st − 1|t − 1 + ct
equations:
Pt |t − 1 = ΦPt − 1|t − 1Φ′ + Qt
t=T?YesStop
Figure 7.8 Flowchart of the Kalman ﬁlter
In order to clarify the sequence of the Kalman ﬁlter equations, Figure 7.8 presents aﬂowchart for the ﬁlter.The recursive nature of the Kalman ﬁlter is a major computational advantage. It enablesthe model to update the conditional mean and covariance estimates of the states at timet based on the sole estimate obtained at time t − 1. Although it takes into account theentire history, it does not need an expanding memory.For an intuitive understanding of the principles involved, let us consider the exampleof the time-varying sensitivity (which follows a random walk) of a dependent variable Ytto a single independent variable Xt . The familiar regression equation is:Yt = bt Xt + εt
for t = 1, . . . , T
(7.32)
which can be represented in a state space form, using the random walk speciﬁcation:Yt = bt Xt + εtbt = bt−1 + ηt
for t = 1, . . . , T
(7.33)
with Var(εt ) = R and Var(ηt ) = Q. We can recognise the observation and transitionequations (7.14) and (7.15) with st = bt , Ht = Xt ,  = 1, d = 0, ct = 0.Upon presentation of a new observation Xt of X, the model expects to observe a valuebt|t−1 Xt of Y based on its predicted state18 bt|t−1 . However, the observed value Yt of the18As the sensitivity coefﬁcient b is assumed to follow a random walk the predicted state is equal to the lastestimate bt−1|t−1 of the state resulting from all available information at time t − 1.
Time-Varying Factor Sensitivities
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dependent variable Y is likely to be different from the model’s expectation. This forecasterror may result from two sources:• Some unpredictable force temporarily affects the dependent variable. This random noiseshould not affect the state.• The true value of the sensitivity of Y to X has changed (structural change). This is nota temporary disturbance and affects the state.The purpose of the Kalman ﬁlter is to attribute some share of the prediction error to eachof these two sources, i.e. separate the signal from the noise. The relative magnitude of thesystem noise variance Q and the observation noise variance R is therefore an importantparameter of the model. The following equations correspond to the various stages of theKalman ﬁlter for the random walk speciﬁcation.19The prediction equations:b̂t|t−1 = b̂t−1|t−1(7.34)Pt|t−1 = Pt−1|t−1 + QThe updating equations:b̂t|t = b̂t|t−1 + kt (Yt − b̂t|t−1 Xt )Pt|t = (1 − kt )Pt|t−1
(7.35)
Pt|t−1 Xt2kt =Pt|t−1 Xt2 + RThe sensitivity estimate b̂t|t is updated by taking into account the forecast error. Thefraction of forecast error that is added to the previous estimate of b is the Kalmangain kt . Its value is in the interval [0,1], with zero corresponding to Q = 0 and onecorresponding to R = 0. The Kalman gain depends on the relative value of observationand system noises and on the estimated variance of the state. For small values of kt (largeobservation noise R compared to system noise Q and/or small uncertainty about the stateestimate), considerable credibility is given to the previous sensitivity and as a result toremote observations. The sensitivity coefﬁcient bt evolves smoothly over time.In contrast, if kt takes larger values (i.e. large system noise Q compared to observationnoise R and/or large uncertainty about the state estimate), then more credibility is givento recent observations and therefore less weight is given to recent sensitivity estimates.In this case, the sensitivity coefﬁcient bt evolves quickly over time and its volatilityincreases.In order to understand well the effect of the ratio between system and observation noisevariance, let us consider an example where the sensitivity coefﬁcient, beta, is constantover the data set except for a short time period (5 days) during which it takes an unusuallyhigh value. This pattern may be created, for example, by incorrect market expectationsmotivated by unfounded rumours. A Kalman ﬁlter is used to estimate the time-varyingsensitivity. Three different signal to noise ratios (Q/R) are applied. For a ratio of 119
The notation âb|c denotes an estimate of a at time b conditional on the information available at time c.
