statistical surveillance of volatility forecasting models

JournalofFinancialEconom etrics, 2011, Vol. 0, No. 0, 1–31

Statistical Surveillance of VolatilityForecasting Models

VASYL GOLOSNOY

Institute of Statistics and Econometrics, University of Kiel

IRYNA OKHRIN

European University Viadrina

WOLFGANG SCHMID

European University Viadrina

ABSTRACTThis paper elaborates sequential procedures for monitoring the va-lidity of a volatility model. A state-space representation describes dy-namics of daily integrated volatility. The observation equation relatesthe integrated volatility to its measures such as the realized volatility orbipower variation. On-line control procedures, based on volatility fore-casting errors, allow us to decide whether the chosen representationremains correctly specified. A signal indicates that the assumed volatil-ity model may no longer be valid. The performance of our approach isanalyzed within a Monte Carlo simulation study and illustrated in anempirical application for selected U.S. stocks. ( JEL: C22, C53, G17)

KEYWORDS: control charts, integrated volatility, jumps, realized volatility,state-space model

This paper concentrates on modeling and forecasting daily integrated volatility ofasset returns relying on intraday information. A parsimonious linear state-spacerepresentation is selected in order to reflect the log-volatility dynamics. This repre-sentation has, however, a rather rigid structure, so its ability to provide propervolatility forecasts should be questioned at every new point in time. For thispurpose, we suggest a methodology for on-line decisions to check whether thepredetermined volatility model remains valid. Our methodology is related to the

The authors are grateful to Yacine Aıt-Sahalia, Francis Diebold, Eric Ghysels, Roman Liesenfeld,Nour Meddahi, Per Mykland, Uta Pigorsch, George Tauchen, and two anonymous referees fortheir helpful comments and suggestions, which improved the paper significantly. Address corre-spondence to Wolfgang Schmid, Department of Statistics, European University Viadrina, GrosseScharrnstr. 59, 15230 Frankfurt (Oder), Germany, or e-mail: [email protected].

doi: 10.1093/jjfinec/nbr017

c© The Author 2011. Published by Oxford University Press. All rights reserved.

For permissions, please e-mail: [email protected].

Journal of Financial Econometrics Advance Access published December 20, 2011 at L

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2 Journal of Financial Econometrics

monitoring approaches of Horvath, Kokoszka, and Zhang (2006) and Andreou andGhysels (2008). A signal would indicate that the model fails to produce appropri-ate volatility forecasts.

There are various approaches for modeling volatility dynamics, which as-sess the well-documented stylized facts found in the daily volatility series (Tsay2005, 135). The ARFIMA (Andersen et al. 2003) or AR models of higher order(Aıt-Sahalia and Mancini 2008) explicitly account for the long memory volatilityproperty. Alternatively, the models of Engle and Rangel (2008) and Engle, Ghysels,and Sohn (2009) separate a slowly evolving long-run volatility component fromshort-run fluctuations. Other promising approaches, such as MIDAS (Ghysels,Santa-Clara, and Valkanov 2006) or HAR (Corsi 2009), use variables sampled atdifferent frequencies to explain volatility dynamics. All these flexible but rathersophisticated methods provide a plausible description of daily volatilities.

The long memory volatility feature may point, however, to possible structuralchanges in the model parameters. A mixture of simple models can provide volatil-ity patterns similar to those of the more advanced approaches (Chen, Hardle, andPigorsch 2010). Following this argument, we suggest the use of a parsimoniousmodel to capturing the dynamics of integrated volatility. This model should beexploited until it fails to provide proper volatility forecasts, so it is necessary tomake a decision about model validity at each new point in time. Such sequentialtask corresponds to the problem of on-line discrimination between locally validvolatility models (Hardle, Herwartz, and Spokoiny 2003).

This paper exploits a linear state-space representation for the log-volatilityprocess. The daily volatility measures, based on intraday information, enter the ob-servation equation. The log-volatility dynamics is assessed with the AR(1) processin the state equation. Such a simple model is justified by the data analysis from theinitial estimation period in our empirical study. This model, however, lacks flexi-bility in accounting for changing economic situations. For this reason, the processof volatility forecasting errors should be used for sequential monitoring of modelvalidity.

Control charts, borrowed from statistical process control, are appropriate sta-tistical decision rules for the surveillance of volatility models (Montgomery 2005).A control chart consists of a control statistic and critical limits, which depend onthe presumed (known) stochastic properties of the monitored process. The charttriggers a signal if the control statistic crosses the critical limits for the first time.A signal indicates the possibility that assumptions concerning the process of inter-est are no longer satisfied and questions the overall model validity. A good controlchart provides no (false) signal for a long time if the model remains correct andtriggers a signal immediately after a change occurs. The reason for each obtainedsignal should be investigated carefully. An isolated signal could be interpreted asan outlier. A sequence of signals within a short period of time, however, wouldpoint to the possibility that the chosen model fails to provide proper volatilityforecasts. In this paper, we apply different types of control charts for volatilitymodel surveillance. Their detecting ability is investigated both in Monte Carlo

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GOLOSNOY ET AL. | Statistical Surveillance of Volatility Forecasting Models 3

simulations and in empirical study. Note that this sequential approach differs fromthe fixed sample tests for model stability of Andrews (2003) and Maheu, Reeves,and Xie (2010) because it is unknown how many periods elapse until a signal oc-curs. Thus, it could be seen as an extension of sequential procedures proposedby Chu, Stinchcombe, and White (1996), Andreou and Ghysels (2006, 2008), andHorvath, Kokoszka, and Zhang (2006).

The suggested monitoring methodology is illustrated in the empirical study,which considers four highly liquid stocks traded on the New York Stock Exchange(NYSE). The dataset covers the intraday observations for 458 trading days fromDecember 2003 until November 2005. The state-space volatility model is estimatedon the basis of the first 100 days, and the rest of the sample is used for on-line moni-toring of its validity. Although we obtain some separate signals, there is no strikingevidence that the chosen model is invalid for three of four considered stocks. Thecharts applied to the General Motors (GM) stock, however, provide a large numberof highly clustered signals starting from March 2005. This indicates that the linearstate-space volatility model with AR(1) state equation fails in the GM case. Ourfindings are supported by the publicly available information concerning the GMstock during the considered time period. Jumps in the price process are found forall stocks, but jump dates usually do not coincide with signal dates. This suggeststhat jumps and monitoring signals could be of a different nature, similar to thefindings of Choi, Yu, and Zivot (2010), Perron and Qu (2010), and Maheu, Reeves,and Xie (2010). Thus, our methodology enables differentiation between valid andinvalid volatility models even in the presence of jump components in the priceequations.

The rest of the paper is organized as follows. Section 1 introduces basic defini-tions and observable measures of the daily volatility, which rely on intraday infor-mation. Section 2 presents the linear state-space representation for log-volatilityand derives stochastic properties of the corresponding forecasting errors. More-over, it introduces control charts for volatility model surveillance. In Section 3, weinvestigate the detecting ability of control schemes in a Monte Carlo simulationstudy. Section 4 presents an empirical application, while Section 5 concludes thepaper. The estimation procedure details and proofs are given in the Appendices.

