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COMPARING AND RANKING COVARIANCE STRUCTURES OF M-GARCH VOLATILIYTY MODELS

COMPARING AND RANKING COVARIANCE STRUCTURES OF M-GARCH VOLATILIYTY MODELS
COMPARING AND RANKING COVARIANCE STRUCTURES OF M-GARCH VOLATILIYTY MODELS

COMPARING AND RANKING COV ARIANCE STRUCTURES OF M-GARCH

VOLATILITY MODELS

S′e bastien Laurent1,Jeroen V.K.Rombouts2,Annastiina Silvennoinen3and Francesco Violante4

October4,2006

Abstract

A large number of di?erent parameterizations have been introduced to model conditional

variance dynamics in a multivariate framework.A major problem with these models is the

large number of parameters that have to be estimated and the many constraints,often di?cult

to make explicit,that have to be imposed to ensure semi-positive de?niteness of the covari-

ance matrices.This paper compares and ranks di?erent multivariate GARCH type models

in terms of their ability to estimate the in-sample conditional variance-covariance structure

and the accuracy of their out-of-sample variance forecasts.The models are compared using

Hansens Superior Predictive Ability test rede?ned in a multivariate framework by providing

three alternative metrics to evaluate the distance between variance matrices.In this paper we

also test the preliminary version of our MGARCH Package for Ox,extension of Laurent and

Peterss G@RCH Package.Up to date,the package includes:BEKK models(Full,Diagonal

and Scalar),CCC-DCC models and RiskMetrics.

Keywords:Volatility,Multivariate GARCH,Covariance Models,SPA test,Financial econo-

metrics.

JEL Classi?cation:C10,C32,C51C52,C53,G10

1CeReFim,Universit′e de Namur,and CORE,Universit′e Catholique de Louvain.

2HEC Montr′e al,CIRANO,CIRPEE and CREF.

3Stockholm School of Economics.

4FUNDP Namur and CORE,Universit′e Catholique de Louvain.

The authors would like to thank Kilian Mie for useful suggestions and comments.

Correspondence to S′e bastien Laurent,FUNDP,Rempart de la Vi′e rge,8,B-5000Namur,Belgium.Telephone:+32 81724869.Fax:+3281724840.E-mail:https://www.doczj.com/doc/88893316.html,urent@fundp.ac.be

or to Francesco Violante,FUNDP,Rempart de la Vi′e rge,8,B-5000Namur,Belgium.Telephone:+3281724810. Fax:+3281724840.E-mail:violante@core.ucl.ac.be

Contents

1Introduction1

2Multivariate GARCH:an Overview2

2.1BEKK-GARCH(1,1) (2)

2.2RiskMetrics (3)

2.3CCC-GARCH(1,1) (3)

2.4DCC(1,1)-GARCH(1,1) (3)

3Model Comparison:distance metrics4

3.1Frobenius Metric (4)

3.2Eigenvalue Metric (4)

3.3Foerstner and Moonen Metric (5)

3.4Cosinus Mass Metric (5)

4Realized Volatility7 5Model Comparison:Hansen SPA Test8 6Data and Empirical Results10 7Conclusions13

1Introduction

Since the?rst seminal paper of Kraft and Engle(1982)an increasing interest in modelling con-ditional covariances has developed a large number of multivariate volatility models,see Bawens, Laurent and Rombouts(2006)for an extensive survey.

In principle all these models su?er from the increase of the cross-sectional dimension due to the large number of parameters to be simultaneously estimated which increases at least quadratically1, even when parameter restrictions,tipically parameter pooling,are imposed.Yet a further compli-cation rises from the necessity to impose non-linear parameter constraints to ensure semi-positive de?nite conditional covariance matrices.

The aim of this paper is to compare and rank,in terms of ability to estimate the sample covari-ance structure and forecast accuracy,some of the most well known and widely used multivariate GARCH models.The speci?cations cosidered include the BEKK(Engle and Kroner1995),CCC (Bollerslev1990),DCC(Engle2001),RiskMetrics(JPMorgan1996).