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(Figure 7.9), the model rapidly adapts to the jump in sensitivity. However, this quickreaction comes at a cost of increased volatility in the beta estimates and of a largestandard error. If the signal to noise ratio is decreased to a value of 0.1 (Figure 7.10), thebeta estimates become smoother and the conﬁdence bands narrower. The model seems tobe more reliable, although it does not adapt quickly enough to the shift in beta occurringat time t = 90. If Q/R is set to a small value, such as 0.02 (Figure 7.11), beta estimatesbecome very smooth and the conﬁdence bands become very small. However, with such asmall ratio, the model does not manage to properly track the sensitivity when it jumps toa high level. It is also interesting to see how the standard error, i.e. Pt|t , decreases aftera few observations. This is because the initial value of Pt|t−1 (i.e. P1|0 ) was deliberatelychosen to be large so that the system could rapidly adjust the value of beta in order tomatch the observations.This model is available in the sensitivity estimation tool pack. Select “Stochastic parameter regression” in the main menu to enter the submenu displayed in Figure 7.12. Select“Random walk”. Make sure that the “Estimate parameters” check box is unchecked.The stochastic parameter regression menu proposes both a random walk model and arandom trend model for the underlying sensitivities. The system and observation noisevariance can be set by the user (option B), in which case a reference to the cells containingthe values has to be entered in the corresponding text box.20 The system noise variancescan also be set to a multiple of the observation noise variance (option A). In this case,the user sets the desired signal to noise ratio in the edit box. An alternative way is to letthe system estimate the optimal values of the parameters from the data. The estimationSignal to noise ratio = 1
Sensitivity
Actual betaEstimated betaConfidence bands: ± 1 SE
1
26
51
76
101Time
126
151
176
Figure 7.9 Actual versus estimated beta. The system noise to observation noise ratio, Q/R, is 1.This ratio allows the Kalman ﬁlter to rapidly adapt to the jump in sensitivity. However, this quickreaction comes at a cost of increased volatility in the beta estimates and of a very large standarderror20Unless the user has a good prior knowledge of the values to be entered in these cells (on the “Example”sheet, range “F2:F5”), we do not recommend using this option.
Time-Varying Factor Sensitivities
231
Signal to noise ratio = 0.1
Sensitivity
Actual betaEstimated betaConfidence bands: ± 1 SE
1
26
51
76
101Time
126
151
176
Figure 7.10 Actual versus estimated beta. Q/R is now 0.1. The beta estimates are smoother andtheir standard errors smaller. However, the model seems to adapt more slowly to the sensitivityjump
Signal to noise ratio = 0.02
Sensitivity
Actual betaEstimated betaConfidence bands: ± 1 SE
1
26
51
76
101Time
126
151
176
Figure 7.11 Actual versus estimated beta. For Q/R = 0.02, beta estimates are very smooth andtheir standard errors are much smaller. This is because variations in co-movements of Xt and Ytare attributed to ﬂuke rather than to true variations in beta
procedure is described later in the chapter. Figure 7.13 shows the results of the randomwalk model for a signal to noise ratio of 1.While models 1 and 2 use values for the R and Q variances that are constant overthe data set, model 3 (“Intervention analysis”) enables users to apply their own timevarying R and Q values. This is achieved by specifying the range containing additivenoise variances (i.e. R and Q time series). This enables the user to force the model to
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Figure 7.12
Sensitivity
The stochastic parameter regression menu of the tool pack
Actual Sensitivities
Alpha
X1
X2
1.00
0.50
0.00
Time
−0.50
Figure 7.13 Stochastic parameter regression example. The underlying sensitivities are assumedto follow a random walk (model 1) and the signal to noise ratio is set to 1
rapidly adapt to important events (by increasing Q during this period) or, on the contrary,to make the model insensitive to irrelevant events (by increasing R during this period).For instance, if the system noise variance Q is increased for variable X2 at time t = 52(cell “Q54” is set to a large number (e.g. 50)), the model adapts rapidly to the level shiftin β2 and models almost perfectly the actual factor sensitivities. The result of this modelis displayed in Figure 7.14.2121Model 3 should only be used to force the model to take prior knowledge into account. In general columnsN to Q in the “Example” sheet should be set to zero unless the user wants to modulate the adaptation speedof the model by increasing the credibility attributed to some observations (by putting a positive number for theadditive Q values) or by decreasing this credibility (by putting a positive number for the additive R value).The values to be entered are discretionary and depend on the user’s perception of the event’s importance. Thebigger the value the more it will alter the model’s adaptation speed for the corresponding period.
