Multivariate ARCH Models: Finite Sample Properties of QML Estimators and an Application to an LM-Type Test

Transcription

1 Multivariate ARCH Models: Finite Sample Properties of QML Estimators and an Application to an LM-ype est E M. I and G D.A. P University of Alicante Cardiff University April,004 Abstract his paper provides two main new results: the first shows theoretically that large biases and variances can arise when the QML estimation method is employed in a simple bivariate structure under the assumption of conditional heteroscedasticity; and the second examines how these analytical theoretical results can be used to improve the finite sample performance of a test for multivariate ARCH effects providing an alternative to a traditional Bartlett type correction. We analyse two models: one proposed in Wong and Li 997 and another proposed by Engle and Kroner 995 and Liu and Polasek 999, 000. We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations which leads to the two series having correspondingly different volatilities in the restricted case, may produce very large variances in some QML estimators in the first model and very severe biases in some QML estimators in the second. Later we use our bias expressions to propose an LM type test of multivariate ARCH effects and show through simulations that small sample improvements are possible when we bias correct the estimators and use the expected hessian version of the test. JEL-code: C3, C3. Keywords: Multivariate GARCH, QML, Finite Sample Properties. Corresponding author. Dpt. Fundamentos del Análisis Económico. University of Alicante. Campus de San Vicente, 03080, Alicante, Spain. address: emiglesias@merlin.fae.ua.es. Both authors thank H. Wong for providing us with the Gauss program to simulate the Wong and Li 997 model. We also thank the comments received at seminars given in Cardiff University, University of Exeter and Michigan State University. We acknowledge gratefully as well the financial support from an ESRC grant Award number:

2 Introduction he multivariate-arch autoregressive conditional heteroscedastic model was first introduced by Kraft and Engle 983, and Bollerslev, Engle and Wooldridge 988. Since then, new combinations of this specification in the variance equation with different structures in the mean equation have been proposed: for example Baba, Engle, Kraft and Kroner 99, Harmon 988, and Engle and Kroner 995 introduced the theoretical framework of simultaneous equation models and Calzolari and Fiorentini 994 have considered some cases of non-linear simultaneous equations, while Polasek and Kozumi 996 proposed the VAR-GARCH structure. he multivariate model implies that the conditional variance - covariance matrix H t of the disturbances ε t depends on the information set I t. If we assume normality in the conditional distribution following Engle and Kroner 995, the multivariate-garch Generalised-ARCH model can be written as: ε t /I t N 0,H t he main problem to be faced in this specification is the relatively large number of parameters that are involved. here are, however, many possible parameterisations for H t in order to reduce the number of parameters to estimate. We begin by considering the vech vec-half representation, which in the simple bivariate case becomes: vechh t = h t h t h t = c 0 c 0 c 03 + g g g 3 g g g 3 g 3 g 3 g 33 + a a a 3 a a a 3 a 3 a 3 a 33 h,t h,t h,t ε,t ε,t ε,t ε,t For the estimation of this model, it is necessary to restrict the number of parameters still further. Another possible specification is the diagonal representation, where each element of the covariance matrix h jk,t is a function, only, of past values of itself and past values of ε j,t ε k,t. In the case of the bivariate model, it becomes: vechh t = h t h t h t = c 0 c 0 c 03 + g g 0 + a a g a 33 h,t h,t h,t ε,t ε,t ε,t ε,t he drawback is that we must still ensure that H t is a positive definite matrix for all values of the ε t, and it can be a difficult task to check this in the previous

3 specifications. his is why Engle and Kroner 995 proposed a new parameterisation: the BEKK Baba, Engle, Kraft and Kroner 99 representation, where...it includes all positive definite diagonal representations, and nearly all positive definite vech representations... Engle and Kroner 995. In the simple bivariate case, it becomes: a H t = C0 C 0 a + a a + g g g g ε,t ε,t ε,t ε,t ε,t ε,t g H g t g g a a a a Hence, this solution is not as restrictive as the diagonal representation, but...comparing this model to the vech form of the model, we see that this model economises on parameters by imposing restrictions both across and within equations... Engle and Kroner 995. Nowadays there exists an extensive literature about multivariate-arch models that have been applied to different varieties of data. Most of them use Quasi- Maximum Likelihood QML-ML as the estimation procedure. However, there are relatively few theoretical papers that examine the consequences of doing that. he relevant part of the conditional log-likelihood function in these models is denoted by: L y, θ = L t y t,θ= t= log H t t= t= y t µ t H t y t µ t. Liu and Polasek 999 gave the following representation of the conditional information matrix of the ML estimator I θ in a general multivariate heteroscedastic model: I θ = t= vechht θ D H t H D vechh t t θ + t= µt H θ t µ t θ. see Liu and Polasek 999, page 03, where µ t = E y t /I t is an M conditional mean vector, H t = var y t /I t is an M M conditional variance matrix, D is the M M M +/ duplication matrix and indicates Kronecker product. hey argue that this formula corrects the initial work by Wong and Li 997 who omitted 3

4 the last expression of the equation. However, Liu and Polasek 999 themselves introduced an error when they applied this expression to a VAR-VARCH Vector- AR-Vector-ARCH model, because they neglected the influence of changes in the parameters in the variance equation on the own disturbance see Liu and Polasek 999, page 05, which illustrates the difficulties of dealing with these models from the theoretical point of view. Regarding asymptotic theory, uncer 994, 000, Bauwens and Vandeuren 995 and Comte and Lieberman 003 have established the strong consistency of the Quasi-Maximum Likelihood estimator QMLE in a simple multivariate-arch model. Asymptotic normality is proved provided that the initial state is either stationary or fixed. More recently, Ling and McAleer 003 have shown the asymptotic normality in a VARMA-GARCH model requiring only the existence of the second-order moment of the unconditional errors, and a finite fourth-order moment of the conditional errors, which represents an important advance. On the other hand, Hafner 000, analyses the fourth moment in this model. However, these papers do not consider the issue of the sample size needed in order to have confidence in the asymptotic result, and in this paper we provide results which go some way towards addressing this. In relation to finite samples, in a more recent paper, Liu and Polasek 000 have compared through Monte Carlo simulation the biases that are generated using the Splus+GARCH program package of MathSoft 996, the BASEL package of Polasek 999 and the application of the method of scoring for MLE using the exact information matrix given previously. he generated biases are seen to be striking, and the Bayesian method seems to be the best alternative there are in fact some recent papers that analyse different types of Bayesian bivariate-arch models applied to economic data, such as Osiewalski and Pipien 004. For a sample size of 00 observations, these results show the existence of severe biases, and this is precisely what has motivated the work in our present paper. On the other hand, Wong and Li 997 reported through Monte Carlo simulation that in their model, the biases in the parameters were very small see Wong and Li 997 pages 9-. In this paper we are interested in a further analysis of these two models. It is interesting to note as well that in a recent paper, Jensen and Rahbek 004 prove how in univariate ARCH processes the QMLE is always asymptotically normal providing that the fourth moment of the innovation process exists, whether or not the process is stationary. his gives support to the estimation of ARCH processes without being subject to contraints, and in this paper we carry out the estimation through unrestricted QML. he plan of the paper is as follows. In the next section we will begin analysing a bivariate model under two specifications that have been proposed in the literature so far: the one given in Wong and Li 997, where they allow the two disturbances to be dependent but not correlated, and the one proposed in Engle and Kroner 995 and Liu and Polasek 999, 000, where linear dependence between the disturbances is introduced. We provide theoretical results of the O biases for the QML esti- 4