After checking the consistency of the procedure-i.e.estimation,forecast,comparison-using ad hoc simulated data we compare di?erent model speci?cations using daily returns for two stocks traded on the NYSE for the period1980-1999,while the out-of-sample realized volatility has been computed using5-minute data from1995to1999.

Models are evaluated using a loss function,which is applied both to the sample conditional covariance estimation and to the sequence of one step ahead covariance forecasts.The latent conditional covariance in the loss function is substituted,depending on the context,either by simulated data or by realized volatility.

The sequence of conditional covariance forecasts has been obtained by means of a moving window of dimension k within which at each iteration the one step ahead forecast has been computed replacing the expectations for the squared innovation with its observed value.The coe?cients are updated every k observations by restimating the model including the last k out of sample observations into the dataset.

Since the objective of this paper is to compare covariance matrices the choice of an appropriate loss function has to be compatible to the multivariate framework.In this paper we investigate the problem providing three di?erent notions of distance between matrices.The goodness of?t of each model to the latent conditional covariance is then tested using Hansen(2001)Superior Predictive Ability Test(SPA).This test has the advantage that it properly accounts for the full set of models and is therefore likely to detect a superior model when such exist.In fact,SPA test evaluates whether a particular model(benchmark)when compared to the true data generating process(DGP)-simulated data or realized volatility-is signi?cantly outperformed by other models,while taking into account the whole set of models that are being compared.

In this paper we also introduce the?rst version of a new OxMetrix application to estimate and forecast multivariate GARCH models.The objective is to extend the work done by Laurent 1Engle and Sheppard(2005)

and Peters(2006)with their G@RCH package dedicated to univariate ARCH-type models.Other than including procedures to estimate and forecast BEKK,CCC,Engle’s DCC and RiskMetrics speci?cations for the conditional covariance,it provides also some features to evaluate distance metrics,such as Frobenius norm,Eigenvalue metric(spectral norm),Foerstner and Moonen metric and Cosinus Mass metric.

The paper is organized as follows.Section2provide a brief overview of several GARCH speci?cation considered in this paper,while the loss functions are described in Section3.Section 4and5,contain some details on realized volatility and of the Hansen(2001)SPA test.Empirical results are presented in Section6.Finally,Section7contains some concluding remarks.

2Multivariate GARCH:an Overview

Consider a vector price process,p t of dimension(N×1),let de?ne the compounded daily return by r t=log(p t)?log(p t?1).The process r t is a vector stocastic process conditioned on the sigma?eld, t?1generated by the information avaliable at t?1.We denoteμt≡E(r t| t?1)the conditional mean vector and H t≡E(r t r t| t?1)the conditional variance matrix.The speci?cation used for the mean equation is:

r t=μt+ t(1)

z t(2)

t=H1/2

t

is a(N×N)positive de?nite matrix and z t is i.i.d random vector with E(z t)=0 where H1/2

t

and V ar(z t)=I N.

In this paper we consider six speci?cations for H t,namely the BEKK by Engle and Kroner, 1995,toghether with the diagonal and the scalar variants,the RiskMetrics by JPMorgan,1996, the Constant Conditional Correlation(CCC)by Bollerslev,1990and the Dynamic Conditional Correlation(DCC)by Engle,2001.In the following Section we will brie?y describe the models formulation2.

2.1BEKK-GARCH(1,1)

The BEKK-GARCH(1,1)is de?ned as follows:

H t=C? C?+A? t?1 t?1A?+G? H t?1G?,(3) where C?,A?and G?are(N×N)matrices and C?is upper triangular.The number of parameters to be estimated is N(5N+1)/2.

2See Bawens,Laurent and Rombouts(2006)for further detail

To reduce this number we can impose the diagonality of the parameter matrices A ?and G ?-i.e.Diagonal BEKK speci?cation -or impose that A ?and G ?are equal to a scalar times the identity matrix -i.e.Scalar BEKK.

2.2RiskMetrics

The RiskMetrics is an integrated Scalar-BEKK speci?cation -i.e.A ?A +G ?G =I k ;where I is the identity matrix and k is the number of series -where the variance equation parameters are ?xed and chosen ex-ante (does not require numerical optimization).The value assigned to the

parameters is a 2ij =0.06and g 2ij =0.94?i =j and 0elsewhere.