Time-Varying Factor SensitivitiesSensitivity
233Actual Sensitivities
Alpha
X1
X2
1.00
0.50Time
0.00
−0.50
Figure 7.14 Example of the stochastic parameter regression using model 3. The underlying sensitivities are assumed to follow a random walk with system noise added at t = 52 (time of levelshift in b2 ). The signal to noise ratio is still set to 1. The ﬁt is almost perfect as the model isforced to adapt to the level shift
The reader should now be familiar with the functioning of the Kalman ﬁlter. Thisadaptive procedure optimally computes the current state of the model (i.e. the factor sensitivities and their covariance matrix) given the previous state and some hyper-parameterssuch as system and observation noise variances. Two important questions remain unanswered: (i) how do we determine the initial state of the model?, (ii) how do we best setthe values for the hyper-parameters?The initial distribution of the state vector (mean s and covariance P0 ) can be setin several ways. The simplest is to use prior knowledge. For instance, one can startwith a prior value of 1 for a stock’s beta, and a prior value of 0 for the sensitivity ofthe return to exchange rates. A popular alternative is to estimate sensitivities (and theircovariance matrix) from a subsample of the data by OLS. However, this may use toomany observations from the data sample and we have already shown the shortcomingsof OLS estimation in cases where sensitivities are not constant. The model provided inthe tool pack uses a diffuse prior (see Harvey (1989)). A diffuse prior corresponds to aninﬁnite covariance matrix P0 (i.e. P−10 = 0) so that no credibility is given to the initialstate estimate, letting the Kalman ﬁlter estimate the states from the data. Practically,however, a large initial covariance matrix is used rather than an inﬁnite one. P0 is givenby equation (7.36):P0 = κI
(7.36)
κ being equal to a large but ﬁnite number (e.g. 100 000). In the case of a diffuse prior,the mean of the initial state distribution, s, is irrelevant as long as it is ﬁnite. We use avalue of zero for all state variables. The use of a diffuse prior can be viewed as a way tolet the ﬁlter estimate the initial values from the ﬁrst observations.Because a diffuse prior is used, little information needs to be supplied by the user. Oncethe time series model for the sensitivities and the values for the hyper-parameters , Q,R, d and c have been chosen, the adaptive model optimally estimates the sensitivities.The values for these parameters can be set by the user but they can also be estimated
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from the data. The model provided in the tool pack supports an optimisation algorithmin order to ﬁnd the “best” parameters.22Given the hyper-parameters, the Kalman ﬁlter derives recursive values for the factorsensitivities and their covariance. Hence, the model provides predictions of the dependentvariable and residuals can be computed, conditional on the hyper-parameter values. Asuitable maximisation routine can then be used to maximise the likelihood of observingthe data.Maximum likelihood estimation consists of ﬁnding an estimate of the unknown parameter vector θ such that it maximises the likelihood of generating the data that wereactually observed. In other words, given the sample of observations Yt , ﬁnding a solutionfor θ that maximises the joint density probability function L(Y, θ ). As observations arenot independent in a time series model, the joint density function is not the usual productof probability density function corresponding to each observation. Instead, conditionalprobability density functions are used to write the joint density function:T
L(Y, θ ) =
p(Yt |Y1 , Y2 , . . . , Yt−1 )
(7.37)
t=1
The disturbances and the initial state vector following a multivariate normal distribution byassumption, the conditional density functions p(Yt |Y1 , Y2 , . . . , Yt−1 ) are also normal. Themeans Ŷt|t−1 and variances ft|t−1 of these normal distributions are given by the Kalmanﬁlter equations (7.26) and (7.28). The model having Gaussian residuals et , the logarithmof the likelihood function L(Y, θ ) – i.e. the log-likelihood function log L(Y, θ ) – can bewritten in terms of the prediction errors et and their variances ft|t−1 that have been deﬁnedin equations (7.27) and (7.28):TTT1  et21log L(θ |Y1 , Y2 , . . . , YT ) = − ln(2π) −ln(ft|t−1 ) −22 t=12 t=1 ft|t−1
(7.38)
The maximum likelihood estimate θ̂ of the parameter vector θ is the value of θ thatmaximises log-likelihood, i.e.:θ̂ = arg max log L(θ |Y1 , Y2 , . . . , YT )θ
(7.39)
The Kalman ﬁlter algorithm and equation (7.38) are used to compute the log-likelihoodfor a given value of θ . A numerical optimiser is employed to ﬁnd the vector θ of hyperparameters that maximises the log-likelihood function.The log-likelihood function in equation (7.38) is a non-linear function of the hyperparameters and, unfortunately, there is no analytical solution θMLE to the equation∂ log L=0∂θ22
“Best” according to criteria such as prediction mean square error or a likelihood function.