5 mators in each specification under the assumption of conditional heteroscedasticity. We impose the restriction that the variance parameters are zero; hence following an approach that can be found in a number of other studies see Engle, Hendry and rumble 985 and Linton 997. In effect, we consider the case where conditional heteroscedasticity is assumed when, in fact, it is absent. For easy of manipulation, we assume as well that the intercept in the mean equation is known, although more complicated structures could be analysed following the same methodology. In fact, the results given in Iglesias and Phillips 003 can be applied as well in this setting so that we could allow for the inclusion of any number of exogenous variables in the conditional mean equation. Although our theoretical results are obtained in a very restricted model, we are able to prove how in the Wong and Li 997 model the variances for some estimators can be large when there is a relatively large difference between the intercepts in the variance equations they only showed results for cases when the intercept parameters had very similar numerical values and in the second model which is examined in Liu and Polasek 999, 000 we show that a large difference in the intercepts under the null of no ARCH-effects when the two series can have very different volatitilies can produce very large biases in some of the QMLEs. We show as well theoretically how in the Wong and Li and Liu and Polasek models some assumptions should be imposed for the QML estimator to be well-defined. We provide evidence that the biases can be very different depending on both the structure we impose on the model, and on the combinations of the parameters we study. We also analyse some invariance properties extending the Lumsdaine 995 work in univariate framework. Later, in Section 3, we show how the bias approximations obtained in the null case can be used to improve the finite sample performance of a test for multivariate ARCH effects. here are many papers that propose improving the finite sample performance of LR tests by using Bartlett-type corrections see for example very recently Johansen 00; however, such a correction is not available for the LM test. We show that, in the context of an LM test, the novel approach of bias correcting the QML estimators may be a suitable alternative. Finally, Section 4 concludes. A theoretical approach of a bivariate-arch model. Case : Allowing for dependent but uncorrelated disturbances We begin by analysing the framework proposed in Wong and Li 997 for the variance equation, where the model is specified as: y t = β + ε t. 5

6 where y t =y t,y t, ε t =ε t,ε t,eε t =0, and we assume the intercept vector β =β 0 β 0 to be known. We could allow for the estimation of the intercept and the introduction of any number of exogenous variables in the mean equation by applying directly the results of Iglesias and Phillips 003. he conditional variance equation follows the structure: H t = where: ht 0 0 h t h t = E ε t/i t h t = E ε t/i t Expressions. and.3 can be re-written as: = α 0 + α ε t + α ε t. = γ 0 + γ ε t + γ ε t.3 ε t = α 0 + α ε t + α ε t + η t ε t = γ 0 + γ ε t + γ ε t + η t where, due to the uncorrelatedness of the epsilons: E η t =E η t =0; E η t η t =0 E η t After some algebra, we find: = E h ; E η = E h t t t E ε t = α 0 γ +α γ 0 γ α γ α ; E ε t = γ 0 α +γ α 0 γ α γ α From the above we may deduce the following restrictions on the variance equation parameters: γ < ; α < ; γ α γ α > 0 Our objective is to analyse the QML biases of O in this simple model. he methodology we will use has been proposed by Cox and Snell 968, where they 6

7 β showed that for independent, but not necessarily identically distributed observations, the bias b of the MLE of β reduces to: b s = E βs β s for s =,...,p, where k ij = E = p i,j,l= L β i β j k si k jl { k ijl + k ij,l } + O.4, k ijl = E 3 L β i β j β l, k ij,l = E L β i β j L, β l for i, j, l =,..., p L denotes the Log-likelihood function. he total Fisher Information matrix and its inverse are defined by K = { k ij } and K = { k ij } respectively. he formula is valid, even for non-independent observations, provided that all k s are of O see Cordeiro and McCullagh 99, and this justifies the application of the methodology in our case. Because under the null, QML is the same as ML, we are, in fact, obtaining the biases of the two estimators in this case under the null of no-arch effects, the formula of McCullagh 987 in terms of cumulants to capture the bias of the QML estimator equals the one of Cox and Snell 968. In order to proceed to obtain the expectations of the second and third order derivatives, we can follow the matrix differential calculus techniques of Magnus and Neudecker 99. Liu and Polasek 999 provided the expression of the conditional information matrix I θ of a general VARk VARCHq model for y t =yt,yt,...,y Mt, by specialising.: I θ = I I I I with I = t= W t H t W t, I = I =0 and I = V D t H t t= H t DV t where W t =I M,X t,...,x t k, V t =I N,Z t,..., Z t q, X t i = diag yt i,yt i,..., y Mt i, for i =,...,k, Z t j = diag ε t j,ε t jεt j,...ε Mt j, for j =,...,q. Note that I M and I N are an M M and an N N identity matrices respectively and D is the duplication matrix defined in.. 7

8 However, if we follow their analysis see Liu and Polasek 999, page 05, the previous expression is found to be in error because it neglects the dependence of the disturbances in the variance equation with respect to the parameters in the mean equation in the maximisation procedure. heir formula is valid only in the situation where there are no parameters to estimate in the mean equation, which is precisely our case. We extend the work by Liu and Polasek 999 to include all the cumulants we need for our analysis and Appendix provides the expressions for the second and third order derivatives of. in our model on applying the differential matrix calculus. Once we know the expressions of all the k components, we are in the position to apply expression.4, obtaining the bias results and the variances given by the information matrix as presented in the heorem below. Unfortunately, although Iglesias and Phillips003 were able to find a bias approximation for the variance parameter estimators in a univariate ARCH model without imposing restrictions, the additional complexity of the multivariate model prevents similar derivations unless restrictions are imposed. o make progress under the assumption that we specify the conditional variance structure given in. and.3, we impose the restrictions that α = α = γ = γ =0. his type of restriction, has been imposed in many of the theoretical analyses that have been carried out in univariate ARCH models so far eg. Engle, Hendry and rumble 985, Linton 997, and it facilitates, especially here, the analysis and the interpretation of the results. It is to be noted that if, for example, it is demonstrated through the bias approximations that severe biases and/or large variances are possible in the restricted model, then these characteristics will surely be found in unrestricted models. Of course, if such problems do not arise in the restricted case the same may be true in the unrestricted model but it need not be so. Hence care is needed in drawing conclusions from the approximations..: If y t = ε t where y t =y t,y t, ε t =ε t,ε t is a vector of random variables that has the structure given in. and.3, with α = α = γ = γ =0, then the biases and the variances of the QML estimators to order are given by: E α 0 α 0 = α 0 + o E γ 0 γ 0 = γ 0 + o E α α = + o E α α =o E γ γ =o E γ γ = + o var α 0 = 4α 0 + o var γ 0 = 4γ 0 + o var α = + o var α = α 0 γ + o 0 var γ = γ 0 α + o var γ = + o 0 Proof. Given in Appendix. Notice that the biases in the restricted model are relatively small suggesting that estimation bias may not be a particular problem in this model. However, it is interesting to note how, when the intercept parameters α 0 and γ 0 differ substantially, 8