2.3CCC-GARCH(1,1)

The CCC is de?ned as follows:H t =D t RD t = ρij h iit h jjt (4)

where

D t =diag (h 1/211t ...h 1/2NNt )

(5)

h iit can be de?ned as any univariate GARCH model,and

R =(ρij )

(6)is a symmetric positive de?nite matrix with ρii =1,?i .

R is the matrix containing the constant conditional correlations ρij .In this paper we specify the univariate process as a GARCH(1,1)speci?cation for each conditional variance in D t :

h iit =ωi +αi 2i,t ?1+βi h ii,t ?1(7)2.4DCC(1,1)-GARCH(1,1)

The DCC(1,1)model by Engle (2002)can be speci?ed as:

R t =diag (q ?1/211,t ...q ?1/2NN,t )Q t diag (q ?1/211,t ...q ?1/2NN,t )

(8)

where the (N ×N )symmetric positive de?nite matrix Q t =(q ij,t )is given by:

Q t =(1?θp ?θq )Q +θp u t ?1u t ?1+θq Q t ?1(9)with u t = i,t / (h ii,t ).Q is the (N ×N )unconditional variance matrix of u t ,and θp and θq are

nonnegative scalar parameters with θp +θq <1.

3Model Comparison:distance metrics

It is not obvious which loss function is more appropriate to evaluate the distance between two matrices as required by the problem we are facing of comparing covariance matrices-i.e.the latent (Σt)and the estimated(H t)covariance matrices sequences.In this paper we use the following four loss functions which,for their properties,seems to be appropriate for our purposes.

3.1Frobenius Metric

The idea behind Frobenius norm is very similar to that of the Euclidean norm on R n.The Frobenius distance between two matrices(Σt and H t)is de?ned as:

d2t=

i,j

|σt,ij?h t,ij|2?i,j(10)

=T race[(Σt?H t)(Σt?H t)?]

= rows(Σt?H t)

i=1

SV i(Σt?H t)

where(.)?denotes the conjugate transpose of(Σt?H t)and SV(.)are the singular values of (Σt?H t)which are de?ned as the eigenvalues of[(Σt?H t)(Σt?H t)?]1/2.Since(Σt?H t)?has only real entries and it is symetric then it is de?ned as self-adjoint or Hermitian matrix,that is (Σt?H t) =(Σt?H t)?.Therefore we can compute the Frobenius norm as the square root of the sum of element-wise squared di?erences,that is:

d t=

i,j

(σt,ij?h t,ij)2?i,j(11)

Given the second equality in Eq.10this distance metric is also called trace norm which is de?ned as the sum of all the singular values of(Σt?H t).

3.2Eigenvalue Metric

The Eigenvalue metric,or spectral norm,is based on the concept of a standard principal component analysis.AssumingΣt and H t to be two covariance matrices,therefore symetric and with real entries,then the di?erence matrix at time t is Hermitian and the distance is given by:

d t=

λ1max(Σt,H t)(12)

whereλ1max(Σt,H t)represents the largest eigenvalue of the matrix(Σt?H t)times its transpose -i.e.(Σt?H t)(Σt?H t) .

SinceΣt and H t are covariance matrices,therefore symetric and semi-positive de?nite,standard principal component analysis implies that the largest eigenvalue is non-negative and captures the di?erence in form ofΣt and H t completely,providing a metric for the largest amount of error between these two matrices.

Alternatively,the spectral norm can be de?ned as the lagest singular value of the semi positive de?nite matrix (Σt ?H t ),i.e.:

d t =λ1max (Σt ?H t )(Σt ?H t ) (13)

where once again λ1max (.)represents the largest eigenvalue of (.).

