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1 Introduction, Asset correlations are well known to be unstable over time Therefore in financial modelling. capturing the dynamic covariance and correlation structure has become a central issue In the. literature of Multivariate Stochastic Volatility MSV several recently proposed models sug. gest using Wishart processes to characterize the evolution of covariance matrices For example. Philipov and Glickman 2006 and Asai and McAleer 2009 henceforth A M adopt an inverse. Wishart specification for the covariance matrices so that statistical inference and model estima. tion can be effected through Bayesian Markov chain Monte Carlo MCMC methods Following. A M this class of models is termed the Wishart Inverse Covariance WIC. Since the WIC model for time series data is built upon latent variables and its estimation. relies on MCMC approaches the computation is expensive To deal with this type of model. OX would be an ideal choice as it is fast and it has extensive and well developed packages for. Bayesian inference A problem arises here in A M s model settings the degrees of freedom df. parameter in the Wishart distribution is not restricted to be integer valued it could be fractional. non integer However the generation of Wishart matrices with fractional df is not available. in OX The existing OX function ranwishart will simply take the floor of any fractional df. as the input argument and thus the matrix that has been sampled is in fact from a Wishart. distribution with integer df which is not correct In such cases where the integer df is used. our simulation shows that the estimation will be seriously misled To fix the problem in OX we. develop the Wishart package WishPack in which fractional df is considered. Moreover when the df is less than the dimension of the scale matrix parameter the Wishart. distribution is known to be singular In this situation the Wishart matrix is still well defined. and the probability density function with respect to Lebesgue measure on an appropriate space. is explicitly given by Srivastava 2003 Practical uses of the singular Wishart matrices and. distributions are revealed in Uhlig 1997 Kubokawa and Srivastava 2008 and Kunno 2009. among others In response to those needs Wishpack also takes into account the singular Wishart. cases This special feature concerning modern Wishart research has not yet been seen in other. statistical packages Finally as a parallel product Wishpack also provides an inverse Wishart. generator with fractional df and the density function of the inverse Wishart distribution. The rest of the paper is organized as follows In Section 2 we motivate the generation of. Wishart matrices with fractional df The generation of the singular Wishart matrix is also. discussed Section 3 illustrates the algorithm for generating Wishart random matrices with frac. tional df by applying Bartlett s decomposition A short discussion of another popular method. namely the Odell Feiveson s 1966 algorithm is given as well Some concluding remarks are. summarized in Section 4,2 Motivation, Let W be the Wishart distribution A M s dynamic correlation MSV model is given by. t k St 1 W k St 1,St 1 Q AQt 1, where Qt is a positive definite matrix by standardizing which we obtain the correlation matrix of. interest and k and St 1 are the df and the time varying scale matrix of the Wishart distribution. respectively The matrix A is the intertemporal sensitivity parameter which is symmetric and. positive definite and d is the persistence parameter which accounts for the memory of the. matrix process Qt The matrix Qt 1 is defined by a singular value decomposition. In the MCMC estimation procedure the full conditionals of Q 1. t and A are proportional to, the Wishart densities with fractional df k and respectively To demonstrate why fractional. does matter we begin with the simulation example 3 3 in A M in which the sample size T 500. and the true parameters are,1 0 3 1 10 0 33,A d 0 8 k 10.
0 3 1 0 33 1 10, Let df denote the set k Table 1 shows the results of the MCMC estimation when df is taken. to be integer valued and to be fractional respectively The 95 intervals are obtained using. the 2 5th 97 5th percentiles Note that for the integer df case the OX ranwishart function. is used while for the fractional df we apply WishPack The MCMC simulation is conducted. with only 1000 iterations since a systematic collapse will occur very soon in the integer df case. The first 400 draws are discarded and the remaining 600 are kept Here we have to emphasize. that the purpose of this simulation study is to illustrate the problem not to estimate the model. hence the numerical standard errors are not provided and the convergence diagnostic is not. implemented From Table 1 we can see that the estimates for A and k are totally misleading. for the integer valued df As a result a systematic failure will happen in this case due to the. repeatedly incorrect updates On the other hand if a fractional df is adopted then the MCMC. estimation procedure will provide a fairly good result. Figure 1 shows the trace plots of the entire chains of k and A under different types of df. Here denotes the determinant operator It is readily seen that in the integer df case A. goes down to nearly zero within the first 300 iterations and never moves back This finally leads. us to a systematic error On the contrary for the fractional df after the burn in period k and. A both stay stable around their true levels This simple simulation clearly demonstrates the. need for the Wishart generator with fractional df in real applications As a matter of fact since. the set of natural numbers is a subset of the positive fractional numbers we can always apply. fractional df in the Wishart sampling, Table 1 MCMC results using the integer and the fractional df parameters respectively. Integer df Fractional df,Parameters Mean 95 Interval Mean 95 Interval. a11 0 100 0 072 0 148 1 010 0 940 1 092,a12 0 027 0 053 0 014 0 374 0 581 0 255. a22 0 103 0 069 0 155 1 161 1 090 1 239,d 0 797 0 728 0 858 0 763 0 638 0 831.
k 4 955 4 179 5 842 7 574 5 667 9 533, 3 Generating Wishart Random Matrices with Fractional df. 3 1 Non singular Wishart Matrices, Let W k S be the Wishart distribution with the fractional df k and the p p scale matrix S. Also let 12 be the gamma distribution with rate 12 If k p then we can generate random. matrices from W k S with the following algorithm,1 Generate the random matrix B by. i Bii k i 1 21 for 1 i p,ii Bij N 0 1 1 j i p and Bij 0 o w. 2 Take Cholesky decomposition for the scale matrix S AAT. 3 W ABB T AT AB AB T W k S, This algorithm is based on Bartlett s decomposition Anderson 2003 To verify that W is.