1 INTEGRATED VOLATILITY AND ITS MEASURES

Let the log asset price y(t), t > 0 follow a univariate jump-diffusion stochasticprocess

dy(t) = μ(t)dt+ σ(t)dW(t) + dJ(t), (1)

where μ(t) is a locally bounded predictable drift process, σ(t) is a strictly positivespot volatility process, and W(t) is a standard Brownian motion. The processesμ(t) and σ(t) are presumed to be stochastically independent of W(t). The com-ponent J(t) is a compound Poisson process. In the case of no jumps, the log asset

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price y(t), t > 0 simplifies to an Ito process, given by

dy(t) = μ(t)dt+ σ(t)dW(t). (2)

The log returns RtΔ are calculated as RtΔ = y(tΔ) − y((t − 1)Δ) for a fixed in-terval of length Δ ⊂ R+, t ∈ N. For the day t, the log return is defined asRt = y(t) − y(t − 1) with Δ = 1. It follows a conditional normal distribu-tion Rt|μt, σ2

t ∼ N (μt, σ2t ) in case of no jumps. The expectation is defined as

μt =∫ t

t−1 μ(u)du, while the integrated volatility σ2t is given by

σ2t =

∫ t

t−1σ2(u)du. (3)

This study focuses on the daily integrated volatility σ2t , which is not directly ob-

servable. Fortunately, availability of ultrahigh frequency intraday observations al-lows to construct its precise estimators. Andersen and Bollerslev (1998) popular-ized the realized volatility RVt measure, which is defined as the sum of squaredintraday returns

RVt =M

∑m=1

R2t,m =

M

∑m=1

[y(tm)− y(tm−1)]2, t− 1 = t0 < t1 < ∙ ∙ ∙ < tM = t.

(4)

The quantity RVt converges in probability to the integrated volatility σ2t with

maxm(tm − tm−1) → 0, M → ∞ under certain regularity conditions (Andersenand Bollerslev 1998). Barndorff-Nielsen and Shephard (2002) derive the asymp-totic distribution of RVt, assuming that the log price follows an Ito process as inEquation (2). Moreover, they consider the log transformation of RVt and provideits asymptotic distribution:

[log(RVt)− log(σ2t )]

/(vlogRV

t

)1/2 L−→ N (0, 1), with (5)

vlogRVt = 2M−1

∫ t

t−1σ4(u)du

/(∫ t

t−1σ2(u)du

)2

.

A feasible estimator of the quantity vlogRVt can be obtained from intraday

observations by

vlogRVt = 2

3

M∑

m=1R4

t,m

/(M∑

m=1R2

t,m

)2

. (6)

RVt remains a consistent estimator of the integrated volatility in the absence ofjump components. RVt appears to be inconsistent, however, for a nonzero jump

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component at day t owing to plimM→∞RVt = σ2t + ∑Nt

k=1 κ2t,k, where {κt,k}

Ntk=1 are

the jump sizes and Nt denotes the number of jumps at day t (Andersen, Bollerslev,and Diebold 2002). To overcome this problem, Barndorff-Nielsen and Shephard(2004) introduce the bipower variation BVt measure, given as

BVt =π

2M

M− 1

M−1

∑m=1

|Rt,m||Rt,m+1|, (7)

and derive its asymptotic distribution. The bipower variation remains a consis-tent estimator of σ2

t even in the presence of a nonzero jump component, so thatplimM→∞BVt = σ2

t . Note that RVt is a more efficient estimator than BVt if jumpcomponents are absent. The asymptotic distribution of the log(BVt) is given as (cf.Bickel and Doksum 2001, 461):

[log(BVt)− log(σ2t )]

/(vlogBV

t

)1/2 L−→ N (0, 1), with (8)

vlogBVt =

(π2

4+ π − 3

)

M−1∫ t

t−1σ4(u)du

/(∫ t

t−1σ2(u)du

)2

.

A feasible estimator of the quantity vlogBVt is given by

vlogBVt =

(π2

4+ π − 3

)π2

4M

M− 3

M−3

∑m=1

|Rt,m||Rt,m+1||Rt,m+2||Rt,m+3|

/

BV2t .

(9)

The distance between RVt and BVt can be used for constructing tests for thepresence of a jump component. Barndorff-Nielsen and Shephard (2004) propose,among others, a test for jumps using the difference log(RVt) − log(BVt), whichis mostly consistent with our modeling in Section 2.1. The test statistic entails themaximum adjustment in the denominator

Qt = [log(RVt)− log(BVt)]

/[(π2

4+ π − 5

)

M−1 max

(

1,TPt

BV2t

)]1/2

, (10)

where TPt is the realized tripower quarticity statistic, given by

TPt =M4

[ √π

Γ(7/6)

]3 MM− 2

M−2

∑m=1

|Rt,m|4/3|Rt,m+1|4/3|Rt,m+2|

4/3, (11)

where Γ(∙) is the Gamma function. The statistic Qt follows asymptotically the stan-dard normal distribution under the null hypothesis of no jump component, that is,

QtL−→ N (0, 1). Alternative tests and a further discussion about jump detection

can be found in Huang and Tauchen (2005) and Barndorff-Nielsen and Shephard(2006).

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Another serious problem in measuring σ2t is the presence of market

microstructure noise (MMN), which prevents usage of most frequent tick obser-vations directly for estimation purposes and should be accounted by volatilityforecasting (Ghysels and Sinko 2011). The impact of MMN can be reduced by de-creasing the intraday sampling frequency (Hansen and Lunde 2006). Alternatively,the MMN impact can be mitigated by using staggered returns for calculating thegeneralized bipower variation (SBVt) (Huang and Tauchen 2005):

SBVi,t =π

2M

M− 1− i

M−(1+i)

∑m=1

|Rt,m||Rt,m+(1+i)|, i > 0, (12)

where the nonnegative integer i denotes the offset. SBVt is immune against au-tocorrelated MMN because of skipping highly correlated neighborhood obser-vations. For example, the choice i = 1 would be helpful against an MA(1)MMN process. Hereafter, we set i = 1 and skip the index i for notation sim-plicity. SBVt remains consistent in the presence of jumps with the asymptoticdistribution:

[log(SBVt)− log(σ2t )]

/(vlogSBV

t

)1/2 L−→ N (0, 1), with (13)

vlogSBVt =

(π2

4+ π − 3

)

M−1∫ t

t−1σ4(u)du

/(∫ t

t−1σ2(u)du

)2

.

The corresponding estimator of the quantity vlogSBVt is given by

vlogSBVt =

(π2

4+ π − 3

)π2

4M

M− 6

M−6

∑m=1

|Rt,m||Rt,m+2||Rt,m+4||Rt,m+6|

/

SBV2t .

(14)

2 MONITORING DAILY VOLATILITY MODEL

In general, the volatility consists of both continuous and jump components, whichare often presumed to be independent of each other. Jumps introduce discontinu-ities into the price process (Merton 1976). Since the number of jumps per day andtheir sizes are random, it is difficult to detect, measure, and predict them. This pa-per models only the continuous volatility component. The presence of jumps istested later in the empirical Section 4.

2.1 State-Space Representation of Daily Volatility

The unobservable daily integrated volatility σ2t is measured by RVt, BVt, or SBVt.

A simple state-space representation for ωt = log(σ2t ) and its feasible measure st is

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given by:

ωt+1 − a= φ(ωt − a) + εt+1, εt+1 ∼ N (0, q), |φ| < 1, (15)

st = ωt + γt, γt ∼ N (0, vt), (16)

where st is log(RVt), log(BVt), or log(SBVt). The vector of unknown model pa-rameters θ = (a, φ, q)′ includes the unconditional expectation of the log inte-grated volatility a, the innovations variance q in the state equation, and the AR

factor φ. The variance vt is considered to be observable via the measures vlogRVt in

Equation (6), vlogBVt in Equation (9), or vlogSBV

t in Equation (14).The state equation (15) assumes AR(1) dynamics for ωt, whereas the observa-

tion equation (16) provides measurement errors γt. The innovations {εt} and {γt}are assumed to be uncorrelated with each other and not autocorrelated. Zero au-tocorrelation in error components presumes that the impact of MMN is negligiblysmall owing to the properly constructed volatility estimators. The assumption ofuncorrelated measurement errors and innovations is quite strong. A suitable exten-sion by assuming nonzero correlations, however, would lead to a time-dependentcorrelation structure. This problem is not easy to handle in a general case, so wemake this simplification in this paper. The introduced representation (15)–(16) re-sembles one of the simple approaches considered by Fleming and Kirby (2003),where a set of volatility models is compared extensively within both simulationand empirical studies. Of course, the dynamics of ωt in the state equation canbe modeled by any other ARMA linear time series. Since the innovations are as-sumed to follow a Gaussian distribution, the system parameters can be estimatedby a usual maximum likelihood (ML) approach. Moreover, the quasi-ML estima-tion procedure would provide standard errors, which are robust for nonnormalerror components. The estimation details are discussed in Appendix A.

