric controlchart with an appropriate stopping rule (see Bhattacharya and Frierson, 1981, p.546-7, fordetails).

22

parametric scheme is superior to the scheme based on cumulative sums bothin the sense of asymptotic power and the expected stopping time.

Gordon and Pollak (1994, 1995) also present two sequential nonparamet-ric schemes for detecting an unknown change-point in the distribution of i.i.d.observations based on the method of sequential likelihood ratios of signs andranks using the Shiryaev-Roberts approach. The process X1,X2, ... is as-sumed again to be an independent sequence of random variables and the DFof the random variables is unknown, continuous and symmetric about theorigin. The density prior the change-point is f0 (x) = 1

2exp (− |x|) , while

after the change-point it is f1 (x) = pα exp (−αx) Ix>0 + qβ exp(βx)Ix<0.The parameters α, β, p and q are assumed to be positive and p+ q = 1. Notethat the exponential distribution is merely an artifice to define the statistic.

Define ζ i = IXi>0 giving the sign of the ith observation. The number of

positive observations among Xk, ...,XT is denoted by U (k, T ) =∑n

j=kζj and

the corresponding count of negative observations is V (k, T ) =∑T

j=k(1− ζj).The rank of the absolute value of the ith observation among the first T ab-

solute values observed is: ρ (i, T ) =∑T

j=1 I|Xj|≤|Xi|. Let ZT

i= ((ρ (i, T )), ζ

i),

then ZT1, ..., ZT

T contain all the information. Because it is assumed that the

distributions of Xi are continuous, the ranked absolute values of the first T

observations determine the random permutation ρ (·, T ). The inverse permu-

tation is denoted as τ (·, T ), so that ρ (τ (i, T ) , T ) = i for i = 1, ..., T. Nextdefine: γ(j, k) = 1, α, β, if j < k, if j ≥ k and ζj = 1, if j ≥ k and ζj = 0,respectively. By conditioning on the signs of X1, ...,XT , Gordon and Pollak(1994, Lemma 2.1, p.766) obtain an explicit likelihood hk(ZT

1, ..., ZT

T ) for thesigns and ranks of absolute values when the change-point is k, the prechangedensity is f0 and the postchange density is f1. The nonparametric likelihoodratio based on signs and ranked absolute values is:

ΛTk =

hk(ZT1 , ..., Z

TT )

h∞(ZT

1 , ..., ZTT )

= (2p)U(k,T )(2q)V (k,T )

T∏

i=1

γ (τ (i, T ) , k)

[1/ (T − i + 1)]∑T

j=i γ (τ (i, T ) , k)(5.3)

for 1 ≤ k ≤ T+1. The nonparametric analog of the Shiryaev-Roberts (NPSR)statistic is:

RT =

∑T

k=1

ΛT

k. (5.4)

The standard index for the rate of false alarms is E∞NA which is typically

23

controlled by considering only stopping rules that satisfy: E∞NA ≥ A forsome specified level A. In Gordon and Pollak (1994) E

∞NA = A is indepen-

dent of the actual distribution of the observations as long as that distributionis continuous and symmetric about the origin. Their Theorem 2.2 (p.768)shows that the NPSR statistic (5.4) shares very similar false alarm rates withthe parametric version, when the false alarm rate is required to be low. Theasymptotic approximations are also valid for small average run lengths andearly change-points. In addition, the relative efficiency of these schemes (withrespect to a normal parametric shift detection policy) is very high, makingthem a robust alternative to parametric methods.

We note that the test procedures described are mainly applied to qualitycontrol where one often monitors one-sided changes of say an abrupt perma-nent increase at a single change-point. In financial markets we acknowledgethat abrupt drops in standardized returns have significant investment conse-quences. In addition, financial markets are characterized by infrequent jumpsof either direction that have a transitory effect on the location of the distrib-ution as opposed to a single disorder of permanent character (e.g. breakdownof a machine). Therefore the Gordon and Pollak test is performed for bothranks and reverse ranks whereas the Bhattacharya and Frierson statistic ismonitored for a two-sided alternative. These points are addressed in thesimulation experiment where we examine the finite-sample properties of thetests for standardized returns, in the presence of both transitory disruptionsas well as multiple change-points in the location of the distribution of Xt. Inthe latter case we allow for frequent and multiple small changes and appraisethe test procedure over shorter time intervals.

6 Quantiles and Tails of the location-scale

family of distributions

In this section we focus on two issues specifically of interest to risk manage-ment: (1) quantiles and (2) tails. The previous sections suggested tests forbreaks in the conditional distribution of returns without any specific atten-tion to the extreme event outcomes of the distribution. In this section wefocus specifically on testing for breaks in quantiles and tails. The latter isdone in the context of the location-scale family of distributions.

24

6.1 Quantile empirical processes for change-points in

the tail

To be included

6.2 Tails and location-scale family

The analysis of the previous sections holds for continuous distributions andfor financial asset returns this can be the class of the location-scale fam-ily of distributions introduced by Fisher (1934) defined by: f(r;µ, σ) =1/σf ((r − µ)/σ; 0, 1) with f(.) a known density function and the two pa-rameters that correspond to the location (µ) and scale (σ). The pivotalquantity is X = (r − µ)/σ whose distribution does not depend on eitherthe parameters or the random variable. Some members of this family are theNormal, Generalized Extreme Value (GEV) such as the Frechet distributionsand stable distributions such as the Levy distribution.