(7.40)
Time-Varying Factor Sensitivities
235Given values of the hyper-parametersq = (Φ, d, ct , Rt , Qt)
Run the Kalman filter for t = 1,…, TCompute the model’s residuals et
Choose newvalues of qthat increaselog (q)
Compute the log-likelihood functionlog (q) conditional on the values ofthe unknown hyper-parameter vector q
No
log (q) = maximum?
YesStop
Figure 7.15
The maximum likelihood estimation procedure
A numerical optimisation method for ﬁnding the maximum likelihood parameters is therefore employed. This iterative procedure is represented in Figure 7.15.The critical stage of this procedure is the algorithm used to determine the parametervector that maximises log-likelihood. There are numerous methods for maximising sucha non-linear function. However, they are unequal in terms of convergence speed anddependence on initial parameter values. The optimiser used by the tool pack is based ona quasi-Newton method (see for instance Greene (1977, p. 204)). It usually converges toa minimum within a few iterations.By checking the “Estimate parameters” check box, the software optimises the R andQ parameters and saves them in the range indicated in the text box. The “Estimationoption” button enables the user to select the optimisation criterion (Figure 7.16), meansquared error or log-likelihood.
Figure 7.16 The optimisation menu for the stochastic parameter regression. System andobservation noise variances can be estimated from the data
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Applied Quantitative Methods for Trading and InvestmentSensitivity
Actual Sensitivities
Alpha
X1
X2
1.00
0.50
Time
0.00
−0.50
Figure 7.17 This is the result of the stochastic parameter regression using model 3 with optimised parameters. The underlying sensitivities are assumed to follow a random walk with systemnoise added at t = 52 (time of level shift in b2 ). The R and Q values have been computed bymaximum likelihood estimation
Once the hyper-parameters of the model are estimated, the procedure runs the Kalmanﬁlter one last time to compute the sensitivities. The R and Q values are saved on thespreadsheet. The result of the optimised model is displayed in Figure 7.17.Given the optimised hyper-parameters, the stochastic parameter model can be expressedas:Y (t) = α(t) + β1 (t)X1 (t) + β2 (t)X2 (t) + ε(t)where α(t) = α(t − 1) + η1 (t), Var(η1 ) = 1.11E-04 (cell “F3”)β1 (t) = β1 (t − 1) + η2 (t), Var(η2 ) = 6.02E-04 (cell “F4”)β2 (t) = β2 (t − 1) + η3 (t), Var(η3 ) = 8.73E-08 (cell “F5”)Var(ε) = 1.30E-08 (cell “F2”)For instance for t = 50, we have:Y = −0.19 + 0.11X1 + 0.50X2 + ε (cells J52:L52)The reader will ﬁnd it a useful and rewarding exercise to play with the provided example.Experimenting with different hyper-parameter values or injecting additional system noisethrough the intervention analysis will enable the user to get a good understanding of thefunctioning of the Kalman ﬁlter. Comparing the estimates from the various models (i.e.unweighted rolling regression, weighted regression and stochastic parameter regression)will give the reader a good understanding of the strengths and weaknesses of the variousapproaches.
7.6 CONCLUSIONThis chapter has described three methods for estimating time-varying factor models. Thethird approach, based on Kalman ﬁltering, is clearly superior to rolling regressions ortime-weighted regressions. As the reader would probably have realised when using thesoftware provided, the computational complexity of the Kalman ﬁlter can easily be handed
Time-Varying Factor Sensitivities
237
over to a computer program. As a result, the stochastic parameter regression model canbe used as easily as the less accurate rolling regression approach.The advantage of adaptive models such as the stochastic parameter regression modelresides in its capability to estimate the dynamics of the underlying factor sensitivities. Thisenables one to predict ahead risk exposures rather than merely measure their past average.Given the many applications of factor modelling and time series regression analysis inequity management, stochastic parameter regression models should prove a useful additionto the asset manager’s toolbox.