From here to the end of this paper and in our empirical analysis we will always refer to the de?nition provided in Eq.12.3.3Foerstner and Moonen Metric

This metric has been introduced by Foerstner and Moonen (1999)as a generalization of the idea introduced in Section 3.2.The distance between two symetric semi-positive de?nite matrices Σt and H t is de?ned as the sum of the squared logarithms of the eigenvalues,that is:d t = n i =1ln 2(λ1i (Σt ,H t ))

(14)

with eigenvalues λ1i (Σt ,H t )from the system |λΣt ?H t |=0.The square root of summing squares closely resemble the Euclidean metric.Again,since the covariance matrices are symetric and semi-positive de?nite,the eigenvalues are ensured to be positive 3.

3.4Cosinus Mass Metric

This measure is quite uncommon with respect to metrics commonly used in the econometrics literature but its simplicity appeals in our framework.Since Σt and H t are symetric semi-positive de?nite matrices the distance can be de?ned as:d t =cos ?1 (LT Σt ) (LT H t ) [(LT Σt ) (LT Σt )]·[(LT H t ) (LT H t )]

(15)where LT Σt =vech (Σt )and LT H t =vech (H t )and vech (·)denotes the operator that stacks the lower triangular portion of a (k ×k )matrix as a (k (k +1)/2×1)vector.

Intuitively,the distance is measured by the angle between the two vectors de?ned by the the stacked lower triangulars of Σt and H t ,and hence the smaller is the angle,the more Σt and H t are closer.

Given these four de?nitions we can de?ne the ?t of a multivariate GARCH-type model by the average distance between the matrices Σt and H t over t :

ModelF itness =1T T t =1d t (16)

To provide an example we generate two random variables with mean zero,conditional variance that follows individually an univariate GARCH (1,1)process and conditional correlation generated by the DCC(1,1)process by Engle (2002).The paramaters of the univariate models and of the unconditional correlation equation are:

3Further details on the properties of this metric are in Foerstner and Moonen (1999)

Table1:DGP speci?cation for GARCH(1,1)-DCC(1,1)process

Sample size:20000Number of series:2

GARCH(1,1)DCC(1,1)

Equation forσ21Equation forσ22Correlation Equation

ω10.2ω20.3ˉρ0.6

α10.1α20.2θp0.2

β10.7β20.5θq0.7

We will now estimate the simulated process using a CCC and a DCC model respectively and then we will compute the distance between the sequences of both in-sample conditional variance (H t)and conditional correlation(R t)sequences and the underlyingΣt andΨt using the four distance metrics illustrated in the beginning of this Section.Table2and3contain the estimation results.

Table2:Estimation results for GARCH(1,1)-CCC

Sample size:20000Number of series:2

GARCH(1,1)

Equation forσ21Equation forσ22

Coe?cient Std.Error t-value Coe?cient Std.Error t-value ω10.1788020.0242987.359ω20.2899790.01958614.81

α10.0897940.008086111.10α20.1877510.009884318.99

β10.7334490.02976924.64β20.5254220.02462721.33

CCC

Coe?cient Std.Error t-value

ρ0.462270.005046991.596

The results in Table4con?rm that DCC model better?ts the true DGP outperforming the CCC whether we consider the Frobenius norm,Eigenvalue metric,Forstner and Moonen or Cosinus Mass metric.The di?erence in the scale is obviously due to the di?erent intuition behind each metric.

If on one side this provides a simple way to rank di?erent models,based on sample per-formances,whether we compare in sample estimation or out of sample forecasts,in the next sections we will describe a tool to test whether the di?erences between these loss functions-i.e. X t=(Σt?H t,0)?(Σt?H t,i)?i-signi?cantly di?ers from zero and therefore if one model statistically outperforms the all the others.