Extreme Value and Stable distributions have been proposed for modelingspeculative prices (e.g. Mandelbrot, 1963, Mittnik and Rachev, 1993a) inan i.i.d. context as well as in an ARCH context (Borkovec, 2000, de Haan,1989 and Mittnik and Rachev, 1993b). Similarly there is a plethora of re-cent research in using Extreme Value Theory (EVT) in financial markets tocapture rare events (see for instance, Embrets et al., 1997) and estimate thetail index. The de-volatilization methods suggested in sections 4.1 and 4.2are valid for popular members of the location-scale family of distributionssuch as the Normal, Levy and Pareto processes. McNeil and Frey (2000) useEVT-based parametric methods to estimate the tails of of the distributionof the residuals of a GARCH model.11 Minimal assumptions are imposedabout the DF of the underlying innovation and a two-stage method is pro-posed according to which in the first stage the GARCH-type model for rtis estimated using pseudo MLE and the fitted model residuals ( ˆXt) whichfollow a WN. In the second stage the residuals are used to estimate the tailof the EVT model. A related advantage of examining the extremes in theprocess ˆXt instead of rt refers to the relatively simple dependence structureof the former.

Within the EVT there are many approaches for estimating the tail indexof a distribution, both parametric and nonparametric. In most approaches

11This approach has also been proposed in Diebold et al. (1999).

25

the extremal behavior obtained from estimating the tail index is based onthreshold or block minima (or maxima) methods which have traditionallybeen used to obtain the sample of extreme observations. The approach fol-lowed here is to consider the underlying sequences of block minima (denotedby ˆX

min

n,t ) or number of exceedances over a threshold (denoted by ˆXξn,t)

(Davidson and Smith, 1990, Smith, 1989) that represent the backbone of tailindex estimation. The analysis does not involve tail estimation (parametricor noparametric) since this presupposes distributional homogeneity which ifignored may lead to spurious estimation and inference results. Instead we usethe underlying processes ˆX

min

n,t and ˆXξn,t to evaluate possible structural

changes in the laws governing extreme observations.According the block maxima method we use the de-volatilized returns

process, ˆXt, which represents an approximately independent process, and di-vide the sample into k non-overlapping blocks with approximately n days ineach block. For an arbitrary block we obtain the minima ˆXmin

n = min(X1, .., Xn)and construct a sequence of minima denoted by Xmin

n,t . According to theEVT a GEV distribution can be fitted to the sample of k independent blockminima realizations of ˆX

min

n,t . This is due to the extremal types theorem in

Fisher and Tippett (1928). The process ˆXmin

n,t has a regular limiting be-havior in the sense that there exist sequences of real constants bn and an > 0

suuch that

limn→∞

P(Xmin

n,t − bn)/an ≤ x = limn→∞

F n(anx+ bn) = H(x)

for a nondegenerate DF H(x). If this condition holds F is said to be in the

maximum domain of attraction of H and the GEV has DF:

Hα(x) =

exp(−(1 + αx)−1/α α = 0exp(−e−x) α = 0

where α < 0, α = 0, α > 0 correspond to the Weibull, Gumbel and Frechetdistributions, respectively. Maximum likelihood and regression methods fortail estimation are based on Xmin

n,t as well as nonparametric methods suchas Hill and Pickland tail estimators also depend on the subsample of thesmallest ordered sample of Xn,t. Applying distributional change-point teststhat do not assume a particular distribution function to the sample of minimawill examine whether the probability law governing extremes has been stable.Another approach to EVT is to focus on exceedances of the measurement oversome high threshold and the times at which the exceedances occur (Davidson

26

and Smith, 1990, Smith, 1989). Let ξ be a prespecified high threshold (say−2.5%) and let the ith exceedance in normalized returns occur at day ti (i.e.X

ξti≤ ξ). The sequence (ti, X

ξti− ξ) represents the intensity of extremes (ti)

where a cluster of ti indicates a period of market declines and the exceedingnormalized returns (exceedance) X

ξti− ξ which will represent the process

monitored for breaks in extremes. The process ti, ˆXξti−ξ is governed by the

probability law of a two dimensional Poisson process. Hence the EDF- and

rank-based mentioned above can not be applied for the threshold sequenceˆXti − ξ which requires discrete distribution function procedures.

The corresponding statistical inference based on ˆXmin

n,t and ti, ˆXξti− ξ

depends on the choices of the block size k and the threshold level ξ, respec-tively. Their choice will affect the tail index estimation and change-pointinference. Both statistical and financial arguments are useful for the choicesof k and ξ. Danielsson and de Vries (1997b) present a statistical procedurefor tje choice of ξ. Financial arguments are also useful as they relate to differ-ent institutions or investors risk tolerance or risk aversion. Similarly ξ maydepend on financial positions and financial market characteristics such asliquidity. Similarly for the choice of k can depend on the sampling frequencyand on short or long financial posisitions (e.g. k = 250 trading days for dailydata to denote jumps every year in a 10 year period or k = 21 trading daysfor intraday day when the risk manager is interested in monitoring the asset’srisk exposure every day). Hence, sensitivity analysis for the choices of k andξ is useful and is performed in the simulations and empirical analysis.