REFERENCESBollerslev, T. (1986), “Generalised Autoregressive Conditional Heteroskedasticity”, Journal ofEconometrics, 31, 307–327.Gouriéroux, C., A. Monfort and G. Gallo (1997), Time Series and Dynamic Models, CambridgeUniversity Press, Cambridge.Greene, W. H. (1977), Econometric Analysis, 3rd edition, Prentice Hall, Englewood Cliffs, NJ.Harvey, A. C. (1989), Forecasting, Structural Time Series Models and the Kalman Filter, CambridgeUniversity Press, Cambridge.Kalman, R. E. (1960), “A New Approach to Linear Filtering and Prediction Problems”, Journal ofBasic Engineering, Transactions ASMA, Series D, 82 (1), 35–45.Pindyck, R. S. and D. L. Rubinfeld (1998), Econometric Models and Economic Forecasts, 4th edition, McGraw-Hill, New York.Rosenberg, B. (1973), “Random Coefﬁcients Models: The Analysis of a Cross Section of TimeSeries by Stochastically Convergent Parameter Regression”, Annals of Social and EconomicMeasurement, 2 (4), 399–428.Schaefer, S., R. Brealey, S. Hodges and H. Thomas (1975), “Alternative Models of SystematicRisk”, in E. Elton and M. Gruber (eds), International Capital Markets: An Inter and Intra CountryAnalysis, North-Holland, Amsterdam, pp. 150–161.
8
Stochastic Volatility Models: A Survey withApplications to Option Pricing and Value atRisk∗MONICA BILLIO AND DOMENICO SARTORE
ABSTRACTThis chapter presents an introduction to the current literature on stochastic volatilitymodels. For these models the volatility depends on some unobserved components or alatent structure.Given the time-varying volatility exhibited by most ﬁnancial data, in the last twodecades there has been a growing interest in time series models of changing variance andthe literature on stochastic volatility models has expanded greatly. Clearly, this chaptercannot be exhaustive, however we discuss some of the most important ideas, focusing onthe simplest forms of the techniques and models used in the literature.The chapter is organised as follows. Section 8.1 considers some motivations for stochastic volatility models: empirical stylised facts, pricing of contingent assets and risk evaluation. While Section 8.2 presents models of changing volatility, Section 8.3 focuses onstochastic volatility models and distinguishes between models with continuous and discrete volatility, the latter depending on a hidden Markov chain. Section 8.4 is devotedto the estimation problem which is still an open question, then a wide range of possibility is given. Sections 8.5 and 8.6 introduce some extensions and multivariate models.Finally, in Section 8.7 an estimation program is presented and some possible applicationsto option pricing and risk evaluation are discussed.Readers interested in the practical utilisation of stochastic volatility models and in theapplications can skip Section 8.4.3 without hindering comprehension.
8.1 INTRODUCTIONIn the last two decades there has been a growing interest in time series models of changing variance, given the time-varying volatility exhibited by most ﬁnancial data. In fact,the empirical distributions of ﬁnancial time series differ substantially from distributionsobtained from sampling independent homoskedastic Gaussian variables. Unconditionaldensity functions exhibit leptokurtosis and skewness; time series of ﬁnancial returns show∗
We gratefully acknowledge help from Michele Gobbo, Massimiliano Caporin and Andrea Giacomelli.