Table3:Estimation results for GARCH(1,1)-DCC(1,1)

Sample size:20000Number of series:2

GARCH(1,1)

Equation forσ21Equation forσ22

Coe?cient Std.Error t-value Coe?cient Std.Error t-value ω10.1788020.0242987.359ω20.2899790.01958614.81

α10.0897940.008086111.10α20.1877510.009884318.99

β10.7334490.02976924.64β20.5254220.02462721.33

DCC(1,1)

Coe?cient Std.Error t-value

ρ0.518910.01148145.197

θp0.173230.004844935.755

θq0.743300.007657897.064

Table4:Distance Metrics

Sample size:20000-Number of series:2

Frobenius MetricΣt?H t,DCC0.13564Ψt?R t,DCC0.13953

Σt?H t,CCC0.37177Ψt?R t,CCC0.36814

Eigenvalue MetricΣt?H t,DCC0.10710Ψt?R t,DCC0.098662

Σt?H t,CCC0.27548Ψt?R t,CCC0.26032

Forstner MetricΣt?H t,DCC0.23337Ψt?R t,DCC0.22773

Σt?H t,CCC0.56888Ψt?R t,CCC0.56565

Cos Mass MetricΣt?H t,DCC0.062785Ψt?R t,DCC0.062947

Σt?H t,CCC0.16320Ψt?R t,CCC0.16641

4Realized Volatility

In our empirical analysis we substitute the realized variance matrix to the latent conditional covariance when evaluating the goodness of the endogeneously generated volatility implied by the di?erent M-GARCH-type models.The daily realized variance is computed from the intraday returs and is de?ned as:

r t+?,?≡p t+??p t(17) for t=?,2?,...,T and?=1/m,where m is the number od daily intervals.

The corresponding daily return,for t=1,2,...,T over the time interval of length?=1/m,is

de?ned by:

r t+1≡

m

i=1

r t+i?,?≡r t+?,?+r t+2?,?+...+r t+1,?(18)

Then,starting fron the(T m×n)matrix of intraday returns,we de?ne the daily realized

volatility as:

V t+1≡1/?

i=1

r t+i?,?r t+i?,?=R t+1R t+1(19)

where the(m×n)matrix R t+1is de?ned by R t+1=(r t+?,?,r t+2?,?,...,r t+1,?).The matrix V t+1represents the empirical counterpart to the daily quadratic return variation.In fact as the sampling frequency of the intraday returns increases,?→0-i.e.m→∞,V t+1converges almost surely to the quadratic variation(QV):

?→0,V t+1(?)as

?→

t+1

t

Σu du≡QV t(20) Therefore the realized volatility V t becomes a less noisy measure for the latent volatility as the intraday frequency?reduces4.As shown in Andersen et al.(2002),to ensure positive semi de?niteness of the realized covariance matrices it su?ces that the columns in the matrix of returns R t are linearly independent.However this condition,which could eventually be preserved even for high dimensional problems,will be violated when the cross-sectional dimension increases with respect to sampling frequency of the intraday return.The condition n

5Model Comparison:Hansen SPA Test

A way to test model di?erences across models is the asimptotic test by Diebold and Mariano (1995)or the non parametric test by Pesaran and Timmermann(1992).However the limit of these tests is that they only implement pairwise comparisons across models.Since our aim is to test whether a model,when compared to the true DGP,is outperformed by all the other models, we need to test their performances together.An alternative method(Andersen et al.,2002),in the tradition of Mincer-Zarnowitz(1969),is to evaluate volatility forecasts,and thus their relative performances,by projecting the realized volatility on a constant and the various model forecasts. The Mincer-Zarnowitz regression takes the form:

vech(V t+1)=b0+b1vech(H t+1,i)+b2vech(H t+1,j)+u t+1(21) where b0is expected to be equal to(0,...,0)and b1=(1,...,1)and b2=(0,...,0)-or alternatively b1=(0,...,0)and b2=(1,...,1)-de?nes the best model forecast for the realized volatility.The R2of the regression represents the goodness of?t and therefore the term of comparison between di?erent models.However,the R2of a Mincer-Zarnowitz regression is not an ideal criterion for comparing volatility models,because it does not penalize a biased forecast5.

4Note that the optimal choice for the intraday frequency needs to take into account the distortion due to microstructure e?ects

5See Hansen and Lunde(2005)

In this paper,we use the Test for Superior Predictive Ability by Hansen (2001)which is based on the Reality Check Test by White (2000)6and the work of Diebold and Mariano (1995)and West (1996).This test controls for the whole set of models to be compared.