A related interesting aspect of financial asset returns is the infinite vari-ance syndrome (Mandelbrot, 1963). There are numerous empirical resultson the stylized fact of infinite variance for speculative prices measured bythe volatility persistence of ARCH-type processes and the characteristic ex-ponent of stable processes. GARCH processes exhibit persistence for high-and daily-frequency speculative returns where

∑bj = 1 yields an Integrated

ARCH (IARCH) process or if the volatility follows a GARCH(1,1) processσ2

t = b0 + b1r2

t−j + γ1σ2

t−1 then b1 + γ1= 1 implies an IGARCH process

(Engle and Bollerslev 1986). An example of the IGARCH process is theRiskMetrics estimator which is widely used by professionals in the financialindustry, particularly for risk management purposes. The RiskMetrics is anexponentially weighted moving average filter for r

2

taccording to which for

daily frequency b1 = 0.06 and γ1= 0.94 derived from MSE optimization of

a wide class of financial assets (see for instance JPMorgan Manual). The

27

IGARCH process implies that unconditional variance of the returns process,var(rt), is not finite and consequently the finiteness of up to the fourth mo-ment condition in Horvath et al. (2001a) and Koul (2002) is not satisfied.It is worth mentioning that although an IGARCH models implies that theprocess is second-order nonstationary, it is paradoxically strongly stationary(Nelson, 1990a). Hence it is finiteness of moments in volatility processes andfilters that must be established before the application of the EDF based sta-tistics discussed in the previous section. Similarly for the class of α-stableprocesses an infinite variance is implied when α < 2 and an infinite meanwhen α ≤ 1. Mandelbrot (1963), for instance, presents evidence of α < 2

for speculative returns. Hall and Yao (2002) show that the QML estimatorsof GARCH models with stable heavy tailed distributions (and infinite oreven finite fourth moment) are not only non-normal but very difficult to esti-mate directly using standard parametric methods. Their results suggest thatwhen the GARCH process is driven by stable innovations with infinite mo-ments the t-percentile subsample bootstrap method can yield estimators thatare Normal and consistent. The effects of volatility persistence for GARCHprocesses as well as moments finiteness in α-stable processes for change-pointestimation are examined in the simulation analysis. A related point for theexistence IGARCH or infinite variance effects in speculative returns is theargument that these could be the spurious effects of ignored change-pointsin the conditional variance dynamics (see for instance Diebold, 1986, Lam-oureux and Lastrapes, 1990, Andreou and Ghysels, 2002b). As mentionedabove the normalized returns transformation is robust to change-points inthe conditional variance dynamics for multiplicative heteroskedastic modelssuch as ARCH-type models and hence allows us to focus on the homogeneityof the conditional distribution of the process.

7 Monte Carlo Analysis

7.1 The Monte Carlo design

The simulated returns process r(m),t sampled at frequency 1/m, is generatedby a GARCH(1,1) model (e.g. Bollerslev et al, 1994):

ln pt − ln pt−1/m ≡ r(m),t = µ(m),j,t + σ(m),j,t · z(m),j,t + sj,tσ2(m),j,t = b0,(m),j + b1,(m),jr

2(m),t−1/m + γ(m),jσ

2(m),t−1/m, t = 1, ..., T.

(7.1)

28

where z(m),t is i.i.d.(0, 1) (and the missing j index indicates homogeneity),sj,t is a jump process and σ2

(m),j,t is the volatility process. Drost and Werker

(1996, Corollary 3.2) derive the mappings between GARCH parameters cor-responding to processes with r(m),t sampled with different values of m. Usingthe estimated GARCH parameters for daily data with m = 1, one can com-pute the GARCH parameters α(m), β(m), φ(m), for any other frequencym. Themodels used for the simulation study are representative of the FX financialmarkets, popular candidates of which are taken to be returns on DM/US$,YN/US$ exchange rates. We take the daily results of Andersen and Boller-slev (1998) and compute the implied GARCH(1,1) parameters b0,(m), b1,(m)

and γ(m) for the 5-minute frequency, m = 288 for the 24-hour traded markets,using the software available from Drost and Nijman (1993). The disaggre-gated GARCH coefficients for these models can be found in Andreou andGhysels (2002a, Table 1, p.368). Sample sizes of 1,2 and 5 years are con-sidered that yield daily samples of Tdays = 250, 500, 1250, respectively, andintraday samples of T5min s = 72000, 144000, 360000 5-minute observations,respectively.

The process in (7.1) driven by a homogeneous white noise (z(m),j,t = z(m),t)with neither breaks in the conditional variance (σ(m),j,t = σ(m),t), nor change-points in the jump process (sj,t = st), nor level shifts (µ(m),j,t = 0), denotesthe process under the null hypothesis. The simulated process (7.1), underthe null hypothesis, can be considered as an extension of the ARCH modelin section 3, equations (3.6)-(3.7), to the above GARCH process with jumps.The stochastic process returns is de-volatilized using methods discussed insection 3. Hence we evaluate via simulations whether the theoretical assump-tions of approximate independence or weak dependence are supported for thedaily de-volatilized returns, Xt ≡ rt/σt, under the null hypothesis, using anumber of statistical tests. The distributional properties of de-volatilized re-turns are also evaluated. Complementary simulation and empirical evidenceregarding the data-driven de-volatilized returns in a multivariate frameworkis presented in Andreou and Ghysels (2002c).12 The results here extend toother estimators of σt and functions of ˆXt and the relevant simulation andempirical results are discussed below.

The properties of EDF- and rank-based change-point tests are also exam-

12The interested reader may refer to the aformentioned paper Tables 1 and 5 which

present simulation evidence and Table 7 that provides empirical results, regarding the

distributional and dependence properties of data-driven devolatilized returns.

29

ined following a comprehensive simulation study. Under the null hypothesiswe simulate the performance of the Kolmogorov-Smirnov (KS), Cramer-VonMises (CVM), Bhattacharya and Frierson (BF) and Gordon and Pollak (GP)test procedures and obtain evidence relating to their size. Under the alter-native hypothesis the de-volatilized returns process is assumed to exhibitchange-points (or disruptions) and four independent simulated processes aregenerated in the context of (7.1) in order to assess the power of the abovetests. This experiment extends some of the simulation evidence in Gor-don and Pollak (1994) for permanent shifts in the mean as well as in Bhat-tacharya and Frierson (1981) that address scale shifts in the variance of i.i.d.processes. Here we deal with strongly dependent time series and data-drivende-volatilized processes that yield weakly dependent or approximately inde-pendent sequences which represent the basis for testing change-points in theconditional (and unconditional) distributions. In addition, the simulationanalysis evaluates the theoretical results in Horvath et al. (2001a) and Koul(2002) for different ARCH-type processes and volatility estimators. Someinteresting extensions and supportive simulation evidence for financial timeseries and risk management applications are presented. The objective of thissimulation exercise is the evaluation of the above statistical procedures forrisk management quality control applications.