The simple model introduced above is assumed to be locally valid. Advancedvolatility forecasting models, such as ARFIMA (Andersen et al. 2003), MIDAS-family models (Ghysels, Santa-Clara, and Valkanov 2006), higher-order AR models(Aıt-Sahalia and Mancini 2008), and HAR-models (Corsi 2009), are able to accountfor long memory, seasonality, and other important stylized facts in daily volatilityseries. Alternatively, long memory may be an indicator of the presence of struc-tural breaks in the model parameters (Martens, van Dijk, and de Pooter 2009), sim-ilar to the long memory and/or unit root problems in linear models (Banerjee andUrga 2005). A mixture of simple autoregressive models with different parameterscan provide the proper unconditional volatility distribution with slowly decayingautocorrelations and other important empirical features (Hardle, Herwartz, andSpokoiny 2003). Such locally constant volatility models are suitable for producingshort-term forecasts (Chen, Hardle, and Pigorsch 2010). Consequently, we advo-cate the local AR(1) structure of the state equation (15), which is justified by theempirical modeling. This local model should be used as long as it is able to provideproper volatility forecasts. Its simplicity makes further analysis and interpretationof results more transparent.

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2.2 Distribution of Forecasting Errors

The model, introduced in Equations (15) and (16), should be exploited forvolatility forecasting purposes. Its linearity enables straightforward forecastsfor the day t via a recursive representation given the information set It−1 ={s1, v1, . . . , st−1, vt−1}. Then, the best linear forecasts ωt|t−1 and st|t−1 of ωt andst, respectively, are their projections onto It−1 (Hamilton 1994, 134):

ωt|t−1 = a+ φ(ωt−1|t−2 − a) +φpt−1|t−2

pt−1|t−2 + vt−1(st−1 −ωt−1|t−2), (17)

st|t−1 = ωt|t−1, (18)

with s1|0 = ω1|0 = a. The conditional variance pt|t−1 = var(ωt − ωt|t−1) isupdated as

pt|t−1 = φ2 pt−1|t−2

pt−1|t−2 + vt−1vt−1 + q, (19)

with the starting value p1|0 = q/(1 − φ2). The observable forecasting errorsηt are calculated as the difference between the measure st and its conditionalforecast st|t−1:

ηt = st − st|t−1. (20)

The stochastic properties of {ηt} are derived in Proposition 1, which is proven inAppendix B.

Proposition 1. Assuming the price equation (2), the state-space representationas in Equations (15)–(16) and defining var(ωt − ωt|t−1) = pt|t−1, the forecastingerrors ηt = st − st|t−1 have the conditional expectation E(ηt) = 0 and variancevar(ηt) = pt|t−1 + vt for all t ∈N and are not autocorrelated. Moreover, assumingthat ω1, εt, and γt are uncorrelated and normally distributed for all t, ηt follows aconditional normal distribution ηt ∼ N (0, pt|t−1 + vt) for all t.

The observable forecasting errors {ηt} are exploited to monitor the validityof the model in Equations (15)–(16). It is believed to be an appropriate volatilitymodel as long as the observed stochastic properties of {ηt} remain in line with theresult in Proposition 1.

2.3 Sequential Monitoring of Model Validity

A sequential control procedure is required for monitoring the validity of the state-space model in Equations (15)–(16). A signal from the control procedure suggeststhat the distributional properties of {ηt} may deviate significantly from those inProposition 1, so that the volatility model may be misspecified. The standardizedvolatility forecasting errors Xt = ηt/(pt|t−1 + vt)1/2 are suitable for monitoring

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purposes. Assume that the state-space representation with the known parametersθ = (a, φ, q)′ is valid at t = 0. A control chart starts at t = 1 for making on-linedecisions between the hypotheses on every new day t > 1

H0,t : E(Xt) = 0 versus H1,t : E(Xt) 6= 0. (21)

If H0,t is valid for all t > 1, the monitored process is said to be in control, otherwiseit is denoted as out of control Woodall (2000). A good control chart should providea signal immediately after H0,t becomes invalid.

A control chart consists of a control statistic Zt, which depends on the process{Xt}, and a critical limit c > 0. A signal is given if the absolute value of the controlstatistic |Zt| exceeds the critical limit c for the first time. Although the monitoringconcentrates on the forecasting error process, we interpret a signal as evidence thatthe model (15)–(16) no longer produces proper volatility forecasts. This interpreta-tion corresponds to an econometric model validation test, which is replaced hereby a sequential monitoring procedure (Chu, Stinchcombe, and White 1996). Sincethere is a possibility of a false signal being received, each alarm must be carefullyinvestigated with respect to its causes and consequences.

The choice of the critical limit c depends on the distribution of the run lengthL, which is defined as the number of periods (days) until the first signal occurs:

L(c) = inf{t > 1 : |Zt| > c}.

The critical limit c is usually obtained by setting the in-control average run length(ARL) equal to a (large) predetermined value A, that is, E(L(c) |H0,t ∀t) = A.This means that on average, the first false signal appears after A periods. A largervalue of the critical limit c leads to a larger ARL E(L(c)). Common choices for thein-control ARL are A ∈ {120, 250, 500} (Montgomery 2005), which roughly cor-respond to half a year, a year, and two years of daily observations, respectively.Alternatively, the limit could be determined by fixing the asymptotic signal prob-ability α equal to the required value, for example, 10%, for the in-control process(Chu, Stinchcombe, and White 1996).

There are various control charts suitable for making decisions between the hy-potheses in Equation (21). The exponentially weighted moving average (EWMA)control chart of Roberts (1959) is popular in practical applications primarily be-cause of its simplicity. EWMA control charts for financial volatility processes wereproposed by Schipper and Schmid (2001). The EWMA control statistic is given by(Montgomery 2005)

Zt = (1− λ)Zt−1 + λXt, t > 1, (22)

where λ ∈ (0, 1] is a smoothing memory parameter. The starting value is usuallyset to Z0 = E(Xt) = 0. Since we are primarily interested in robust detection oflarge shifts which are of greatest economic relevance, the no-memory control chartof Shewhart (Sh) with λ ≡ 1 is chosen as an appropriate scheme for this task. Then,the control statistic is given by Zt = Xt. According to the result of Proposition 1, the

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critical limit for the special case of the two-sided Shewhart chart can be explicitlycalculated by c∗ = Φ−1(1− 1/(2A)). Note that the ratio 1/A corresponds to thetype I error probability of a significance test.

Cumulated sum (CUSUM) control charts, devised by Page (1954), are anotherpopular family of monitoring procedures, which exhibit a set of optimal detect-ing properties (cf. Moustakides 1986, 2004). The CUSUM-type charts of Horvath,Kokoszka, and Zhang (2006), Andreou and Ghysels (2008), namely the CUSUM(CS), fluctuation sum (FS), and recursive residual (RR) schemes, are also con-sidered for monitoring the daily volatility model in both the simulation and theempirical studies.