Since our aim is to compare the performances of two or more multivariate models,while Hansen’s SPA Test is originally designed for the univariate framework 7,?rst we have to introduce new notions of distance metrics -i.e.the distance metrics di?ned in Section 3-to ?t the test to the multivariate setting 8.However,since the test in itself is based on the average amount of distance between pairs of points in time,which is a scalar regardless of the dimension,it re-mains substantially the same but for some algebrical adaptation to the higher dimensional setting.Let n =0,1,...,N denote the number of estimated models for which we compute one-step-ahead covariance forecasts and let denote H n,1,...,H n,T the sequence of covariance matrices forecasts for model n which will be compared to the latent covariance matrix Σt by mean of a prede?ned loss function L n,t ≡L (Σt ,H n,t ),t =1,...,T .Yet,let denote with n =0the model that is chosen as the benchmark and that is compared to the set of competing models n =1,...,N .By using SPA Test we can analyse whether any of the competing models outperforms the benchmark model in terms of predictive ability.

The relative performance of model n (n =1,...,N ),with respect to the benchmark model (n =0)at time t ,is de?ned as:

X t,n =L 0,t ?L n,t ;n =1,...,N ;t =1,...,T (22)

Hansen (2001)showed that under reasonable assumptions the moment λn ≡E [X t,n ]is well-de?ned for n =1,...,N .Therefore,a λn >0implies that model n outperforms the benchmark and allows for a formulation of the null hypothesis of the SPA test.More clearly,under the null hypothesis we will verify if λn ≤0for n =1,...,N .If one of the competing models outperforms the benchmark,the null hypothesis will be rejected.Focusing just on the model which yields the highest λn gives us the following null hypothesis:

H 0:λs max ≡max n λn ωnn ≤0(23)

where λs max denotes the best standardized relative performance and ωnn the asymptotic vari-

ance of λn .Since the sample average ˉX t,n =t ?1 T i =1

X i,n is a consistent estimate for λn the corresponding test statistic is:

T sm t =√t ˉX s t,max (24)

where ˉX s t,max =max n ˉX t,n ?ωnn

and ?ω2nn is a consistent estimator of ω2nn ≡lim n →∞V ar (√n ˉX t,n )and is 6Hansen’s

Spa Test demostrates better power properties.See Hansen (2001)for further details 7In the multivariate framework arrays of covariance matrices rather than vectors of variances are compared 8The choice of an appropriate loss function in the univariate case usually boils down to some measure of squared errors.The test in fact provides two prede?ned loss functions:Mean Squared Error (MSE)and Mean Absolute Deviation (MAD)

estimated using the bootstrap method9.Hence,T sm

t

represents the largest t-statistic(of relative performance)where the superscript sm stands for’standardized maximum’.Under regularity conditions,stated above:

T sm t =max n

ˉX

t,n

?ωnn

p

?→max n

λn

ωnn

(25)

which is lower then0if and only ifλn≤0for some n.More clearly,the question is whetherˉX s t,max is large enough in order to reject the null hypothesis ofλs max≤0.Therefore,the SPA test,using the bootstrap technique in order to estimate the distribution ofˉX s t,max under the null hypothesis, allows to determine critical values as thresholds whereˉX s t,max becomes too large to be consistent with the null hypothesis.Rather than giving a detailed description of the bootstrapping10we will focus on the general idea behind this method.Given the previous assumptions-i.e.λn≡E[X t,n] is well-de?ned for n=1,...,N-it holds that:

ˉX?λ)d?→N(0,?)(26) whereλ=(λ1,...,λN) ,ˉX=(ˉX1,...,ˉX N) and?=E[n(ˉX?λ)(ˉX?λ) ].The covariance matrix?will be estimated by bootstrapping methods.More precisely,the test uses the observed sample in order to generate k random resamples-i.e.iid bootstrap method-from the distribution ofˉX.For each of these resamples an estimate ofω2nn is computed and therefore the distribution of

ˉX s

t,max

approximated.Finally,critical values and p-values can be calculated from the estimated distribution ofˉX s t,max.