The analysis examines the power of the de-volatilization procedures andnonparametric tests in detecting disruptions in returns of financial assetsthat have both permanent and temporary effects, that cause changes in thelocation, the scale and the shape of the distribution, that yield a change inthe probability law of extremes, as well as parameter instability in the dy-namics of the second conditional moment.13 We start by examining breaksin the conditional variance dynamics (σ(m),j,t) which can also be thought aspermanent regime shifts in volatility at change points πT (π = .5, .75) anddenoted by HA

1 . Such breaks may be due to an increase in the intercept,b0,(m),j, or a shift in the volatility persistence, b1,(m),j + γ(m),j. The purposeof this alternative hypothesis is to establish that the de-volatilized returnsprocess is relatively robust to structural changes in the conditional varianceparameters. This is a useful result which suggests that in the presence ofstructural change in the conditional volatility parameters (of the ARCH-type

13Change-point tests that focus on the conditional variance dynamics are recently de-veloped, for instance, in Kokoszka and Leipus (2000) and Lavielle and Moulines (2000)and extended and applied for financial time series in Andreou and Ghysels (2002b, 2002c).

30

processes considered) the standardized returns transformation would yield ahomogeneous process. Hence risk management measures based on the con-ditional (rather than the unconditional) distribution would be more robustto structural change. Second, we consider a change in the tails of the d.f.from z(m),j,t ∼ N(0, 1), at t = 1, ..., πT , to either z(m),j,t ∼ N(0, η), (whereη = 1.1,1.2, 1.5, 2 ), or z(m),j,t ∼ t(0, 1;ν), or z(m),j,t ∼ χ(0, 1;ν), ν = 3,6at t = πT + 1, ..., T . We denote this alternative hypothesis as HB

1 . This isan interesting alternative for financial stock returns for at least two reasons:The large volume of the empirical evidence for different heavy-tailed distrib-utions proposed to model stock returns. Also the common practice is to fitsuch alternative distributions to asset returns assuming (and rarely testing)that the sample is homogeneous whereas it might be that certain distrib-utional characteristics (such as heavy tails or asymmetries) may spuriouslyexist due to breaks. Third, we examine a permanent level shift in returns,where µ(m),j,t = 0 at t = 1, ..., πT and there is a level shift to µ(m),j,t = 0.1, 0.5,

at t = πT + 1, ..., T . This alternative hypothesis (HC

1) is often used as the

benchmark for comparing most nonparametric change-point tests. Hence itis included for comparison purposes especially with jump-size change-pointswhich have a transitory character on returns and are often encountered infinancial markets. This brings us to the last source of disruption (HD

1) which

is a change-point in the jump-process, st. We consider both random anddeterministic jump processes, the latter for purposes of fixing (and thus con-trolling) the monitoring aspect of the nonparametric control chart. Therandom jump sequence sr

tfollows a Poisson process (P ) with jump intensity

λ. We consider two types of changes, from N(0, 1) to N (0, 1) + P (λ) as wellas from N(0, 1)+P (λ1) to N(0,1)+P (λ2) where λi denotes the frequency of

the jump such that there is a jump every 250 and 500 days. The size of the

jump remains constant for evaluating the power of the tests to detect breaks

in the jump intensity for alternative transformations of Xt, such as X2

tand

(Xmin

t)2. Change-points in the size of the jump δt are also examined where

δt = eYt− 1 and Yt ∼ N(µ

0, σ0). The deterministic jump process sdt has

jump size µ = 2, 3 and jump frequencies at given regular dates in the dailysample, ∆·tj, (where ∆ = 250, 500 and tj = 1, 2, ...,∆/T ) and zero otherwise(see, for instance, Drost et al (1998) for related empirical evidence in dailyUS$ denominated FX returns). This procedure stops at ∆ and restarts at∆ + 1 and the relevant test outcome is reported. In our experiment thissimple jump process would facilitate the evaluation of the test’s power on a

31

nonparametric control chart (with known change-times).A number of alternative volatility filters, σi

t, are considered below which

differ in terms of the estimation method, sampling frequency and informa-tion set. The data-driven variance filters belong to two classes of volatili-ties: (i) The Range process, R(m),t = supt−1/m<τ<t pτ − inft−1/m<τ<t pτ doesnot involve parameter estimation (ii) The intraday volatility estimates ofthe Quadratic Variation such as the one-day Quadratic Variation filter alsocalled Integrated or Realized Volatility (e.g. Andersen and Bollerslev, 1998,Andersen et al., 2001) defined as the sum of the log of squared returnsr(m),t for different values of m, to produce the daily volatility measure:

σQV 1t =

∑mj=1 r

2(m),t+1−j/m, t = 1, ..., ndays, where for the 5-minute sam-

pling frequency the lag length is m = 288 for financial markets open 24hours per day (e.g. FX markets). Andreou and Ghysels (2002a) also pro-pose a class of quadratic variation filters defined as the Historical QuadraticVariation (HQV) which is the sum of m rolling QV estimates for one day:σHQV 1t = 1/m