3 SIMULATION STUDY

A Monte Carlo simulation study compares the control charts with respect totheir detecting ability. The study investigates two important issues for each givenchange in the volatility model parameters. First, it quantifies the impact of falselyspecified model parameters on forecasting losses. Second, it provides evidenceabout the average detection time for various types of changes in the initial pa-rameter vector θ. Moreover, the impact of nonnormality of forecasting errors isassessed by considering t-distributed innovations.

The simulations are conducted with B independent runs of T0 + T observa-tions each. The in-control model is generated as in Equations (15)–(16), where themeasures st and vt are the only observable variables. The in-control model param-eters are chosen as a = 0, φ = 0.40, q = 0.2, and vt ≡ v = 0.04, similar to theempirical estimates in Section 4. The first T0 = 100 observations are skipped inorder to mitigate the impact of the starting values ω1|0 = a and p1|0 = q/(1− φ2).A change occurs at t = T0 + 1, whereas only one parameter in the vector θ alterseach time, while the others remain unchanged. These parameter changes are mod-eled as a1 = a + Δa, φ1 = φ + Δφ, and q1 = qΔq, with Δa ∈ {−1,−.875, . . . , 1},Δφ ∈ {−0.3,−0.25, . . . , 0.5}, and Δq ∈ {0.67, 0.7325, . . . , 1.67}. They correspond tothe parameter fluctuations in the empirical study, see Figure 5.

Calculating average forecasting losses is important in order to assess the im-pact of misspecified model parameters on the goodness of forecasts. For thispurpose, the volatility measures are generated from the model based on θ1 andcompared with the misspecified volatility forecasts, which rely on the model withthe prechange parameter vector θ. The forecasting errors ηt = st − st|t−1 and their

standardized counterparts Xt = ηt/(pt|t−1 + vt)1/2 form the processes of inter-

est. Given the vector θ1, the mean square forecasting error L1,b = T−1 ∑T0+Tt=T0+1 η2

t

and the mean square standardized forecasting error L2,b = T−1 ∑T0+Tt=T0+1 X2

t arecalculated for each run b ∈ {1, . . . , B}. Then, the average losses, obtained asLi = B−1 ∑B

b=1 Li,b with i ∈ {1, 2}, quantify the consequences of neglectingchanges in the model parameters. Table 1 reports the average losses calculatedfor T = 5000, B = 105 with both normally and t8-distributed innovations.

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Tab

le1

Sim

ula

ted

aver

age

fore

cast

ing

non

stan

dar

diz

edL

1an

dst

and

ard

ized

L2

loss

esfo

rn

orm

ally

dis

trib

ute

dan

dt-

dis

trib

ute

din

nov

atio

ns

and

dif

fere

nt

typ

esof

chan

ges

inth

ep

aram

eter

vec

tor

N( 0

,1)

t 8N( 0

,1)

t 8N( 0

,1)

t 8a 1

L1

L2

L1

L2

φ1

L1

L2

L1

L2

q 1L

1L

2L

1L

2

−1.

000

0.65

72.

679

0.69

82.

846

0.10

00.

256

1.04

30.

341

1.39

10.

134

0.17

90.

730

0.23

90.

973

−0.

875

0.56

12.

286

0.60

22.

452

0.15

00.

252

1.02

70.

336

1.36

90.

147

0.19

20.

781

0.25

51.

041

−0.

750

0.47

71.

945

0.51

82.

111

0.20

00.

249

1.01

40.

332

1.35

10.

159

0.20

40.

832

0.27

21.

110

−0.

625

0.40

61.

656

0.44

71.

823

0.25

00.

246

1.00

40.

328

1.33

90.

172

0.21

70.

883

0.28

91.

178

−0.

500

0.34

81.

420

0.38

91.

586

0.30

00.

245

0.99

90.

327

1.33

10.

184

0.22

90.

934

0.30

61.

246

−0.

375

0.30

31.

236

0.34

41.

403

0.35

00.

245

0.99

70.

326

1.32

90.

197

0.24

20.

986

0.32

21.

314

−0.

250

0.27

11.

105

0.31

21.

272

0.40

00.

245

1.00

00.

327

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As shown in Table 1, different types of changes in θ cause different averagelosses, whereas the evidence for nonstandardized L1 and standardized L2 av-erage losses is quite similar. A moderate increase in the AR-parameter φ leadsto larger losses compared with a decrease, where the loss functions remain al-most unchanged. An increasing variance q causes much larger losses than in-creases in φ. Decreasing variances even reduce forecasting losses compared withno change situation. Changes in the mean a lead to the most pronounced losses.Considering t-distributed innovations leads to findings similar to those in thenormality case.

The detecting ability of the control charts (Sh, CS, FS, RR) is investigated usingthe process of the standardized forecasting errors {Xt}. All charts are calibratedto provide the in-control ARL A = 120 assuming normally distributed innova-tions. The control statistics and critical bounds of CS, FS, RR charts are provided inHorvath, Kokoszka, and Zhang (2006) in equations (3.13), (3.15), and (3.29), respec-tively. The required (prerun) in-sample period length is chosen as m = 100 as inthe empirical study in Section 4.2. The in-control ARL is equal to 120 for the chartparameters aCS = 0.559, aFS = 0.332, and aRR = 0.228 and for the Shewhart chartc∗ = 2.638. The out-of-control ARLs, presented in Table 2, are calculated as the av-erage number of periods until the first signal for both normally and t-distributedinnovations.

Table 2 shows that changes causing the largest average forecasting losses (cf.Table 1) could be detected quickly by all considered charts. The CS, FS, RR schemesshow quite similar detecting abilities. They are better than the Shewhart chart atdetecting changes in the mean a and worse at detecting increases in the varianceq. Given a change in the mean, the out-of-control ARLs for the CS, FS, RR schemesa are much smaller compared with those of the Shewhart chart. This result is tobe expected because the charts of Horvath, Kokoszka, and Zhang (2006) exhibit aset of optimality detection properties. Given an increase in the variance, however,the out-of-control ARLs of these charts are larger than the Shewhart one. Thesefindings hold both for normally and t-distributed innovations, whereas the latterlead to much lower in-control ARLs. The obtained results allow us to conclude thatthe considered charts are suitable for detecting important changes in the volatilitymodel parameters.

4 EMPIRICAL STUDY

The empirical study illustrates the application of the volatility monitoring method-ology to the real data. First, we describe the dataset and provide investiga-tion of its stochastic properties. Then, the estimates of the model in Equations(15)–(16) are compared for both in-sample and full sample periods. Finally, thecontrol charts are applied to monitoring the model validity. The dates of sig-nals from the monitoring procedures are contrasted with the dates of detectedjumps.