9 the above model can generate severe and large variances in the QML estimators of the α and γ parameters at least in one of them. In practical applications that fit a model with this specification to real data, one should be mindful of this fact when interpreting the estimation results. It is very easy to find an interpretation in this situation: under the null of no-arch effects, the two intercepts reflect the two unconditional volatilities of the two series. So our results show that the severe variances result in this case when the two series have very different volatilities. able. shows the standard errors of O, and a comparison with the simulated errors for different combinations of the intercepts of the conditional variance equation, confirming the results shown previously. able.: Approximate Standard Errors when we overspecify the multivariate ARCH effects. = 400. α 0 =0.8 α 0 =0.04 γ 0 =0.04 γ 0 =0.04 α α α γ γ γ Simulated values are given in brackets for 0000 replications On the other hand, the bias and variances of the QML estimators to O in a univariate ARCH model, E ε /I t t =α + α ε t, when nothing is estimated in the mean equation while α = 0, are given by see Engle, Hendry and rumble 985 and Iglesias and Phillips 003: E α α = α + o E α α = + o var α = 3α + o var α = + o Comparing these biases with those of heorem., it is seen that, in the new bivariate specification, the biases in the parameters that are common have the same structure while, on the other hand, there is a loss of estimation efficiency to O in the intercept parameter estimator, and no gain or loss in efficiency for the estimator of the ARCH parameter. Extending the work in Lumsdaine 995, the representation of the relevant part of the log-likelihood involves: L t = log h t +logh t + ε t h t + ε t h t Using the same argument as the one given in Lumsdaine 995, page 0, we can prove that if α 0 and γ 0 change in the same proportion, the biases and t-statistics 9

10 in α, α, γ and γ will remain invariant. his result matches with the bias and variance results obtained in heorem.. However, if α 0 and γ 0 vary in different proportions, the invariance property does not hold.. Case : Allowing for dependent and correlated disturbances We analyse now the variance-specification proposed by Engle and Kroner 995 and Liu and Polasek 999,000, given by the bivariate model: y t = β + ε t.5 where y t =y t,y t, ε t =ε t,ε t,eε t =0, and we assume again the intercept vector β = β 0,β 0 to be known. We allow for possible misspecification of the marginal distribution of the errors; thus we allow in for the QML estimator to be used. he variance representation implies a diagonal structure for the disturbances following an ARCH process: ht h where var ε t /I t =H t t = and: h t h t h t = hen, it follows that: α 0 α 0 α 30 h t h t α α α 33 ε,t ε,t ε,t ε,t.6 E ε t ε s /I t =α 0 + α ε t ε s,t = s.7 0 otherwise E ε t/i t = α 0 + α ε t.8 E ε /I t t = α 30 + α 33 ε t he unconditional expectations become E ε = α0, E t α ε = α30, E ε t tε t = α33, and the unconditional correlation coefficient between both disturbances is α0 α α0 α α33 α. his implies the restriction that in this model, in order to guarantee that the correlation coefficient is absolutely smaller than α0α30 : 0

11 α 0 α 0 α 30 < α α α 33 When α = α = α 33 = 0, then α 0 α0α30 <. In addition, α 0,α 30 > 0, while 0 <α,α 33 <. Our objective is again to analyse the biases of O in this simple model when we use the QML estimation procedure, under the assumption that we specify a diagonal structure in the conditional variance, when, in fact, the true model is the one for which we have α = α = α 33 =0. Using again the methodology proposed by Cox and Snell 968 Appendix gives the expressions for the second and third order derivatives, the bias results are given in heorem...: If y t = ε t where ε t =ε t,ε t is a vector of random variables that has the structure given in.6 with α = α = α 33 =0, then the biases and the variances of the QML estimators to order are given by: E α 0 α 0 = α 0 α30α 0 α 30 +α 0 α 0 +α 0 α 30 +α0 α 0 α30 α4 0 + o α 0 α 30 +4α 0α 0 α 30+α 4 0α0α30 α 0 var α 0 = α 03α 3 0 α α 0 α 0 α 30 +0α 0α 4 0 α 30+α o α 3 0 α α 0 α 0 α 30 +5α 0 α 4 0 α 30+α 6 0 E α 0 α 0 = α 0α 4 0 α 30 +α 0 α α 0 α 0 α 30 +α4 0 α 0 +α4 0 α 30 α o α 0 α 30 +4α0α 0 α30+α4 0α0α30 α 0 var α 0 = α3 0 α α 0 α 0 α 30 +5α0α4 0 α30+α6 0 + o α 0 α 30 +4α0α4 0 α30+α4 0 E α 30 α 30 = α0α 30α 0 α 30 +α 0 α 0 +α 0 α 30 +α0 α 0 α30 α4 0 + o α 0 α 30 +4α 0α 0 α 30+α 4 0α0α30 α 0 var α 30 = α 303α 3 0 α α 0 α 0 α 30 +0α 0α 4 0 α 30+α o α 3 0 α α 0 α 0 α 30 +5α 0 α 4 0 α 30+α 6 0 E α α = α 0α30α 0 α 30 +α 0 α 0 +α 0 α 30 +α 0α 0 α 30 α o α 0 α 30 +4α 0 α 0 α 30+α 4 0α0 α30 α 0 var α = α 0 α 30α0 α30+3α 0 α 3 0 α α 0 α 0 α 30 +5α0α4 0 α30+α6 0 + o E α α = α4 0 α 30 +α 0 α α 0 α 0 α 30 +α 0 α4 0 +α 30 α4 0 α 6 0 α 0 α 30 +4α0α 0 α30+α4 0α0α30 α 0 var α = α 0 α 30 +α4 0 α 0 α 30 +4α 0 α 4 0 α 30+α o E α 33 α 33 = α 0α30α 0 α 30 +α 0 α 0 +α 0 α 30 +α 0α 0 α 30 α 4 0 α 0 α 30 +4α 0 α 0 α 30+α 4 0α0 α30 α 0 var α 33 = Proof. Given in Appendix. α 0 α 30α0 α30+3α 0 α 3 0 α α 0 α 0 α 30 +5α0α4 0 α30+α6 0 + o + o + o