∑mj=1QV 1(m),t+1−j/m, t = 1, ..., T as well as the Exponentially

weighted Historical Quadratic Volatility (EHQV), which involves exponen-tial declining weights: σEHQV 1t = A

−1∑m

j=1 a−jQV 1(m),t−1−j/m, t = 1, ..., T

(where a = 0.999 for daily filters). Finally, A is a scaling constant to guar-antee that the filter weights sum to one. These filters will be denoted as(E)(H)QV k with k equal to 1, 2 and 3 days. Themodel-based volatility filters

are: (i) The Exponentially Weighted Moving Average volatility or RiskMet-rics (RM) defined following the industry standard introduced by J.P. Morgan(see Riskmetrics Manual, 1995) as: σRM

t= λσRM

t−1+(1 − λ) r2

t, t = 1, ..., Tdays,

where λ = 0.94 for daily data, rt is the daily return and Tdays is the num-ber of trading days. This represents the IGARCH process. (ii) Quasi MLEestimates of GARCH, σQMLE

t (see, for instance, Gourieroux, 1997) (iii) M-and m.d.-volatility estimates, σMD

t , such as least squares and least absolutedeviations filters denoted by σLSt and σLADt , respectively. First, we examinethe temporal dependence and distributional properties of Xt(i), where i rep-resents the volatility process, under the null hypothesis. Second, given thatthe tests for change-points in Xt(i) involve volatility estimation they therebyraise some interesting questions: Does the estimation method of the varianceaffect the power of these tests? Does the conditioning information set affectthe power of the tests? How do robust volatility estimators affect the powerof change-point tests?

32

7.2 The Monte Carlo results

In this section we evaluate the dependence and distributional properties of thetwo classes of data-based and model-based de-volatilized returns as discussedin sections 3 and 7.1 and we assess the properties of the change point testsdiscussed in sections 4, 5 and 7.1. The simulated Normal and Student’s tGARCH processes are designed after DGPs of representative empirical FXreturns discussed above. Each experiment is performed with 500 replications.Table 1 presents the simulation results for the Bera-Jarque Normality testfor all the standardized returns series. The results in Andreou and Ghysels(2002c) focus on the high-frequency quadratic variation estimators. Underthe Normal-GARCH process, Table 1 presents simulation evidence in favorof the Normality hypothesis for the standardized returns series (at the 5%significance level). This result extends under the t-GARCH hypothesis forrisk-adjusted returns by intraday volatility filters only. Table 2 presentssimulation evidence for first- and second-order temporal independence inall de-volatilized returns based on intraday volatility filters (as opposed toX(RM)), under both Normal or Student’s t GARCH processes. Generallythe simulation results support the assumptions required for the applicationof the above sequential change-point tests in the daily de-volatilized returnsusing intraday volatility filters at the 5-minute sampling frequency.

The properties of the change-point tests are discussed based on the sim-ulation results reported in Tables 3 and 4 for EDF and rank-based statistics,respectively. The simulated statistic in Table 3a refers to the Kolmogorov-Smirnov statistic KS3 defined in (4.3).14 The simulation results for KS3applied to ˆX2

t≡ (rt − µ(m),t)

2/σ2(m),t are summarized as follows:

15

(i) The test has good size properties even for relatively small sample

14It is important to note at the outset that the asymptotic critical value of c0.05 = 1.63

yields a seriously undersized tests (with extremely low power). Similar results on thedifference between the asymptotic and simulated critical value of the Kolmogorov Smirnovstatistic for goodness of fit purposes and quantiles processes are reported in Horvath,Kokoszka and Teyssière (2001a) and Horvath, Jach and Kokoszka (2001) for ARCH aswell as i.i.d. processes. Instead Horvath, Jach, Kokoszka (2001) show that the simulatedcritical value for the Kolmogorov-Smirnov statistic is c0.05 = 0.775 which is adopted inTable 3a.

15The power of the tests are better for the squared de-volatilized process X2

trather than

Xt mainly because we are dealing with change points in the tails of distributions. Thisresult complements the theoretical results of Horvath et al. (2001a) and Koul (2002) whoconsider the empirical process for X2

tin ARCH type models.

33

T = 250 and for alternative distributions such as Normal, Student’s t, Chi-square and the Normal process with occasional jumps generated by a Poissonprocess.

(ii) The KS3 test applied to X2

talso enjoys good power properties under

many types of alternatives of changes in the tail of the distribution including achange-point in the process governing jumps. For instance, it detects changepoints in the variance of the Normal distribution even when the size of thechange is small (e.g. 10% increase) particularly for sample sizes greater than500 observations. This result applies if the change point occurs in the middle(0.5T ) or towards the end of the sample (0.75T ). KS3 is very successful indetecting mixtures of normals even for changes in the variance beyond a 20%increase (e.g. from N (0, 1) to N (0, 1.2), defined as HA

1). Similarly, KS3 has

impressive power for tail changes due to genuine distributional heterogeneitysuch as a change from N (0, 1) to t(0, 1; ν) when ν = 3, 6 (HB

1) even for

small samples, T = 250 and any break point, π. However, the power of KS3

appears to be relatively lower for detecting a change-point in the shape of thedistribution from N(0,1) to χ(0, 1; ν) when ν = 3, 5 (HB

1) except for samples

T > 500.(iii) The last alternative hypothesis, HD

1, extends the appropriateness of

KS3 in detecting change-points in the tails even when this is due to occa-sional jumps. Given the interest in jumps and extremes for risk management,we learn from the the simulations that KS3 has very good power in detect-ing a break from N(0,1) to N(0, 1) + Poisson(λ), even for T = 250 andλ = 1 (defined as HD