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Table 2 Simulated out-of-control ARLs of Sh, CS, FS, RR charts (Horvath, Kokoszka,and Zhang 2006) with normally distributed (upper part) and t-distributed (lower part)innovations for changes in the parameters a, φ, and q

a1 Sh CS FS RR φ1 Sh CS FS RR q1 Sh CS FS RR−1.000 11.3 3.5 3.3 3.0 0.10 103.7 207.5 253.8 320.7 0.1340 499.9 178.2 204.0 242.6−0.875 15.2 3.9 3.9 3.5 0.15 109.7 196.6 236.6 289.7 0.1465 356.3 162.3 183.2 213.3−0.750 21.0 4.4 4.8 4.2 0.20 114.7 186.1 213.3 257.3 0.1590 263.3 152.3 167.2 186.6−0.625 29.5 5.2 6.1 5.1 0.25 118.5 172.1 192.7 220.2 0.1715 201.5 143.7 149.8 162.8−0.500 42.2 6.6 8.3 6.7 0.30 120.8 157.8 167.7 188.0 0.1840 158.2 133.1 137.6 141.7−0.375 60.5 9.2 12.7 9.8 0.35 121.4 139.4 145.0 152.8 0.1965 127.4 121.2 122.1 125.4−0.250 84.5 19.0 24.1 19.1 0.40 120 120 120 120 0.2090 104.8 112.4 112.8 110.0−0.125 108.9 66.0 66.5 63.3 0.45 116.2 101.1 97.7 89.5 0.2215 87.8 105.8 101.2 96.70.000 120 120 120 120 0.50 110.2 79.6 75.6 65.6 0.2340 74.5 97.1 91.7 83.40.125 109.0 66.5 66.8 63.2 0.55 102.1 60.7 56.7 46.4 0.2465 64.3 90.8 84.5 74.50.250 84.5 19.2 24.1 19.1 0.60 92.2 43.4 42.6 31.4 0.2590 56.0 85.9 76.5 66.30.375 60.4 9.3 12.7 9.8 0.65 80.7 28.7 30.5 21.2 0.2715 49.4 78.5 70.6 59.40.500 42.1 6.5 8.4 6.7 0.70 68.4 19.5 23.0 15.5 0.2840 44.0 73.1 65.0 52.80.625 29.5 5.2 6.1 5.1 0.75 56.1 13.2 17.7 12.0 0.2965 39.4 68.6 59.1 46.70.750 21.0 4.4 4.8 4.2 0.80 44.6 10.0 14.1 9.9 0.3090 35.6 62.9 53.8 42.10.875 15.2 3.9 3.9 3.5 0.85 34.7 7.8 11.5 8.6 0.3215 32.4 59.0 51.4 38.11.000 11.3 3.5 3.2 3.0 0.90 26.5 6.5 9.7 7.6 0.3340 29.6 55.1 46.0 34.3

a1 Sh CS FS RR φ1 Sh CS FS RR q1 Sh CS FS RR−1.000 8.97 3.5 3.3 3.3 0.10 33.5 166.2 177.0 153.9 0.1340 78.3 133.0 136.9 118.8−0.875 11.2 3.9 3.9 3.9 0.15 34.3 152.7 160.6 140.4 0.1465 65.3 119.9 120.9 104.6−0.750 14.1 4.5 4.8 4.7 0.20 35.0 141.5 143.3 124.9 0.1590 55.4 109.9 105.8 92.0−0.625 17.6 5.2 6.1 6.0 0.25 35.4 126.5 125.7 108.1 0.1715 47.8 98.5 94.6 81.5−0.500 21.8 6.5 8.1 7.9 0.30 35.6 111.9 107.8 91.9 0.1840 41.8 89.9 83.9 72.4−0.375 26.4 9.0 11.9 11.6 0.35 35.7 95.3 88.9 76.8 0.1965 36.9 82.9 74.7 65.8−0.250 30.9 16.5 20.6 19.8 0.40 35.5 78.6 72.0 62.8 0.2090 33.0 75.2 67.2 58.4−0.125 34.4 47.0 45.8 41.3 0.45 35.0 63.1 57.4 50.2 0.2215 29.7 68.1 60.7 51.90.000 35.8 78.2 72.0 63.0 0.50 34.2 49.1 44.8 39.1 0.2340 26.9 62.5 53.8 47.20.125 34.5 46.4 45.2 41.0 0.55 33.1 36.4 34.1 30.7 0.2465 24.6 57.9 48.9 42.30.250 31.0 16.8 20.6 19.8 0.60 31.6 25.4 25.9 24.1 0.2590 22.6 51.6 43.6 38.90.375 26.6 8.9 11.9 11.7 0.65 29.7 17.9 20.1 19.2 0.2715 20.9 47.8 40.2 35.60.500 21.9 6.4 8.1 8.0 0.70 27.5 12.9 16.1 15.5 0.2840 19.5 43.7 37.0 32.50.625 17.7 5.2 6.1 6.0 0.75 25.0 9.8 13.2 12.8 0.2965 18.1 40.2 33.8 30.50.750 14.1 4.5 4.8 4.7 0.80 22.3 8.0 11.1 10.8 0.3090 16.9 36.6 30.9 28.10.875 11.2 3.9 3.9 3.9 0.85 19.4 6.6 9.5 9.2 0.3215 16.0 34.8 28.7 25.71.000 8.95 3.5 3.3 3.3 0.90 16.6 5.8 8.3 8.2 0.3340 15.1 32.6 27.1 24.2

In-control parameters are a = 0, φ = 0.4, q = 0.2, v = 0.04, in-control ARL is A = 120.Number of simulations B = 105. In-control values are bold.

4.1 Data and Descriptive Statistics

The study considers four highly liquid stocks traded on the NYSE, namely theGM Corporation, Hewlett-Packard Company (HPQ), Coca-Cola Company (KO),

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and United Technologies Corporation (UTX). The ultrahigh frequency price ob-servations from Trade and Quote NYSE database are taken for a time period of458 trading days, from December 1, 2003 to November 27, 2005. The daily seriesof realized volatilities RVt, bipower variations BVt, and staggered bipower varia-tions SBVt,i=1 are calculated on the basis of synchronized 5-minute asset returns.This frequency choice is justified by the volatility signature plots, which are notpresented here owing to space limitations; moreover, it appears to be appropriatefor dealing with MMN in highly liquid markets (Hansen and Lunde 2006). Thesampling grid is taken from 9:35 A.M. to 3:55 P.M. by a previous tick interpolationmethod (Dacorogna et al. 2001). The first and last five trading minutes are omittedin order to avoid irregularities at the beginning and end of the trading time.

The full sample of 458 trading days is divided into two parts. The first 100observations from December 2003 to April 2004 are used for the initial model es-timation. The descriptive statistics for the in-sample and full sample daily volatil-ity measures are reported in Table 3. Since RVt, BVt, and SBVt provide similarresults, we present the figures only for the BVt measure. The in-sample and fullsample ACF and PACF functions for log BVt series are shown in Figures 1 and 2,respectively.

The descriptive statistics in Table 3 indicate only minor differences betweenthe alternative volatility measures. As expected, the mean of log(RVt) measure islarger than the means of log(BVt) and log(SBVt) measures for each consideredasset indicating on possible jump components. The full sample means refer tothe interval [−0.5,0.5], whereas the average variances are around [0.035,0.05]. Thecorresponding in-sample values are [−0.4,0.7] and [0.035,0.046], respectively. TheShapiro–Wilk test strongly rejects normality for the full sample but cannot reject

Table 3 Descriptive statistics for in-sample and full sample log(RV), log(BV),log(SBV) volatility measures: means st, average variances vt, and p values of Shapiro–Wilk test for normal distribution

Full sample In samplest vt SW st vt SW

GMLog(RVt) 0.3147 0.0501 0 0.1805 0.0440 0.035Log(BVt) 0.2411 0.0464 0 0.1276 0.0459 0.832Log (SBVt) 0.1843 0.0395 0 0.0739 0.0408 0.583

HPQLog(RVt) 0.4407 0.0428 0 0.6333 0.0385 0.112Log(BVt) 0.3648 0.0413 0.091 0.5778 0.0400 0.069Log(SBVt) 0.3172 0.0370 0.019 0.535 0.0372 0.945

KOLog(RVt) −0.3807 0.0402 0.537 −0.2754 0.0373 0.296Log(BVt) −0.4494 0.0385 0 −0.3237 0.0383 0.009Log(SBVt) −0.4943 0.0351 0.235 −0.3732 0.0365 0.139

UTXLog(RVt) −0.0714 0.0442 0.005 0.1139 0.0424 0.292Log(BVt) −0.1358 0.0416 0.011 0.063 0.0429 0.506Log(SBVt) −0.1806 0.0359 0.026 0.0195 0.0381 0.967

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Figure 1 ACF (left) and PACF (right) of log(BV) for the in-sample period.

it for the in-sample data in the majority of cases. The analysis of the in-sampleACF and PACF plots in Figure 1 shows that a stationary AR(1) process seems tobe appropriate for describing the time-series behavior of the daily volatilities. TheACFs quickly decay to insignificant values, whereas the PACFs are clearly signifi-cant only at lag one. On the contrary, the full sample ACFs and PACFs in Figure 2advocate AR processes of higher order, such as AR(4) in the GM, HPQ, and UTXcases. Thus, starting the analysis from the in-sample AR(1) model, the aim was todetect days where such a parsimonious approach fails to provide proper volatilityforecasts.