12 In spite of the large and tedious expressions we get, it is important to highlight the utility we can get from them, because they enable us to find approximations to the biases for any combination of parameters, and to discover their evolution. Notice that all the coefficient bias approximations contain in the denominator the term α 0 α 30 α 0 which is the determinant of the unconditional covariance matrix of the disturbances. Hence one obvious situation in which large estimator biases are to be expected is when there is high correlation between the disturbances. However, the biases can still be large even when this correlation is relatively modest as will be shown below. hus we can provide theoretical support for the large biases found by Liu and Polasek 000 even though our analytical results are obtained under strong restrictions. An additional use for the approximations is for bias correction under the assumption of overspecification of the conditional process; we can use the expressions for bias correction, substituting estimates for the true values of the expressions. he direct applicability of the results for testing will be shown in the next section of the paper. We have noted that our theoretical analyis supports the results in Liu and Polasek, in the sense that the biases can be very large in these models -even though our setting is different-, but our findings provide evidence that when the disturbances are not highly correlated, the biases are only so large for some combinations of parameters. able. shows how the larger biases are those for the parameters of the ARCHcomponents, especially when there is a large difference between the intercepts of the two conditional variance equations. For example, the approximate bias of the estimator of α increases from around to when the constant terms α 0 and α 30 change from being the same and equal at 0.5 to α 0 being kept constant at 0.5 and increasing α 30 to 5. able.: Biases and variances of O for some different parameter configurations, α 0 =0.5, α 0 =0.05 and =00. α 30 =0.5 α 30 =5 E α 0 α var α E α 0 α var α E α 30 α var α E α α var α E α α var α E α 33 α var α Once we have found the bias expressions of O, we can again extend the

13 work by Lumsdaine 995 to our model. In this case we need to change α 0,α 0 and α 30 in the same proportion to get invariance in the bias and t-statistics of α, α and α 33. Otherwise, the invariance property becomes invalid again, this is consistent with the results in heorem.... Special case when the correlation of the disturbances is overspecified In this case, if we set α 0 =0, heorem. now becomes: C.: If y t = ε t where ε t =ε t,ε t is a vector of random variables that has the structure given in.6 under overspecification of the conditional correlation α 0 =0, then the biases and the variances of the QML estimators to order are given by: E α 0 α 0 = α0 + o E α α = + o E α 0 α 0 =o E α α = α 0 +α 30 + o α0α30 E α 30 α 30 = α 30 + o E α 33 α 33 = + o var α 0 = 3α 0 + o var α = + o var α 0 = α0α30 + o var α = + o var α 30 = 3α 30 + o var α 33 = + o Proof. In the results given in heorem., we set α 0 =0. he expression for the bias of α is now especially easy to interpret and it is easy too to analyse the effect of a large distance between the two intercepts. On the other hand, the bias and variances of the QML estimators in a univariate ARCH model, when nothing is estimated in the mean equation, were given at the end of Section.. So we see that the effect of imposing a correlation between the disturbances, when in fact it does not exist, again does not affect the bias structure, although on the other hand, this time there is neither gain nor loss in efficiency to the order of the approximation. 3 An LM type test allowing for bias correction in the estimators In this section, we examine how the biases of O can be used to improve the finite sample performance of a test for multivariate ARCH effects. We propose that instead of improving the finite sample behaviour of the test by applying a Bartletttype correction Bartlett 937 which, in any case is not available, we proceed by bias correcting the estimates themselves. he justification for this is the following. Let s consider the LM test which takes the form see Harvey 989, page 69: 3

14 LM =D log L Ψ 0 I Ψ 0 D log L Ψ 0 3. Ψ 0 where D log L is the vector of first order derivatives of the log likelihood function evaluated under the null hypothesis, Ψ 0 is the vector of restricted estimates, and I is the estimated information matrix. Ψ0 Harvey 989 notes that D log L Ψ 0 Ψ Ψ 0 D log L Ψ where Ψ is the vector of unrestricted estimates; using this approximation we may write that: LM =D log L Ψ 0 I D log L Ψ0 Ψ 0 Ψ Ψ 0 D log L Ψ I D log L Ψ Ψ Ψ 0 Ψ 0 3. which has an asymptotically equivalent form given by: Ψ Ψ 0 I Ψ0 Ψ Ψ In the case where there are no nuisance parameters and the null hypothesis is that H 0 :Ψ 0 =0, we see that Ψ Ψ 0 =Ψ, in which case the above statistic reduces to Ψ I Ψ 0 Ψ. 3.4 Our proposal is to use a bias-corrected estimate of Ψ, denoted by Ψ BC, in place of Ψ in the LM statistic. Since Ψ BC is second order efficient it is anticipated that the statistics will converge to its limiting distribution faster so the size of the test in small samples will be closer to its nominal level. If the bias correction is non-stochastic, then the information matrix will be unchanged; this is the case for the situation we shall consider below.hus, if in 3.3 Ψ 0 is replaced by Ψ 0, the bias approximation for the unrestricted estimate obtained when assuming the null is true, then Ψ Ψ 0 =Ψ BC so that the above statistic is modified to: Ψ BC I Ψ BC 3.5 Ψ0 Under the alternative, the bias correction that is used is incorrect so that as well as improving the size some improvement in power seems likely. he statistic we actually use is 4

15 D log L Ψ 0 I D log L Ψ 0 Ψ which is asymptotically equivalent to 3.5. o see this note that D log L Ψ0 Ψ Ψ0 D log L Ψ = Ψ BC D log L Ψ On substituting from this approximation into 3.6 we may deduce that required result. So far we have assumed the absence of parameters not subject to test; however the basic argument is unchanged when such parameters are present. We can choose to ignore them and use the form of the test which tests only a subset of the complete parameter vector, or we can include them in which case they too can be evaluated at their bias corrected values. he argument for including them turns mainly on the possibility that the power may be increased since the bias correction is invalid under the alternative. We show now in more detail through simulation how the bias-correction procedure works. For ease of application we use the Wong and Li model Case in the previous section as an example. In particular, since in the LM procedure estimation is conducted only under the null, the bias approximations which for the conditional variance parameters are found only in the null case can be employed directly since they are non-stochastic and known.. As was seen in heorem. bias approximations were found for the constant terms in the variance equations. and.3; these are nuisance parameters for the LM test on the variance parameters, since they are not subject to the test. Bias corrected estimates for them are easily found. hese bias corrected estimates will be employed in the LM test. However, as has been noted above, an additional use of the bias approximations for the conditional variance parameters in the null case can also be found. Rather than evaluate these parameters as zero under the null, we may set them at the O biases since the expected values of the QML estimators are not zero but are close to the bias approximation. his yields the test statistic given in 3.6. o analyse the effect of this use of the bias corrections, we shall first conduct simulations with the bias corrected constant terms in the LM while setting the parameters under test to zero. hen in further simulations we both use bias corrected estimates for the constant terms and set the parameters under test to their asymptotic bias values. Hence in this case we are effectively testing a null under which the conditional variance parameters are equal to the expected value of the QML estimator rather than zero. In this case Ψ=α 0,α,α,γ 0,γ,γ is the 6 vector of unknown parameters whereas the null hypothesis we wish to test is H 0 = α,α,γ,γ =0 In what follows we shall consider three versions of the LM test statistic: 5