Applied Quantitative Methods for Trading and Investment. 2003 John Wiley & Sons, Ltd ISBN: 0-470-84885-5
Edited by C.L. Dunis, J. Laws and P. Naı̈m
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evidence of volatility clustering; and squared returns exhibit pronounced serial correlationwhereas little or no serial dependence can be detected in the return process itself.These empirical regularities suggest that the behaviour of ﬁnancial time series maybe captured by a model which recognises the time-varying nature of return volatility,as follows:εt ∼ IID(0, 1), t = 1, 2, . . . , Tyt = µt + σt εtwhere yt denotes the return on an asset. A common way of modelling σt is to express itas a deterministic function of the squares of lagged residuals. Econometric speciﬁcationsof this form are known as ARCH models and have achieved widespread popularity inapplied empirical research (see Bollerslev et al. (1992, 1993), Bera and Higgins (1993)).Alternatively, volatility may be modelled as an unobserved component following somelatent stochastic process, such as an autoregression. The resulting models are calledstochastic volatility (SV) models and have been the focus of considerable attentionin recent years (Taylor, 1994; Ghysels et al., 1996; Shephard, 1996). These modelspresent two main advantages over ARCH models. The ﬁrst one is their solid theoreticalbackground, as they can be interpreted as discretised versions of stochastic volatilitycontinuous-time models put forward by modern ﬁnance theory (see Hull and White(1987)). The second is their ability to generalise from univariate to multivariate seriesin a more natural way, as far as their estimation and interpretation are concerned. Onthe other hand, SV models are more difﬁcult to estimate than ARCH models, due tothe fact that it is not easy to derive their exact likelihood function. For this reason, anumber of econometric methods have been proposed to solve the problem of estimationof SV models.The literature on SV models has expanded greatly in the last 10 years, reaching considerable proportions; this chapter cannot therefore be exhaustive. We prefer to discuss someof the most important ideas, focusing on the simplest forms of the techniques and modelsused in the literature, referring the reader elsewhere for generalisations and technicalities.In the organisation of the structure of the present chapter, we have been inspired by thepaper of Shephard (1996), who gave a very interesting survey on SV models updatedto 1995.To start, we will consider some motivations for stochastic volatility models: empiricalstylised facts, pricing of contingent assets and risk evaluation.8.1.1 Empirical stylised factsTo illustrate the models and to develop the examples we will work with three Europeanstock indexes: the FTSE100, the CAC40 and the MIB30, which are market indexes forthe London, Paris and Milan equity markets. These series run from 4 January 1999 to12 August 2002, yielding 899 daily observations.Throughout we will work with the compounded return1 on the seriesyt = log(xt /xt−1 )1An advantage of using a return series is that it helps in making the time series stationary, a useful statisticalproperty (see footnote 4).
Stochastic Volatility Models
241
FTSE100
0.10
CAC40
0.10
0.05
0.05
0.05
0.00
0.00
0.00
−0.05
−0.05
−0.05
040
250
500
750
040
Norm. Apx
250
500
MIB30
0.10
0
75040
Norm. Apx
250
500
750
Norm. Apx
20
20
−0.09 −0.05 −0.01 0.03 0.07
−0.10 −0.06 −0.02 0.02 0.06 0.10 −0.09 −0.05 −0.01 0.03 0.07
1.0
1.0
ACF-
0.5
20
1.0
ACF-
0.5
0
10
20
ACF-
0.5
0
10
20
0
10
20
Figure 8.1 Summaries of the daily returns on three European stock indexes: the FTSE100, theCAC40 and the MIB30. Summaries are: time series of returns, non-parametric density estimateand normal approximation, correlogram of squared returns
where xt is the value of the stock index. Figure 8.1 displays some summaries of these threeseries. The raw time series of yt suggests that there are periods of volatility clustering:days of large movements are followed by days with the same characteristics. This isconﬁrmed by the use of a correlogram on yt2 , also reported in Figure 8.1, which showssigniﬁcant correlations existing at quite extended lag lengths. This suggests that yt2 mayfollow a process close to an ARMA(1,1), for a simple AR process cannot easily combinethe persistence in shocks with the low correlation. A correlogram of yt shows little activityand so is not given in this ﬁgure.Figure 8.1 also gives a density estimate of the unconditional distribution of yt togetherwith the corresponding normal approximation.2 This suggests that yt is leptokurtic. Thisis conﬁrmed by Table 8.1, which reports an estimate of the excess of kurtosis with respectto the normal distribution, which is signiﬁcantly positive.Table 8.1 also reports the Jarque–Bera test for normality and the asymmetry coefﬁcientsevidencing that the distributions are negatively skewed, partially due to the period ofanalysis, and for all three the null hypothesis of normality is clearly rejected.These stylised facts can be summarised as follows: non-signiﬁcant serial correlation inthe levels of returns; volatility clustering, which implies a signiﬁcant and positive serialcorrelation in the squares yt2 ; heavy tails and persistence of volatility.2The graphs are produced with the Ox software using some of its basic commands and default options. Seealso Section 8.7.1.
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Table 8.1 Summary statistics for the daily returns in Figure 8.1. In parentheses: the p-value ofthe Jarque–Bera test
MeanStandard deviationAsymmetryExcess of kurtosisJarque–Bera test
FTSE100
CAC40
MIB30
−0.036340.001463−0.25741.54701597 (0.00)
−0.0002190.015630−0.2239984.624507106 (0.00)
−0.043130.001689−0.233832.2683741