1). Nevertheless, the test’s power drops (to 20% for

T > 500) when it is challenged in detecting a change point in the Poissonprocess intensity, λ. Although this is acknowledged as a subtle change-pointprocess we show that alternative transformations of de-volatilized returns canbe more appropriate for change-points in the jump process. For instance, theblock minima transformation ˆX

min

n,t can be used as the underlying processfor estimating tails and extremes. Following the discussion in section 6 weconstruct ˆX

min

n,t by focusing on non-overlapping blocks, n, of 20 tradingdays and constructing the sequence of block minima. The latter can beconsidered sequence of declines in the daily returns of a financial portfolioover monthly horizons which are of particular interest to risk managers, in-vestors and institutions. The simulations for (Xmin

n,t )2 examine if the testshave power in detecting whether the probability law of extreme downwardjumps has changed. It is shown (in the lower panel of Table 3a) that despitethe small sample created by sampling one observation in each block (denoted

34

by Tblock), the KS3 test has good size and power in detecting changes in thejump process when λ2 > 2λ1 and Tblock ≥ 75 observations.

In Table 3b we also show the simulated performance of the Cramer-VonMises test CVM1 defined in (4.5) which shares many similar general featureswith the asymptotic KS3 statistic (c0.05 = 2.884 Blum et al., 1961) i.e. itis undersized except for large samples (T = 1250), lacks power in detectingsmall scale changes in N(0, η) for η < 1.2 change, t(0, 1; ν) and χ(0, 1; ν),ν = 3, 5, for sample sizes T < 500. However its power improves with thesample size which implies that the simulation results support the asymptoticdistribution of CVM1 at T > 1250 (except for the challenging alternative(HD

1) that generates a change point in the intensity of the jump, λ). In terms

of the asymptotic approximations of EDF statistics the simulation results

suggest that CV M1 has overall higher quantitative power than KS3.16

Turning now to the rank-based tests we present the simulation results

of the BF test in Table 4. As before, the experiment simulates a GARCH

process at 5-minute sampling frequency for a 24-hour traded market and an-

nual horizons of 1,2 and 5 years. The daily range process discussed in section

3.1 is employed for de-volatilizing returns. The general conclusion from Table

4a is that the BF for ˆX2

thas good size power in detecting change-points in

the scale of the Normal, in the tails (e.g. from Normal to Student’s t) andshape of distributions (e.g. from Normal to χ2), for small shifts (e.g. 10%increase) and samples T > 500 and any change-point π = 0.5, 0.75. For thebenchmark alternative hypothesis of a location change the BF test appliedto X(m),t has good size and impressive power properties for a even a small(10%) shift in µ(m),j,t. As a final remark on the simulation analysis we showevidence in Table 4b that the de-volatilized returns are robust to parame-ter instability due to either small or large changes in the constant (b0,(m),j)or persistence (γ(m),j) of the GARCH coefficients. These results suggestthat the de-volatilized process is robust to conditional variance structuralparameter instabilities and presents the appropriateness of conditioning onvolatility when estimating the distribution tail probability in financial riskmanagement. Hence the de-volatilized returns process captures not only thenon-linear dependence structure but is also robust to changes in parametersof volatility (shown here for a GARCH model).

16For conciseness we report only the KS3 results in Table 1 for the adjusted simulated

critical level.

35

8 Empirical Illustrations

To be included

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42

Table 1: Monte Carlo Simulations of Normality Test Results of

daily de-volatilized FX returns ˆXt ≡ rt/σt or ˆXt(i), i = σt

N-GARCH t-GARCH

Xt(i) Ex.Krt. Ex.Sk. J-B Ex.Krt. Ex.Sk. J-B

Xt(RM) -0.119 0.002 2.023 -1.245 0.001 81.189

p-value 0.463 0.000

Xt(QV 1) -0.049 0.007 -1.707 -0.030 0.004 1.776

p-value 0.524 0.521

Xt(QV 2) -0.010 0.006 1.715 0.004 0.003 1.824

p-value 0.561 0.521

Xt(QV 3) 0.031 0.006 1.931 0.400 0.003 2.041

p-value 0.562 0.499

Xt(HQV 1) -0.005 0.005 1.741 0.028 0.004 1.974

p-value 0.563 0.507

Xt(HQV 2) 0.048 0.005 2.078 0.091 0.003 2.632

p-value 0.546 0.446

Xt(HQV 3) 0.092 0.006 2.644 0.136 0.003 3.379

p-value 0.512 0.389

Xt(EHQV 1) -0.031 0.005 1.687 -0.008 0.004 1.802

p-value 0.548 0.522

Xt(EHQV 2) 0.044 0.007 1.931 0.304 0.005 8.923

p-value 0.551 0.198

Xt(EHQV 3) 0.122 0.007 2.888 0.685 -0.001 32.162

p-value 0.479 0.019

Note: The simulation design is described in section 3.1. We consider Normal and Student’s t GARCH

processes. The volatility filters are defined in section 2.3.3. The standardised returns are tested for

Normality and the Jarque-Bera (J-B) Normality test statistic is reported with the respective p-values as

well as the excess kurtosis (Ex.Krt.) and skewness (Ex.Sk.). The total sample size is 2500 observations

which is adjusted for the subsample of 2250 due to the standardized returns by rolling volatilities.