16 Model : he nuisance parameters are replaced with uncorrected QML estimates and the parameters under test are set to zero M. his is the standard test statistic given in 3.. Model : he nuisance parameters are replaced with bias corrected QML estimates with the parameters under test set to zero M. his case is considered in this paper for comparison purposes. Model 3: he nuisance parameters are replaced with bias corrected QML estimates and the parameters under test are set to their asymptotic bias values M3. his is the statistic given in 3.6. here are several variants of the LM test and generally they differ only in the estimator of the information matrix; see for example Amemiya 985 and Dagenais and Dufour 99 for some related literature. We may distinguish three types of such estimators; the Outer Product OP matrix of the score vector, the Hessian HES matrix and the expectation of the Hessian ExpHES matrix. A non-operational procedure which we shall examine for comparative purposes, uses the true Hessian ruehes where the actual values of unknown parameters are employed rather than estimates. Each of these four variants of the LM test will be examined in the simulations in the contexts of Models -3. he LM test based upon the expected Hessian is not always available since finding the closed form solution for the expected Hessian may not be possible. In this case, however, it is straightforward. Besides, to find the expected hessian for any higher order specification of the Wong and Li 997 model would be straightforward. From Wong and Li 997 we find on using.,. and.3, that we may write: D log L Ψ = Hessian Ψ = where: hessian i = t= ht ε t ht hessian 0 0 hessian [ ε it h iit dh, ht dhdh ],i =, ε t ht t= h iit dh =,ε t,ε t On taking expectations through Hessian Ψ we have: ExpHES Ψ= with inverse: α 0 α0 γ 0 α 0 α0 3 γ 0 γ 0 α 0 γ 0 α0 α0 3γ 0 α γ α0 γ γ 0 α 0 γ 0 3α 0 γ 0 α 0 γ 0 dh γ 0 α0 γ 0 3 6

17 ExpHES Ψ = 4α 0 α0 α 0 γ 0 α0 α 0 γ 0 0 α 0 γ γ γ 0 α0 γ 0 γ 0 α0 γ 0 α 0 γ We thus have four variants of the LM test. heir size and power are examined in a set of simulation experiments. First the test sizes are examined for sample sizes =50, 00, 00 and 500 where the nuisance parameters are set to α 0 =0.8 and γ 0 =0.04. his choice of parameter values and sample sizes was made to ensure that the small sample biases and variances were not trivial. In the simulations, to examine the power of the tests we considered two sets of values for the variance parameters: i α = α = γ = γ =0.6, and ii α = α = γ = γ =0.49. he first of this represents a moderate departure from the null whereas the second lies close to the stationarity bound and so is a relatively extreme departure. he results on the test size are given in able 3. and for size-adjusted power in able 3.. he first clear result we find is that of the bad size properties in small samples for the HES LM test see able 3., because it is clearly over-sized, even at =500, in marked contrast to the other tests. he OP and the ExpHES have better size properties, although when we check the size-adjusted power of the tests able 3. it reveals the lack of power of the OP test for finite samples. At the more extreme alternative the ExpHES and the ruehes tests have power close to unity at all sample sizes. From the results, the first recommendation in practical applications is to use the ExpHES to test for multivariate ARCH effects. Once we have selected the ExpHES, we can concentrate on the selection among Model, Model or Model 3. Model 3 seems to have much better size properties than Model or Model. Comparing Models and 3 it is interesting to see the marginal effect of introducing the QML biases in place of zeros in specifying the null hypothesis. As it was suggested by our earlier theoretical analysis, the size of the test is clearly improved. Analysing the ruehes, the test presents the best size properties is again Model 3. hus, the use of bias-correction to improve the size of the test, as an alternative to the traditional Bartlett 937 type correction, is supported in our study. If we consider the size-adjusted power, we observe how the test power in Model and 3 improves on that of Model with Model 3 being slightly superior. So the overall conclusion from the simulations is that, of the operational tests, only ExpHES performs well. Its size is approximately correct even at =50 while it has high power against both the moderate and extreme alternatives at all sample sizes considered. It even dominates the non-operational ruehes test for the moderate alternative and has comparable but slightly less power for the extreme alternative. Hence, our simulations support the use of the ExpHES test while bias correcting all the QML estimates. 7

18 able 3.: Size Results based on 5% critical values OP HES ExpHES ruehes M M M3 M M M3 M M M3 M M M3 = = = = he results are based on Monte Carlo replications under the null of no-arch effects. α 0 =0.8 and γ 0 =0.04. able 3.: Power Results based on 5% critical values size-adjusted OP HES ExpHES ruehes When the alternative hypothesis is α = α = γ = γ =0.6 M M M3 M M M3 M M M3 M M M3 = = = = When the alternative hypothesis is α = α = γ = γ =0.49 = = = = he results are based on Monte Carlo replications. α 0 =0.8 and γ 0 = Conclusions In this paper we have provided theoretical evidence of the severe biases and large variances that result from unconstrained-qml estimation of a simple bivariate-arch model under overspecification of the conditional heteroscedasticity processes. When we analyse the model in Wong and Li 997, we find that some of the estimators can have large variances if the difference between the intercepts in the model is relatively large. In the case of the Engle and Kroner 995 and Liu and Polasek 999, 000 specification, we find that strongly contemporaneously correlated disturbances and/or a large difference between the intercepts can produce large biases in the estimators of the ARCH-terms for some combinations of parameters. Under the null of no-arch effects, the intercepts capture the volatility of the series, and then the results of this paper warn about the testing of multivariate ARCH effects among series that may have very different degree of volatilities. One rule of thumb in practical applications, would be to standardize always the volatilities of the series before they are used in a multivariate model, although the best recommendation is to use the bias expressions that are provided in this paper. We believe that the possibility of extreme biases and 8