In other words,the SPA test is a method which yields the probability distribution against which one compares the best performance from those of some competing models.It generates the probability distribution of the model which performs best relative to the benchmark.This distribution is certainly not a standard one which we can look up in a text book but rather strongly depends on the models which enter our model comparison.As a result,the distribution is di?erent in every case and one has to?nd it by means of statistical methods,such as the bootstrap.

6Data and Empirical Results

Our empirical analysis is based on General Electric(GE)and IBM stock returns for the period January1,1981-December31,1999.Returns and realized variances are computed from equally spaced?ve-minute prices from the Trade and Quote(TAQ)database-i.e.78intraday observations. The choice of5-min.intervals strikes a satisfactory trade o?between the accuracy of the continuous time underlying process and the noise due to market microstructure e?ects.

Missing values are obtained by linearly interpolating the closest previous and the?rst following 5-min.price.The?ve-minute returns are computed as the?rst di?erence of the logarithmic prices premultiplied by100.The dataset has been cleaned from weekends and holydays and similarly, 9See Hansen(2001)and Hansen,Kim and Lunde(2003)for further details

10See Hansen(2001)for technical details

we do not consider early closing days.The period from January1,1995to the end of the sample will be used for the out of sample evaluation to compare model forecasts.We?nally end up with3652in sample and1187out of sample observations.Out of sample forecasts have been computed by means of a moving window of dimension k=100within which,at each iteration k,the one step ahead forecast has been computed replacing the expectations for the t+1 t+1 in the variance equation forecast by its observed value.The coe?cients are updated every100 observations restimating the model by including the last100out of sample observations into the dataset.

Estimation results for the six competing models are avaliable on request.The dynamic behind the one step ahead covariance matrices forecast is reported in Figure1.Table5contains the p-values of the out of sample one-step ahead forecasts comparisons under the null hypotesis that the benchmarck model,the BEKK-GARCH(1,1),is the best forecasting model.The consistent p-value-i.e.p?value c-is produced by the SPA test and is asimptotically valid and unbiased11. This p-value informs whether the benchmark model is outperformed,in terms of predictive ability, by one or more competing models.More precisely,a high p-value informs that there is no evidence that one or more competing models outperform the benchmark.The SPA test provides also an upper-i.e.p?value u-and a lower-i.e.p?value l-bound for the consistent p-value.Tables5, 6and7report the p-values-i.e.the consistent and its bounds-the sample performance of the model under the null(benchmark),sample informations and the bootstrap parameters12.

Table5:SPA test p-values-A

Benchmark model:BEKK-GARCH(1,1)

Number of models:n=5Sample size:n=1100

Test Statistic:TestStatScaledMax()

Bootstrap parameters:B=1000(resamples)q=0.5(dependence)

Loss function p?value l p?value c p?value u Sample performance

Frobenius Metric0.480000.716000.93500?3.48049

Eigenvalue Metric0.477000.738000.95500?3.35674

Forstner Metric0.590000.590000.99800?0.84536

Cos Mass Metric0.486000.486001.00000?0.27176

The results in Table5show that there is no evidence that BEKK-GARCH(1,1)is outper-formed by any of the other models.Moreover BEKK-GARCH(1,1)has always the best sample performance measured as the average distance between the out of sample realized variance and the BEKK-GARCH(1,1)one step ahead variance forecasts.

Further evidence is given by the results reported in Table6where the SPA test is performed

11See Hansen,Kim and Lunde(2003)for technical details

12See Hansen,Kim and Lunde(2003)for details

Figure1:Variance matrices forecasts

under the null that the DCC(1,1)-GARCH(1,1)is the best model.Here the null is rejected at least at5%signi?cance level con?rming the results reported above.

Table6:SPA test p-values-B

Benchmark model:DCC(1,1)-GARCH(1,1)

Number of models:n=5Sample size:n=1100

Test Statistic:TestStatScaledMax()

Bootstrap parameters:B=1000(resamples)q=0.5(dependence)

Loss function p?value l p?value c p?value u Sample performance

Frobenius Metric0.006000.006000.00900?3.57184

Eigenvalue Metric0.003000.003000.00500?3.44561

Forstner Metric0.024000.024000.04100?0.86912

Cos Mass Metric0.000000.000000.00000?0.31193

A partially contradicting conclusion is suggested by the results reported in Table7where the Diagonal-BEKK-GARCH(1,1)model has been chosen as benchmark model.This check is performed since the latter speci?cation is a variant of the BEKK-GARCH model,which results in a lower parametrization by means of parameter pooling,and it ranks as second best model when sample performances are considered.