1

Table 2: Monte Carlo Simulations of Temporal Dependence Test results of

daily de-volatilized FX returns ˆXt ≡ rt/σt or ˆXt(i), i = σt

N-GARCH t-GARCH

Xt(i) L-B(15) M-L(15) ARCH(15) L-B(15) M-L(15) ARCH(15)

Xt(RM) 16.095 23.026 1.432 14.700 41.553 3.679p-value 0.454 0.213 0.239 0.516 0.004 0.000

Xt(QV 1) 15.423 14.325 0.948 15.178 14.972 0.988p-value 0.491 0.527 0.534 0.491 0.504 0.512

Xt(QV 2) 15.388 14.432 0.956 15.146 15.077 0.995p-value 0.489 0.519 0.525 0.493 0.499 0.507

Xt(QV 3) 15.351 14.545 0.963 15.139 15.169 1.001p-value 0.489 0.514 0.520 0.494 0.495 0.503

Xt(HQV 1) 15.393 14.360 0.951 15.121 15.082 0.996p-value 0.490 0.525 0.531 0.494 0.499 0.510

Xt(HQV 2) 15.369 14.480 0.958 15.119 15.236 1.260p-value 0.489 0.518 0.524 0.495 0.492 0.327

Xt(HQV 3) 15.375 14.726 0.975 15.126 15.402 1.722p-value 0.487 0.509 0.513 9.495 0.486 0.155

Xt(EHQV 1) 15.362 14.491 0.958 15.141 20.448 1.260p-value 0.490 0.518 0.525 0.493 0.289 0.327

Xt(EHQV 2) 15.349 14.596 0.963 14.965 28.510 1.722p-value 0.490 0.521 0.528 0.505 0.132 0.155

Xt(EHQV 3) 15.347 14.595 0.962 15.157 28.500 1.721p-value 0.489 0.590 0.528 0.495 0.131 0.154

Note: The simulation design is described in section 3.1. We consider Normal and t GARCH processes. The

volatility filters are defined in section 2.3.3. We examine the temporal dependence of standardised returns

using the Portmanteau Ljung-Box (L-B), McLeod-Li (M-L) and ARCH tests. The number in parentheses

refer to the lag length. Similar results were obtained for alternative lag lengths. Similar results were

obtained for alternative lag lengths. The total sample size is 2500 observations which is adjusted for the

subsample of 2250 due to the standardized returns by rolling volatilities.

2

Table 3a: Simulations results for the Kolmogorov-Smirnov Statistic, KS_3

Kolmogorov Smirnov Statistic* Simulated critical value from Horvath, Jach, Kokoszka (2001), c_a=0.775, a=5%

Unweighted KS, KS_3Squared Returns Transformation Squared Returns Transformation

Size Standard Normal Chi-square and Student`s tT 250 500 1250 T 250 500 1250N(0,1) 0.06 0.08 0.10 Chi(5) 0.06 0.08 0.09

t(6) 0.04 0.08 0.06

Power Mixture of Normals Change in dfT 250 500 1250 T 250 500 1250Change-point at 0.5T Change-point at 0.5TN(0,1) to N(0,s1) N(0,1) to Chi-sq & ts1=1.1 0.16 0.26 0.40 Chi(0,1;3) 0.24 0.46 0.96s1=1.2 0.30 0.62 0.96 Chi(0,1;5) 0.10 0.14 0.48s1=1.5 0.94 1.00 1.00 t(0,1;3) 1.00 1.00 1.00s1=2 1.00 1.00 1.00 t(0,1;6) 0.66 0.88 1.00Change-point at 0.75T Change-point at 0.75TN(0,1) to N(0,s1) N(0,1) to Chi-sq & ts1=1.1 0.12 0.12 0.20 Chi(0,1;3) 0.10 0.34 0.78s1=1.2 0.16 0.36 0.76 Chi(0,1;5) 0.06 0.12 0.38s1=1.5 0.64 0.98 1.00 t(0,1;3) 0.98 1.00 1.00s1=2 1.00 1.00 1.00 t(0,1;6) 0.36 0.75 0.98

Squared Returns Transformation Squared Block Minima of De-volatilized ReturnsBlock size=20 obs.

Size SizeN(0,1)+Poisson(lambda)** N(0,1)+Poisson(lambda)T 250 500 1250 T 1500 5000P(1) 0.08 - - Blocks 75 250P(2) - 0.10 - P(1) 0.08 0.06P(5) - - 0.08 P(6) 0.02 0.06

Power PowerChange in Jump Process Change in Jump ProcessT 250 500 1250 T 1500 5000

Blocks 75 250Change-point at 0.5T Change-point at 0.5TN(0,1) to Normal+Jump N(0,1) to Normal+JumpP(1) 0.78 - - P(2) 0.94 1.00P(2) - 1.00 - P(6) 0.96 1.00P(5) - - 1.00Change-point at 0.5T Change-point at 0.5TN(0,1)+Jump1 to N(0,1)+Jump2 N(0,1)+Jump1 to N(0,1)+Jump2P(1),P(2) 0.08 - - P(1),P(2) 0.08 0.14P(2),P(4) - 0.20 - P(1),P(6) 0.16 0.32P(5),P(10) - - 0.20

*Note that the statistics KS_1 and KS_2 have a degenerate asymptotic df.**These random variables are also normalized like the rest in the Table.