19 variances should be taken into account in practical applications when QML is used as the estimation procedure in this model, and this paper provides an analysis of what happens in a simple bivariate process. In the last section of the paper we use our bias results to improve the finite sample performance of an LM type test for multivariate ARCH effects by bias-correcting the estimators of the parameters. We show that this can be considered as an alternative way to improve the finite sample behaviour in testing instead of applying a Bartlett-type correction. he general recommendation from this paper is that when testing for multivariate ARCH effects by performing the LM test, the expected Hessian form should be used and all QML estimators should be bias corrected. 9

20 Appendix he proof of heorem., implies the use of expression.4 to find the k ij, the µ k ijl and the k ij,l components. Using differential matrix calculus, defining Ht = h t h t h t h t, and assuming the parameter vector to be α 0,γ 0,α,α,γ,γ, we obtain: SECOND ORDER DERIVAIVES evaluation evaluation evaluation evaluation k h t k 0 k 3 ε t h t k 4 h t ε t k 5 0 k 6 0 k h t k 3 0 k 4 0 k 5 h t ε t k 6 h t ε t k 33 h t ε 4 t k 34 h t ε t ε t k 35 0 k 36 0 k 44 h t ε 4 t k 45 0 k 46 0 k 55 h t ε 4 t k 56 h t ε t ε t k 66 h t ε 4 t

21 HIRD ORDER DERIVAIVES evaluation evaluation evaluation k h t 3 k 0 k 3 h t 3 ε t k 4 h t 3 ε t k 5 0 k 6 0 k 0 k 3 0 k 4 0 k 5 0 k 6 0 k h t 3 k 3 0 k 4 0 k 5 h t 3 ε t k 6 h t 3 ε t k 33 h t 3 ε 4 t k 34 h t 3 ε t ε t k 35 0 k 36 0 k 33 0 k 34 0 k 35 0 k 36 0 k 44 h t 3 ε 4 t k 45 0 k 46 0 k 44 0 k 45 0 k 46 0 k 55 0 k 56 0 k 66 0 k 55 h t 3 ε 4 t k 56 h t 3 ε t ε t k 66 h t 3 ε 4 t k 333 h t 3 ε 6 t k 334 h t 3 ε 4 t ε t k k k 434 h t 3 ε t ε 4 t k k k 444 h t 3 ε 6 t k k k k k k k k k 555 h t 3 ε 6 t k 556 h t 3 ε 4 t ε t k 656 h t 3 ε t ε 4 t k 666 h t 3 ε 6 t

22 he Cox and Snell 968 expressions that are required apart from the second order derivatives, and the third order derivatives previously given, once we evaluate them when α = α = γ = γ =0, are: eval. eval. eval. eval. + k, 0 + k, k, k, k, k,6 0 + k, 0 + k, k, k, k, k,6 0 + k, 0 + k, k, k, k, k, k 3, α k 3, α 0 γ k 3,3 α k 3,4 γ 0 α k 3,5 γ k 3,6 α k 3, k 3, k 3, k 3, k 3, k 3, k 4, γ 0 α k 4, α k 4,3 γ 0 α k 4,4 γ 0 α k 4,5 α k 4,6 γ k 4, k 4, k 4, k 4,4 0 α k 4, k 4, k 5, k 5, k 5, k 5, k 5, k 5, k 5, γ 0 k 5 + k 5, α 0 k 53 + k 5,3 α 0 k γ k 5,4 k γ k 5,5 α 0 k γ k 5,6 α 0 k γ k 6, 0 k 6 + k 6, 0 k 63 + k 6,3 0 k 64 + k 6,4 0 k 65 + k 6,5 0 k 66 + k 6,6 0 k 6 + k 6, k α 0 γ k 6, k γ k 6,3 k γ k 6,4 k α k 6,5 α 0 γ 0 k 66 + k 6,6 k γ k 33, 3 k α k 33, 3 k γ k 33,3 3 k k 33,4 3γ 0 k k 33,5 3α 0 γ 0 k k 33,6 3 k 43 + k 43, γ 0 k k 43,4 γ 0 α k 43, k 44,4 3γ3 0 α 3 0 α 0 γ 3 0 k 43 + k 43, k α k 43,3 γ 0 α 0 k k 43,6 γ 0 k α k 44, 3γ 0 α 3 0 k k 44,5 3γ 0 k α k 44,6 3γ 0 α k k 35,4 0 k k 35,5 0 k k 44,3 3γ 0 α 0 k 35 + k 35, 0 k k 35,3 0 k 45 + k 45, 0 k 45 + k 45, 0 k k 45,3 0 k k 45,4 0 α 0 k 44 + k 44, 3γ 0 α k 35, 0 k k 35,6 0 k k 45,5 0

23 eval. eval. eval. eval k 45, k 36, k 36, k 36, k 36, k 36, k 36, k 46, k 46, k 46, k 46, k 46, k 46, k 55, 3α k 55, 3α 0 k k 55,3 3α 0 γ 0 k 65 + k 65, α 0 γ 0 k 66 + k 66, 3 α 0 k k 66, k 55,4 3α 0 γ k 55,5 3α3 0 γ k 65,3 α 0 γ k 65,4 k 66 + k 66, 3 γ 0 k k 66,3 3 γ 0 k k 55,6 3α 0 γ 0 k k 65,5 α 0 γ k k 66,4 3γ 0 α 0 γ k 65, γ k 65,6 α 0 γ 0 k k 66,5 3α 0 γ 0

24 Appendix he proof of heorem., implies the use of expression.4 to find the k ij,k ij,l µ and the k ijl components. Using differential matrix calculus, defining Ht h t h = t we obtain: h t h t, and ordering the parameters as: α 0,α 0,α 30,α,α,α 33, SECOND ORDER DERIVAIVES evaluation evaluation evaluation evaluation k ht k h t h t k 3 ht k 4 ht ε t k 5 h t h t ε t ε t k 6 ht ε t k ³h t h t + h t k 3 h t h t ³ k 4 h t h t ε t k 5 ε t ε t h t h t + h t k 6 h t h t ε t k 33 ht k 34 ht ε t k 35 ε t ε t h t h t k 36 ht ε t k 44 ht ε 4 t k 45 ε 3 t ε t h t h t k 46 h ε t ε t k 55 ε t ε t ³h t h t + h t k 56 ε 3 t ε t h t h t k 66 ht ε 4 t