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Figure 2 ACF (left) and PACF (right) of log(BV) for the full sample period.

4.2 Estimation Results

The in-sample estimation of the system (15)–(16) relies on the first 100-day obser-vations. Both in-sample and full sample ML-estimates are reported for log RVt,BVt, and SBVt measures in Table 4 with ML and quasi-ML standard errors. Thereare only small differences between the in-sample ML estimates for the variousvolatility measures. The AR(1) parameter φ in the state equation is positive andsignificantly smaller than unity for all stocks, pointing to the weak stationarity of

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Table 4 In-sample and full sample estimates of parameters a, φ, q, their ML and quasi-ML standard errors

a MLSE QMLSE φ MLSE QMLSE q MLSE QMLSEIn-sample estimates

GMLog(RVt) 0.162 0.051 0.072 0.332 0.078 0.109 0.212 0.026 0.039Log(BVt) 0.115 0.050 0.071 0.339 0.079 0.106 0.198 0.025 0.041Log(SBVt) 0.061 0.049 0.069 0.304 0.077 0.109 0.213 0.026 0.045

HPQLog(RVt) 0.622 0.066 0.096 0.556 0.071 0.101 0.169 0.023 0.036Log(BVt) 0.564 0.076 0.111 0.608 0.066 0.097 0.176 0.024 0.034Log(SBVt) 0.512 0.077 0.113 0.619 0.065 0.089 0.171 0.023 0.037

KOLog(RVt) −0.277 0.052 0.074 0.469 0.078 0.115 0.148 0.02 0.027Log(BVt) −0.332 0.056 0.079 0.441 0.075 0.110 0.186 0.023 0.033Log(SBVt) −0.377 0.055 0.079 0.442 0.076 0.116 0.183 0.023 0.032

UTXLog(RVt) 0.101 0.047 0.067 0.293 0.083 0.120 0.203 0.025 0.032Log(BVt) 0.052 0.047 0.066 0.298 0.081 0.128 0.201 0.025 0.034Log(SBVt) 0.017 0.044 0.062 0.184 0.081 0.132 0.232 0.027 0.037

Full sample estimates

GMLog(RVt) 0.305 0.050 0.073 0.657 0.029 0.045 0.268 0.016 0.028Log(BVt) 0.236 0.049 0.070 0.617 0.029 0.046 0.317 0.018 0.033Log(SBVt) 0.181 0.047 0.067 0.602 0.029 0.046 0.320 0.018 0.031

HPQLog(RVt) 0.433 0.033 0.048 0.581 0.032 0.050 0.173 0.011 0.021Log(BVt) 0.360 0.034 0.048 0.542 0.032 0.051 0.208 0.012 0.023Log(SBVt) 0.314 0.034 0.049 0.558 0.031 0.048 0.205 0.012 0.023

KOLog(RVt) −0.386 0.027 0.038 0.572 0.035 0.053 0.112 0.008 0.011Log(BVt) −0.454 0.027 0.038 0.497 0.034 0.053 0.156 0.009 0.015Log(SBVt) −0.498 0.026 0.037 0.485 0.035 0.052 0.152 0.009 0.013

UTXLog(RVt) −0.082 0.028 0.040 0.535 0.036 0.050 0.146 0.010 0.015Log(BVt) −0.146 0.027 0.038 0.481 0.035 0.054 0.170 0.011 0.017Log(SBVt) −0.184 0.027 0.038 0.445 0.035 0.054 0.189 0.011 0.019

the processes. Table 5 provides the statistical analysis of the in-sample and fullsample standardized forecasting errors.

The in-sample means of the forecasting errors are close to zero, whereas thevariances slightly exceed unity owing to outliers causing heavy tails of the volatil-ity distribution. These outliers can be observed in the QQ-plots for the forecast-ing errors Xt in Figure 3 (left-hand side). There is no significant autocorrelationin Xt series at any lag, see the ACFs and PACFs in Figure 4. The p values ofthe Shapiro–Wilk normality tests for the in-sample Xt are quite large and canbe interpreted in favor of the AR(1) representation. Summarizing, the model inEquations (15)–(16) can not be statistically rejected for the in-sample period forany stock.

The full sample model estimates are shown in Table 4, lower block, and the re-sults for the full sample forecasting errors Xt are reported in Table 5, lower block.The full sample estimates of the AR(1) parameter φ are still significantly smaller

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Table 5 Descriptive statistics for in-sample and full sample standardized forecastingerrors Xt = ηt/(pt|t−1 + vt)

1/2: means, standard deviations, variances, p values ofthree tests: testing H0 : E(Xt) = 0, testing H0 : var(Xt) = 1, and Shapiro–Wilk test fornormal distribution

Xt H0 : E(Xt) = 0 sd(Xt) var(Xt) H0 : var(Xt) = 1 SW Xt ∼ NIn-sample residuals

GMLog(RVt) 0.033 0.767 1.107 1.226 0.002 0.701Log(BVt) 0.023 0.840 1.133 1.284 0 0.265Log(SBVt) 0.024 0.834 1.128 1.272 0 0.091

HPQLog(RVt) 0.028 0.808 1.131 1.279 0 0.152Log(BVt) 0.031 0.790 1.157 1.339 0 0.119Log(SBVt) 0.044 0.705 1.150 1.323 0 0.034

KOLog(RVt) −0.006 0.958 1.167 1.362 0 0.435Log(BVt) 0.011 0.919 1.103 1.217 0.003 0.267Log(SBVt) 0.001 0.992 1.117 1.247 0.001 0.223

UTXLog(RVt) 0.031 0.785 1.129 1.274 0 0.760Log(BVt) 0.026 0.819 1.128 1.273 0 0.629Log(SBVt) 0.007 0.952 1.090 1.187 0.009 0.618

Full sample residuals

GMLog(RVt) 0.028 0.593 1.128 1.272 0 0Log(BVt) 0.017 0.744 1.094 1.197 0.006 0Log(SBVt) 0.014 0.788 1.079 1.165 0.017 0

HPQLog(RVt) 0.019 0.714 1.125 1.265 0 0Log(BVt) 0.013 0.809 1.109 1.230 0.002 0Log(SBVt) 0.010 0.842 1.101 1.212 0.004 0

KOLog(RVt) 0.015 0.788 1.195 1.427 0 0.731Log(BVt) 0.011 0.833 1.099 1.207 0.004 0.504Log(SBVt) 0.010 0.841 1.117 1.249 0.001 0.541

UTXLog(RVt) 0.029 0.594 1.165 1.357 0 0.051Log(BVt) 0.026 0.628 1.131 1.278 0 0.003Log(SBVt) 0.009 0.855 1.096 1.202 0.005 0.007

than unity but larger than their in-sample counterparts. There are also remarkabledifferences between other in-sample and full sample model parameters. The fullsample forecasting errors exhibit zero means, but their standard deviations signifi-cantly exceed unity. The normality assumption cannot be rejected for the full sam-ple Xt for the KO stock. It is clearly rejected, however, for the GM and the HPQcases, where the Shapiro–Wilk p values are close to 0. The full sample QQ-plotsfor Xt in Figure 4 (right-hand side) show many more outliers compared with thein-sample QQ-plots in Figure 4 (left-hand side). There are many outliers for GMparticulary, where severe deviations from normality are clearly observed in theright tail. They indicate a possible model instability and require further analysis.