Table 3b: Simulations results for the Cramer-Von Mises Statistic, CVM1

Cramer Von Mises StatisticSquared Returns Transformation

Unweighted CVM, CVM_1 Asymptotic critical values c_a=2.884, a=5%

Size Standard Normal Chi-square and Student`s t N(0,1)+Poisson(lambda)*T 250 500 1250 T 250 500 1250 T 250 500 1250N(0,1) 0.00 0.00 0.00 Chi(5) 0.00 0.00 0.04 P(1) 0.00 - -

t(6) 0.00 0.00 0.02 P(2) - 0.00 -P(5) - - 0.00

Power Mixture of Normals Change in df Change in Jump ProcessT 250 500 1250 T 250 500 1250 T 250 500 1250

Change-point at 0.5T Change-point at 0.5T Change-point at 0.5TN(0,1) to N(0,s1) N(0,1) to Chi-sq & t N(0,1) to Normal+Jumps1=1.1 0.02 0.02 0.12 Chi(0,1;3) 0.00 0.00 1.00 P(1) 0.14 - -s1=1.2 0.02 0.04 0.76 Chi(0,1;5) 0.00 0.00 0.08 P(2) - 0.68 - s1=1.5 0.16 0.96 1.00 t(0,1;3) 0.90 1.00 1.00 P(5) - - 1.00s1=2 0.94 1.00 1.00 t(0,1;6) 0.06 0.34 1.00

Change-point at 0.75T Change-point at 0.75T Change-point at 0.5TN(0,1) to N(0,s1) N(0,1) to Chi-sq & t N(0,1)+Jump1 to N(0,1)+Jump2s1=1.1 0.02 0.00 0.04 Chi(0,1;3) 0.00 0.00 0.24 P(1),P(2) 0.00 - -s1=1.2 0.02 0.02 0.40 Chi(0,1;5) 0.00 0.00 0.04 P(2),P(4) - 0.00 -s1=1.5 0.04 0.58 1.00 t(0,1;3) 0.44 1.00 1.00 P(5),P(10) - - 0.02s1=2 0.38 1.00 1.00 t(0,1;6) 0.00 0.12 0.88

*These random variables are also normalized like the rest in the Table.

Table 4a: The Rank-based tests for alternative change-pointsin the de-volatilized Returns/sqrt(Range) process

Bhattacharya and Frierson Statistic Critical value c_a=1.645Trials=100, Sample=0.05T:T

De-volatilization: Returns/Range generated by a Normal GARCH(1,1)*Squared De-volatilized Returns De-volatilized Returns Squared De-volatilized Returns

Size Standard Normal Size Standard Normal Chi-square and Student`s tT 250 500 1250 T 250 500 1250 T 250 500 1250N(0,1) 0.05 0.04 0.06 0.04 0.10 0.09 Chi(5)

t(6) 0.06 0.06 0.05Power Mixture of Normals Power Mixture of Normals Power Change in dfT 250 500 1250 T 250 500 1250 T 250 500 1250

Change-point at 0.5T Change-point at 0.5T Change-point at 0.5TN(0,1) to N(0,s1) N(0,1) to N(m1,1) N(0,1) to Chi-sq & ts1=1.1 0.08 0.21 0.44 m1=0.1 1.00 1.00 1.00 Chi(0,1;3)s1=1.2 0.31 0.57 0.88 m1=0.5 1.00 1.00 1.00 Chi(0,1;5)s1=1.5 0.85 1.00 1.00 t(0,1;3) 1.00 1.00 1.00s1=2 1.00 1.00 1.00 t(0,1;6) 0.82 0.97 1.00Change-point at 0.75T Change-point at 0.75T Change-point at 0.75TN(0,1) to N(0,s1) N(0,1) to N(m1,1) N(0,1) to Chi-sq & ts1=1.1 0.08 0.09 0.20 m1=0.1 0.99 1.00 1.00 Chi(0,1;3)s1=1.2 0.14 0.28 0.42 m1=0.5 1.00 1.00 1.00 Chi(0,1;5)s1=1.5 0.46 0.78 1.00 t(0,1;3) 1.00 1.00 1.00s1=2 0.95 0.97 1.00 t(0,1;6) 0.53 0.75 0.96

Squared De-volatilized Returns Squared Block Minima of De-volatilized ReturnsBlock size=20 obs.

Size SizeN(0,1)+Poisson(lambda) N(0,1)+Poisson(lambda)T 250 500 1250 T 500 1500 5000

Blocks 25 75 250P(1) 0.04 - - P(2)P(2) - 0.03 - P(6)P(5) - - 0.04 P(12)Power PowerChange in Jump Process Change in Jump ProcessT 250 500 1250 T 500 1500 5000

Blocks 25 75 250Change-point at 0.5T Change-point at 0.5TN(0,1) to Normal+Jump N(0,1) to Normal+JumpP(1) 0.21 - - P(2)P(2) - 0.22 - P(6)P(5) - - 0.43 P(12)

Change-point at 0.5T Change-point at 0.5TN(0,1)+Jump1 to N(0,1)+Jump2 N(0,1)+Jump1 to N(0,1)+Jump2P(1),P(2) 1.00 - - P(2),P(4)P(2),P(4) - 1.00 - P(3),P(6)P(5),P(10) - - 1.00 P(6),P(12)*Similar results are found for the t-GARCH process and the relevant powerresults are not reported here for economy purposes.

Table 4b: The de-volatilized returns process is robust tochange-points in the conditional variance dynamics.

Bhattacharya and Frierson Statistic Critical value c_a=1.645Trials=100, Sample=0.05T:T

De-volatilization of GARCH(1,1) processSquared De-volatilized Returns

Returns/sqrt(GARCH) Returns/sqrt(RM) Returns/sqrt(Range)

SizeT 250 500 1250 250 500 1250 250 500 1250N(0,1) 0.03 0.00 0.00 0.03 0.00 0.00 0.05 0.04 0.06

PowerChange-point at 0.5TChange-point in the constant of the conditional variance2*b_0 0.01 0.00 0.01 0.01 0.01 0.00 0.06 0.10 0.1010*b_0 0.00 0.00 0.00 0.05 0.01 0.04 0.03 0.08 0.07

Change-point in the dynamics of the conditional variance0.5*gamma 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.000.1*gamma 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.01 0.01