25 HIRD ORDER DERIVAIVES evaluation evaluation evaluation k h t 3 k 4 h t h t k 3 h t h t k 4 h t 3 ε t k 5 4 h t h t ε t ε t k 6 h t h t ε ³ t k h t h t h t +3 h t ³ k 3 h t h t h t + h t k 4 4 h t h t ε t ³ k 5 h t ε t ε t h t h t +3 h t k 6 h t ε t ³h t h t + h t ³ k 4h t 3h t h t + h t ³ k 3 h t h t h t +3 h t k 4 h t ε t ³h t h t +3 h t ³ k 5 4h t ε t ε t 3h t h t + h t k 6 h t ε t ³h t h t +3 h t k 33 h t h t k 34 h t h t ε ³ t k 35 h t ε t ε t h t h t + h t k 36 h t h t ε t k 33 4 h t h t k 34 h t ε t ³h t h t + h t ³ k 35 h t ε t ε t h t h t +3 h t k 36 4h t h t ε t k 44 h t 3 ε 4 t k 45 4 h t h t ε 3 t ε t k 46 h t h t ε t ε t k 44 4 h t ³ h t ε 4 t k 45 h t ε t ε t h t h t +3 h t k 46 h t ε t ε t ³h t h t + h t k 55 h t ε t ε t ³h t h t +3 h t k 56 h t ε t ε 3 t ³h t h t + h t k 66 h t h t ε 4 t k 55 4h t ε t ε t ³3h t h t + h t k 56 h t ε t ε 3 t ³h t h t +3 h t k 66 4 h t h t ε 4 t k 333 h t 3 k 334 h t h t ε t k h t h t ε t ε t k 336 h t 3 ε t k h t ³ h t ε 4 t k 435 h t ε 3 t ε t h t h t + h t k 436 h t h t ε t ε t k 444 h t 3 ε 6 t k h t h t ε 5 t ε t k 446 h t h t ε 4 t ε t k 335 h t ε t ε t ³h t h t +3 h t k h t h t ε t ε 3 t k 455 h t ε 4 t ε t ³h t h t +3 h t k 456 h t ε 3 t ε 3 t ³h t h t + h t k 366 h t 3 ε 4 t k 466 h t h t ε t ε 4 t k 555 4h t ε 3 t ε 3 t ³3h t h t + h t k 556 h t ε t ε 4 t ³h t h t +3 h t k h t h t ε t ε 5 t k 666 h t 3 ε 6 t

26 he Cox and Snell 968 expressions that are required apart from the second order derivatives, and the third order derivatives previously given, once we evaluate them when α = α = α 33 =0, are: evaluation evaluation k + k, 0 k + k, 0 k 3 + k,3 0 k 4 + k,4 0 k 5 + k,5 0 k 6 + k,6 0 k + k, 0 k + k, 0 k 3 + k,3 0 k 4 + k,4 0 k 5 + k,5 0 k 6 + k,6 0 k + k, 0 k + k, 0 k 3 + k,3 0 k 4 + k,4 0 k 5 + k,5 0 k 6 + k,6 0 k 3 + k 3, 0 k 3 + k 3, 0 k 33 + k 3,3 0 k 34 + k 3,4 0 k 35 + k 3,5 0 k 36 + k 3,6 0 k 3 + k 3, 0 k 3 + k 3, 0 k 33 + k 3,3 0 k 34 + k 3,4 0 k 35 + k 3,5 0 k 36 + k 3,6 0 k α 4 + k 0α , α 0 α 30 α 0 3 k 4 + k 4, α 0α 30 α0 α30 α 0 α 30 α 0 3 k 43 + k 4,3 α 0 α 30 α 0 α 30 α 0 3 k 44 + k 4,4 α 0 α3 30 α 0 α 30 α 0 3 k α 45 + k 0 α 30 α 0 α 30 4,5 α 0 α 30 α 0 3 k α 46 + k 0 α3 30 4,6 α 0 α 30 α 0 3 k α 4 + k 0 α 0 α 30 4, α 0 α 30 α 0 3 k 4 + k 4, α 0 α 30α 30 α 0 α 0 α 30 α 0 3 k α 43 + k 3 0 α 30 4,3 α 0α 30 α 0 3 k 44 + k 4,4 α 0 α 0α 30 α 0α 30 α 0 3 k α 45 + k 3 0 α 30α 30 α 0 4,5 α 0α 30 α 0 3 k 46 + k 4,6 α 3 0 α 30 α 0α 30 α 0 3 k α 5 + k 0 α 30α 30 α 0 5, α 0α 30 α 0 3 k α 0 α 30α 5 + k 30 +α 0 α 0 5, α 0α 30 α 0 3 k α 53 + k 0 α 30α 0 α 30 5,3 α 0α 30 α 0 3 k α 54 + k 0 α 0 α 30α 30 α 0 5,4 α 0α 30 α 0 3 k α 55 + k 0 α 30α 30 +α 0 α 0 5,5 α 0α 30 α 0 3 k α 56 + k 0 α 30 α 0 α 30 5,6 α 0α 30 α 0 3 k α 0α 0 α 30α 5 + k +α0α30 0 5, α 0 α 30 α 0 3 k α 5 + k 0 α 0 α 30α +α0α30 0 5, α 0 α 30 α 0 3 k α 0α 30 α 0α 53 + k +α0α30 0 5,3 α 0 α 30 α 0 3 k α 0α 0α 0 α 30α 54 + k +α0α30 0 5,4 α 0 α 30 α 0 3 k α 0α 55 + k 0 α 0 α 30α +α0α30 0 5,5 α 0 α 30 α 0 3 k α 0α 30α 30 α 0α 56 + k +α0α30 0 5,6 α 0 α 30 α 0 3 k 6 + k 6, α 4 0 α 0 α 30 α 0 3 k 6 + k 6, α 3 0 α 30 α 0 α 0 α 30 α 0 3