The differences in the in-sample and full sample parameter estimates in Table 4can be interpreted as evidence of possible changes in the model parameters during

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Figure 3 QQ-plots of the standardized forecasting errors Xt from the state-space representation

for log(BV) for the in-sample (left) and full sample (right) periods.

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Figure 4 ACF (left) and PACF (right) of the standardized forecasting errors Xt from the state-

space representation for log(BV) for the in-sample period.

the out-of-sample period. The time evolution of the model parameters is re-vealed by re-estimating the volatility model using a rolling window of 100 days.The rolling estimates are presented in Figure 5 with plus–minus twice quasi-MLstandard error bounds.

Figure 5 illustrates that the autoregressive parameter φ and the variance q re-main quite stable, but the most obvious changes are observed in the mean pa-rameter a. As shown in the simulation study in Section 3, these are the changeswhich lead to the largest forecasting losses. The model parameters stay relatively

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Figure 5 Rolling window estimates for parameters a (left), φ (center), and q (right) with two-

sigma quasi-ML bounds (gray) of the state-space representation for log(BV) with window range

of 100 days.

stable for the KO and UTX stocks, whereas there are quite large changes in theHPQ and especially in the GM case. In particular, there is a pronounced simulta-neous increase in the variance q and mean a for GM starting from Spring 2005.Control charts are required in order to detect these alterations as soon as theyoccur.

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4.3 Applying Control Charts to the Data

The volatility models are monitored by applying control charts to the forecastingerrors {Xt}. The critical limit of the Shewhart charts are chosen as c∗ = 2.638 andc∗ = 2.865 to provide the in-control ARLs of A = 120 and A = 240 days, re-spectively. The number of signals from the Shewhart chart for both in-sample andfull sample periods are summarized in Table 6. The Shewhart charts for the BVt

measure with A = 120 are shown in Figure 6 (left-hand side) and with A = 240 inFigure 7 (left-hand side). The vertical lines in Figures 6 and 7 (left-hand side) sep-arate the in-sample and out-of-sample periods. The days with signals are shownin Figures 6 and 7 as bold circles. Note that the model is not re-estimated after asignal. The jump days are identified with the test given in Equation (10). Figures6 and 7 (right-hand side) contrast the signal days and the days with a significantjump component. The overall number of jump days is reported in Table 7 below.

The Shewhart charts based on the RVt, BVt, and SBVt measures provide a sim-ilar number of signals for both in- and out-of-sample phases. Moreover, the major-ity of signals occur at the same days for all volatility measure, as shown in Figures6 and 7 (right-hand side). Although the actual number of signals is larger than ex-pected under the model validity for all charts, we suggest differentiating betweenthe isolated and clustered signals. The isolated signals can often be interpreted asoutliers, for example, they can occur because of the heavy tails of distribution ofXt, see the QQ-plots in Figure 4. On the contrary, clustered signals seriously ques-tion the model validity during that period of time. The application of the CUSUMfamily charts (CS, FS, RR) is conducted for the critical limits, which provide thein-control ARL A = 240 or the asymptotic false signal probabilities α = 0.1, 0.2, asin Horvath, Kokoszka, and Zhang (2006). Since we do not re-estimate the model,the mean of forecasting errors after a signal is set equal to 0, that is, X ≡ 0. It wouldalso be reasonable, however, to re-estimate the model after each signal and to cal-culate new means of the forecasting errors. The number of signals for all charts isreported in Table 7. Setting the in-control ARL for CUSUM charts equal to A = 240leads to a similar number of signals to those from the Shewhart chart. Using thefalse signal probability for determining CUSUM critical limits strongly reducesthe number of signals. There remains a large number of signals for the UTX andespecially for GM cases, however.

The volatility representation in Equations (15)–(16) presumes no jumps in theprice equation. Both jumps and parameter changes can be responsible for the vi-olation of the distributional result, given in Proposition 1. Thus, it is importantto differentiate between jumps and structural instability in the model parameters.The number of the control chart signals, given in Table 7 above, is much smallerthan the number of jumps given in Table 7 below. Comparable to the findings ofBarndorff-Nielsen and Shephard (2004, 2006), about 10% of days exhibit jumps atthe significance level of 1%. Figures 6 and 7 (right-hand side) compare Shewhartsignal dates with jump dates. The vertical lines denote days where the jumpsand the signals occur simultaneously. We find that jumps and signals usually

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Figure 6 Left: The Shewhart chart with A = 120 using the standardized forecasting errors Xt

from the state-space model for log(BV). Vertical line divides in-sample and out-of-sample periods;

bold points denote signals. Right: Bold points denote dates of jumps detected to the significance

level α = 0.01 and dates of signals for the log(RV), log(BV), log(SBV) measures. Vertical lines

joint signals and jumps occurring at a same days

occur at different dates. The Cramer contingency coefficient for the dependencebetween the Shewhart signal and jump dates shows almost zero dependence be-tween jumps and signals, whereas the dates of signals from different charts aresignificantly dependent. Thus, we conclude that the control charts react primarily

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Figure 7 Left: The Shewhart chart with A = 240 using the standardized forecasting errors Xt

from the state-space model for log(BV). Vertical line divides in-sample and out-of-sample periods;

bold points denote signals. Right: Bold points denote dates of jumps detected to the significance

level α = 0.01 and dates of signals for the log(RV), log(BV), log(SBV) measures. Vertical lines

joint signals and jumps occurring at a same days

to deviations from model validity and not the jump component. These findingsare in line with the studies of Andersen, Bollerslev, and Diebold (2007), Maheu,Reeves, and Xie (2010), and Perron and Qu (2010), which report expost differencesbetween jumps and structural breaks for volatility series.

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Figure 8 CS, FS, RR charts for BV series of GM calibrated with the false signal probabilities

α = 0.1 (left) and α = 0.2 (right). After a signal, the prerun X is set to 0

The Shewhart charts provide only a limited number of signals for the KO andthe UTX stock during the monitoring phase. All signals are isolated observations,which form no clusters in time. For these reasons, the model is not rejected forthese two stocks at any point in time. There are about 15 signals for A = 120and 10 signals for A = 240 during the monitoring period of the HPQ stock. Thesenumbers are comparatively large because the expected (in-control) number of falsesignals is about seven and four, respectively. Note that these signals occur in differ-ent directions, whereas the number of positive and negative signals is roughly thesame. Thus, there is no clear direction of change in the model mean. These signalscan be attributed to an increase in the HPQ variance in Summer 2005, observedin Figure 5. The CUSUM charts provide many signals for the UTX stock, whichcan be justified by the observed changes in the UTX mean a in Figure 5. These(quite minor) changes in the UTX mean seem to be of mean-reverting nature, sothe Shewhart chart is not flexible enough to capture them.