There is no evidence that,when using the Eigenvalue loss function,the model is outperformed

Table7:SPA test p-values-C

Benchmark model:Diagonal BEKK-GARCH(1,1)

Number of models:n=5Sample size:n=1100

Test Statistic:TestStatScaledMax()

Bootstrap parameters:B=1000(resamples)q=0.5(dependence)

Loss function p?value l p?value c p?value u Sample performance

Frobenius Metric0.286000.286000.67100?3.49623

Eigenvalue Metric0.263000.263000.62500?3.37531

Forstner Metric0.006000.009000.02300?0.86185

Cos Mass Metric0.000000.000000.00000?0.28482

by the others since the SPA test consistent p-value is not negligible,even though if we consider any of the other metrics the null is always rejected.

7Conclusions

In this paper we have estimated and compared six di?erent multivariate GARCH-family volatility models,namely BEKK(Full,Diagonal and Scalar speci?cations),Riskmetrics,CCC and DCC (Engle2001)models in terms of their out of sample conditional covariance forecast ability.The dataset used consists of daily stock returns of GE and IBM.Five-min.returns are used to com-pute the realized variance matrix,which represents a proxy for the latent conditional covariance when evaluating out of sample forecasts model?tness.Our aim was to extend Hansen(2001)SPA test to the multivariate framework by providing four alternative matrix distance metrics,namely Frobenius,Eigenvalue,Forstner and Moonen and Cosinus Mass metric.

The main?nding is that there is no evidence that the BEKK-GARCH(1,1)speci?cation is outperformed by any of the other models,resulting to be the best forecasting model.Further-more there is no evidence of SPA test lacking power in the multivariate setting since all the other models are rejected when the test is performed under the null that Diagonal and Scalar BEKK, Riskmetrics,CCC and DCC are considered as the benchmark13.

With this paper we also provide and test a?rst version of a new OxMetrix application to esti-mate and forecast multivariate GARCH-type volatility models.The package includes estimation and forecast procedure for:BEKK models(Full,Diagonal and Scalar),CCC and DCC models and RiskMetrics.It also includes simulation procedures and some functions to evaluate the distance between arrays of matrices.

13With the exception of Frobenius and Eigenvalue metrics under the null that Diagonal BEKK is the best forecasting model

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蚕蛹蛋白混纺纱的开发

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目录 第一章概述..... .. (1) 第二章产品介绍 (2) 一、产品特性 (2) 二、技术指标 (3) 三、仪表结构 (4) 第三章使用方法 (6) 一、准备工作 (6) 二、开始测试 (7) 三、屏蔽端使用方法 (8) 四、电池充电 (9)

第一章概述 随着我国电力工业的快速发展,电气设备预防性实验是保障电力系统安全运行和维护工作中的一个重要环节。绝缘诊断是检测电气设备绝缘缺陷或故障的重要手段。绝缘电阻测试仪(兆欧表)是测量绝缘电阻的专用仪表。1990年5月批准实施的JJG662-89《绝缘电阻表(兆欧表)》已把它作为强制检定的仪表之一。目前,电气设备(如变压器、发电机等)朝着大容量化、高电压化、结构多样化及密封化的趋势发展。这就需要绝缘电阻测试仪本身具有容量大、抗干扰能力强、测量指标多样化、测量结果准确、测量过程简单并迅速、便于携带等特点。 绝缘电阻测试仪采用超薄形张丝表头、多种电压等级输出、容量大、抗干扰强、交直流两用(C型)、操作简单、具有时间提示功能。是测量变压器、互感器、发电机、高压电动机、电力电容、电力电缆、避雷器等绝缘电阻的理想测试仪器。

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