27 evaluation evaluation k 63 + k 6,3 α 0α 0 α30 α 0 α 30 α 0 3 k 64 + k 6,4 α 0α 4 0 α 0 α 30 α 0 3 k α 65 + k 4 0 α 30 α 0 6,5 α 0 α 30 α 0 3 k α 66 + k 0 α 0 α 30 6,6 α 0 α 30 α 0 3 k α 6 + k 0 α 3 0 6, α 0 α 30 α 0 3 k 6 + k 6, α 0 α 0 α 0 α 30 α 0 α 30 α 0 3 k α 63 + k 0 α 0α 30 6,3 α 0α 30 α 0 3 k 64 + k 6,4 α 0 α3 0 α 0α 30 α 0 3 k α 65 + k 0 α 3 0 α 0 α 30 6,5 α 0α 30 α 0 3 k 66 + k 6,6 α 0 α 0α 30 α 0α 30 α 0 3 k 33 + k 33, 0 k 33 + k 33, 0 k k 33,3 0 k k 33,4 0 k k 33,5 0 k k 33,6 0 k α 43 + k 0 α 0 α 30 43, α 0α 30 α 0 3 k 43 + k 43, α 3 0 α 0 α 30 α 0α 30 α 0 3 k k 43,3 α 4 0 α 0α 30 α 0 3 k k 43,4 α 0 α 0 α 30 α 0α 30 α 0 3 k α k 4 0 α0 α30 43,5 α 0 α 30 α 0 3 k α k 4 0 α30 43,6 α 0 α 30 α 0 3 k 3α 44 + k 0 α , α 0 α 30 α 0 3 k 44 + k 44, 3α 0α 0α 30 α0 α30 α 0 α 30 α 0 3 k 3α k 0α 0 α 30 44,3 α 0 α 30 α 0 3 k k 44,4 3α 3 0 α3 30 α 0 α 30 α 0 3 k 3α k 0 α 0 α 30 α 0 α 30 44,5 α 0 α 30 α 0 3 k k 44,6 3α 0 α 0 α3 30 α 0 α 30 α 0 3 k α 35 + k 0α 0 α30 α0 35, α 0 α 30 α 0 3 k α 0α 0α 35 + k 30 +α 0 α 0 35, α 0 α 30 α 0 3 k α k 0 α 0 α 0 α 30 35,3 α 0 α 30 α 0 3 k α k 0 α 0 α 30 α 0 35,4 α 0 α 30 α 0 3 k α 0α k 0α 30 +α 0 α 0 35,5 α 0 α 30 α 0 3 k α k 0α 0 α30α0 α30 35,6 α 0 α 30 α 0 3 k 3α 45 + k 0α 0 α303α30 α0 45, α 0 α 30 α 0 3 k 3α 0α 30[α 0α 45 + k 0 4α 0α 0α 30 α 0+α 30α 0α 30+α 0 α 4 α30] 0 45, α 0 α 30 α 0 4 k 3α k 0 α 30[α 0α 0 α 30 α 0 α 0 α 30α 0 α 30 +α 0+α 4 0] 45,3 α 0 α 30 α 0 4 k 3α k 0 α 0 α303α30 α0 45,4 α 0 α 30 α 0 3 k 3α k 0 α 30[α 0α 0 4α 0α 0 α 30 α 0+α 30α 0 α 30 +α 0 α 4 0 α 30] 45,5 α 0α 30 α 0 4 k 3α k 0 α 30[α 0α 0 α 30 α 0 α 0 α 30α 0 α 30 +α 0+α 4 0] 45,6 α 0α 30 α 0 4 k α 36 + k 0 α 0 36, α 0 α 30 α 0 3 k α 36 + k 0 α0α30 α0 36, α 0 α 30 α 0 3 k α k 3 0 α30 36,3 α 0 α 30 α 0 3 k k 36,4 α 3 0 α 0 α 0 α 30 α 0 3 k α k 0 α 0 α30 α0 36,5 α 0 α 30 α 0 3 k k 36,6 α 3 0 α 30 α 0 α 30 α 0 3

28 evaluation k α 46 + k 0[α 0 α 30+α 0 α 30 +6α 0α 0α 0 α 30 α 0+6α 4 0 α 30 3α 0 α 30α 0 α 30 +α 0] 46, 4α 0 α 30 α 0 4 k α 46 + k 3 0[ α 0 +α 0α 30 +3α 0 +3α 30α 0 α 30 α 0+3α 0 α 0 α 30 6α α 0 α 30] 46, α 0α 30 α 0 4 k α k 0[α 0 α 0+α 0 α 30+6α 30 α 0α 0 α 30 α 0+6α 4 0 α 0 3α 30 α 0α 0 α 30 +α 0] 46,3 4α 0α 30 α 0 4 k α 0 α k 0[α 0 α 30+α 0 α 30 +6α 0α 0α 0 α 30 α 0+6α 4 0 α 30 3α 0 α 30α 0 α 30 +α 0] 46,4 4α 0α 30 α 0 4 k α k 4 0[ α 0 +α 0α 30 +3α 0 +3α 30α 0 α 30 α 0+3α 0 α 0 α 30 6α α 0 α 30] 46,5 α 0α 30 α 0 4 k α k 0 α30[α 0 α0+α 0 α30+6α30α 0α 0α 30 α 0+6α 4 0 α0 3α30α 0α 0α 30+α 0] 46,6 4α 0 α 30 α 0 4 k α 55 + k 0 +α0α30[α 0 α30+α 30 α0+6α0α 0α 0α 30 α 0+3α 30α 4 0 α0α 0 α30 α 0 α 30] 55, α 0 α 30 α 0 4 k α 55 + k 0 +α0α30[3α0α3 0 α30+α 0 α0α 30 α5 0 α 0α 0 +α0α30+3α 0 +3α 30α 0α 30 α 0] 55, α 0 α 30 α 0 4 k α k 0 +α0α30[α 0 α0+α 0 α30+6α30α 0α 0α 30 α 0+3α 0α 4 0 α0α 0 α30 α 0 α 30] 55,3 α 0 α 30 α 0 4 k α 0α k 0 +α 0α 30[α 0 α 30+α 30 α 0+6α 0 α 0α 0 α 30 α 0+3α 30α 4 0 α 0α 0 α 30 α 0 α 30] 55,4 α 0 α 30 α 0 4 k α 0α k 0 +α 0α 30[3α 0 α 3 0 α 30+α 0 α 0α 30 α5 0 α 0α 0 +α 0α 30 +3α 0 +3α 30α 0 α 30 α 0] 55,5 α 0 α 30 α 0 4 k α 30α k 0 +α 0α 30[α 0 α 0+α 0 α 30+6α 30 α 0α 0 α 30 α 0+3α 0α 4 0 α 0α 0 α 30 α 0 α 30] 55,6 α 0α 30 α 0 4 k 3α 65 + k 0 α 0[α 30α 0 α 30 α 0 α 0 α 30α 0 α 30 +α 0+α 4 0] 65, α 0α 30 α 0 4 k 3α 0 α 0[α 30α 65 + k 30 4α 0α 0 α 30 α 0+α 0α 0 α 30 +α 0 α 4 0 α 0] 65, α 0α 30 α 0 4 k 3α k 0 α 0[α 30α 0 α 30 α 0 α 0 α 30α 0 α 30 +α 0+α 4 0] 65,4 α 0 α 30 α 0 4 k 3α k 0 α0[α30α 30 4α 0α 0α 30 α 0+α 0α 0α 30+α 0 α 4 α0] 0 65,5 α 0 α 30 α 0 4 k k 65,6 3α 0 α 0 α 30 3α 0 α 30 α 0α 30 α 0 3 k 66 + k 66, 3α 0 α 0 α 30 α 0α 30 α 0 3 k 66 + k 66, 3α 0 α 0α 30 α 30 α 0 α 0α 30 α 0 3 k k 66,3 3α 3 0 α 30 α 0α 30 α 0 3 k k 66,4 3α 3 0 α 0 α30 α 0 α 30 α 0 3 k k 66,5 3α 0 α 0 α30α30 α0 α 0 α 30 α 0 3 k k 66,6 3α 3 0 α3 30 α 0 α 30 α 0 3