The on-line monitoring of GM provides a lot of signals clustered in the timeperiod starting from March 2005. For instance, there are only three negative signals

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Table 6 Number of signals (positive, negative, total) from Shewhart charts with in-control ARLs A = 120 and A = 240

Number of signalsA = 120 A = 240

In Out In Out+ − All + − All Total + − All + − All Total

GMLog(RVt) 3 1 4 20 4 24 28 2 1 3 17 2 19 22Log(BVt) 3 1 4 22 4 26 30 3 1 4 19 3 22 26

Log(SBVt) 3 1 4 22 3 25 29 3 1 4 21 1 22 26

HPQLog(RVt) 2 0 2 6 9 15 17 2 0 2 4 7 11 13Log(BVt) 2 0 2 6 7 13 15 2 0 2 5 5 10 12

Log(SBVt) 2 0 2 9 7 16 18 2 0 2 5 5 10 12

KOLog(RVt) 1 3 4 0 8 8 12 0 3 3 0 5 5 8Log(BVt) 0 3 3 0 4 4 7 0 2 2 0 3 3 5

Log(SBVt) 0 3 3 0 3 3 6 0 2 2 0 2 2 4

UTXLog(RVt) 1 3 4 1 12 13 17 0 0 0 1 9 10 10Log(BVt) 1 3 4 2 12 14 18 0 0 0 1 7 8 8

Log(SBVt) 0 0 0 1 8 9 9 0 0 0 1 3 4 4

of the Shewhart chart with A = 240 until the middle of March, 2005, but 19 sig-nals in the positive direction with Zt > c∗ = 2.865 starting from that period.Thus, a remarkable simultaneous increase in the GM mean a and variance q is sup-ported by the evidence of highly clustered signals. These signals could hardly beinterpreted as separate outliers, so we conclude that the in-sample GM volatility

Table 7 Number of signals from Sh, CS, FS, RR charts for the out-of-sample monitoringperiod based on BV and number of jumps with the significance level α = 0.01

Chart Limit GM HPQ KO UTX

Shewhart A = 120 30 15 7 18Shewhart A = 240 26 12 5 8

CS A = 240 20 9 8 15CS α = 0.1 4 0 0 2CS α = 0.2 5 0 0 3

FS A = 240 44 15 10 26FS α = 0.1 16 2 0 7FS α = 0.2 22 4 0 10

RR A = 240 38 14 17 26RR α = 0.1 5 2 1 4RR α = 0.2 7 3 1 6

Number of jumpsIn sample 7 8 11 6Out of sample 46 40 44 43Total 53 48 55 49

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model appears to be no longer valid. The CUSUM charts for the GM case withthe false signal probabilities α = 0.1 and α = 0.2 are presented in Figure 8. AllCUSUM schemes provide positive signals in March 2005. The FS chart providesmany clustered signals starting from March 2005. The detected signals for GM canbe explained by the publicly available information concerning the problems GMfaced at that period. For instance, GM announced the largest losses in 13 years inthe middle of March 2005, experienced Finch’s agency downgrading from BB+ toBBB− in June 2005 and reported disappointing numbers for the first half of 2005in October 2005. The price of the GM stock dropped by almost half owing to theseperturbations between March 2005 and November 2005. Remarkably, the validityof the state-space model for the GM daily volatility appears to be extremely ques-tionable from the middle of March 2005 onwards.

The evidence from the empirical study suggests that the chosen simple state-space volatility model remains appropriate for KO and should be rejected for GMfrom the middle of March 2005 onwards. The signals for the GM stock are highlyclustered and could be explained by means of publicly available information. Theresults for HPQ and UTX are rather mixed, so that a deeper analysis is required forthese stocks. Although many jumps are detected for all stocks, the jump compo-nent can be separated from signals about the model validity because the jumpdates largely differ from the signal dates. Both Shewhart and CUSUM controlcharts lead to qualitatively similar results, being able to provide on-line signalson the same days.

5 CONCLUSIONS

The current paper elaborates tools for monitoring model validity for daily inte-grated volatility. A linear state-space representation is selected to link the observ-able volatility measure, based on the intraday information, to the unobservable logdaily volatility. The latter is assumed to follow an AR(1) process in the state equa-tion. This simple model should be used for volatility forecasting until the modelis statistically rejected. A decision about model validity should be done on everynew day. Both Shewhart and CUSUM control charts are used for monitoring thestochastic properties of the forecasting errors. A signal from the chart indicates thatthe assumptions concerning the forecasting errors may be violated and the chosenvolatility model should be questioned. Since an isolated signal could be an outlier,we suggest interpreting a sequence of several signals within a short time period asa clear piece of evidence for model failure.

The empirical study illustrates our approach based on four highly liquid U.S.stocks. The ultrahigh frequency price quotations serve to calculate the daily real-ized volatility and bipower variation measures. The in-sample data are used formodel estimation, whereas the out-of-sample observations are exploited for se-quential monitoring purposes. The in-sample results do not allow us to reject thestate-space volatility model for any of the four stocks. The out-of-sample on-line

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monitoring clearly rejects the state-space volatility model for GM by providing alot of clustered signals starting from the middle of March 2005. These signals canbe explained by the publicly available information concerning the financial prob-lems of GM at that time. Since the obtained signals and detected jumps mainlyoccur on different days, we suggest that the signals point to possible structuralchanges in the volatility model.

APPENDIX A: STATE-SPACE MODEL ESTIMATION

The parameters of the state-space model are estimated with the numerical maxi-mization of the joint likelihood function. Assuming that the initial state ω1 and theinnovations {γt, εt}T

t=1 are normally distributed, the forecasts ωt|t−1 and st|t−1 arenormally distributed as well

st| It−1 ∼ N (ωt|t−1, pt|t−1 + vt), t = 1, . . . , T.

The parameter vector θ = (a, φ, q)′ for the state-space representation (15)–(16) isestimated with the ML approach. The joint likelihood function of st conditionalon It−1 is given as the product of the conditional densities, as in Hamilton (1994,385). It is maximized with a Newton–Raphson numerical procedure (see Hamilton1994, 137). The estimated parameter values are denoted by θ = (a, φ, q)′.

The ML standard errors are obtained by taking square roots of the diagonal el-

ements of the matrix T−1Ψ−1

, where Ψ is the information matrix estimator. More-over, in the case of nonnormal innovation, we apply the quasi-ML procedure inorder to obtain consistent and asymptotically normal estimates of θ. The quasi-MLstandard errors are obtained as the square roots of the diagonal elements of the

matrix T−1[ΨΨ−1OPΨ]−1, at which the matrix ΨOP is the outer-product estimate of

the information matrix (Hamilton 1994, 389).

APPENDIX B: PROOF OF PROPOSITION 1

To calculate the moments of the forecasting errors ηt, we use the equality ηt =ξt + γt, where ξt = ωt − ωt|t−1, which follows from st|t−1 = ωt|t−1 and Equation(16). Since both ξt and γt have zero expectations, we get E(ηt) = 0. The varianceof ηt is given by

var(ηt) = var(ξt) + var(γt) + 2cov(ξt, γt)

= pt|t−1 + vt + 2cov[ωt, γt]− 2cov[ωt|t−1, γt]. (23)

First, the error γt is uncorrelated with ω1 and ετ for all τ under the state-space representation assumptions. Second, ωt is a linear combination of ετ for allτ < t and ω1. Consequently, γt and ωt are uncorrelated and the first covariance

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in Equation (23) is equal to 0. The quantity ωt|t−1 depends on the information setIt−1, so it is uncorrelated with γt and the second covariance is also 0. Thus, theconditional variance is var(ηt|It−1) = pt|t−1 + vt. Since the forecasting error ηt isuncorrelated with st−i for all i > 0, it is uncorrelated with ηt−i = st−i − st−i|t−i−1,so that cov(ηt, ηt−i) = 0, for all i > 0.

Our state-space modeling implies that if ω1, εt, and γt are normally distributedfor all t, then st|t−1 conditional on It−1 is also normally distributed. Its momentsare the expectation ωt|t−1 and the variance pt|t−1 + vt. Consequently, ηt is condi-tionally normally distributed with zero expectation and the variance as above. �

Received March 19, 2010; revised October 20, 2011; accepted October 21, 2011

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