actually drawn from W k S we conduct a simulation study and check if the results satisfy the. following property Gupta and Nagar 2000 Corollary 3 3 11 1. If W W k S then 2 k c p 1 6 0 1, where 2 k is the chi square distribution with df k For the verification four vectors are. chosen which are c1 1 0 0 T c2 0 1 0 T c3 0 0 1 T and c4 1 1 1 T Next define. 2i cTi W ci, ci Sci i 1 4 and for each i by using WishPack we draw T 50000. random samples Wj from the Wishart distribution with the true parameters. Thus according to 1 2i should follow 2 k Figure 2 shows the empirical distributions of 2i. along with their sample means and sample variances We can see that the sample means are. very close to 10 5 k and the sample variances are approximately 21 2k To have further. verification we also generate the Q Q plots and conduct the Kolmogorov Smirnov KS test. The results are given in Figure 3 It is clear that in the Q Q plots the points all approximately lie. on the 45 degree lines and the KS test results in p values suggest that 2i follows 2 k under. the significance level 0 05 Moreover we obtain the sample mean of the Wishart matrices. T 63 3 21 2 21 0 63 21 21,W Wj 21 2 84 1 10 4 21 84 10 5 kS E W. 21 0 10 4 105 1 21 10 5 105, From the above analysis we can find that Wishpack works nicely the Wishart random. matrices with fractional df are appropriately generated. 3 2 Wishart Matrices with Singularity, When k p the Wishart distribution W k S is singular The singular Wishart matrix does.
have practical applications e g Kubokawa and Srivastava 2008 Kunno 2009 and therefore. its generation is also taken into account in WishPack Nevertheless unlike the non singular. Wishart case Bartlett s decomposition cannot be directly applied in the singular situation. For this reason fractional df is not considered instead we develop the function for generating. singular Wishart random matrices with integer valued df k The procedure is given by. 0 Any input fractional df will be mapped to its floor i e k bkc b c is the floor function. 1 Generate a p k random matrix B with all entries independently from N 0 1. 2 Take Cholesky decomposition for the scale matrix S AAT. 3 W ABB T AT AB AB T W k S, Similar to the study in Section 3 1 we can verify the plausibility of the singular Wishart. random matrices by applying 1 All settings and analyzing steps are the same as those in. Section 3 1 except that here the true df parameter is k 2 5 Note that since k can no longer. be fractional in the singular case the df that is actually considered will be bkc 2 The results. are given in Figure 4 and Figure 5 With similar analyses and discussion we can conclude that. WishPack works well for the generation of Wishart random matrices with singularity Section. 3 1 and 3 2 together illustrate how Wishpack works for the generation of Wishart matrices. 3 3 A Note on Odell Feiveson s Algorithm, We close this Section with the Odell and Feiveson s 1966 hereafter O F algorithm for gener. ating Wishart matrices This method also based on Bartlett s decomposition is popular and. has been widely used e g Liu 2001 and the Matlab package MCMC by Shera 1998 One. should notice that in O F s construction the leading diagonal entry of the Wishart matrix is. a chi square variate with df N 1 where N is the number of independent normally distributed. random vectors It is hence clear that the Wishart matrix has N 1 df not N Consequently. if we would like to employ O F s method to generate a Wishart matrix from W k S in fact. we will end up with a matrix from W k 1 S, To see this by applying the Matlab package MCMC we implement the same analysis as. those in Section 3 1 and 3 2 The only change is that the true df is k 10 The results are. shown in Figure 6 from which we see that obviously each 2i does not follow 2 10 They are. more likely to have a 2 9 distribution since the sample means are almost 9 k 1 and sample. variances are close to 18 2 k 1 This illustration is to provide an example that we may. need to pay extra attention to the use and the explanation of O F s algorithm. 4 Conclusion and Discussion, This paper demonstrates the importance of the Wishart matrix with fractional df in econometric. problems and discusses the generation of Wishart matrices for singular and non singular cases. Since existing OX functions are unable to generate these types of Wishart random samples. we develop this WishPack in order to meet such needs in Bayesian methodology The density. functions for both Wishart and inverse Wishart distributions are also included. Srivastava 2003 and D az Garc a 2007 define different measures for the computation of. Jacobians for deriving the density of a singular Wishart distribution In Wishpack we provide. the calculation of the singular Wishart density based on Srivastava s 2003 version The package. contains several header files and can be downloaded at http www4 ncsu edu yku2 smain html. References, Anderson T W 2003 An Introduction to Multivariate Statistical Analysis 3rd edn John.
Wiley and Sons New York, Asai M and McAleer M 2009 The structure of dynamic correlations in multivariate stochas. tic volatility models Journal of Econometrics 150 2 182 192. Doornik J A 2007 Object Oriented Matrix Programming Using Ox 3rd ed London Tim. berlake Consultants Press and Oxford www doornik com. D az Garc a J A 2007 A note about measures and Jacobians of singular random matrices. Journal of Multivariate Analysis 98 960 969, Gupta A K and Nagar D K 2000 Matrix Variate Distributions CRC Press. Konno Y 2009 Shrinkage estimators for large covariance matrices in multivariate real and. complex normal distributions under an invariant quadratic loss Journal of Multivariate. Analysis 100 2237 2253, Kubokawa T and Srivastava M S 2008 Estimation of the precision matrix of a singular. Wishart distribution and its application in high dimensional data Journal of Multivariate. Analysis 99 1906 1928, Liu J S 2001 Monte Carlo Strategies in Scientific Computing Springer Verlag. Odell P L and Feiveson A H 1966 A numerical procedure to generate a sample covariance. matrix Journal of the American Statistical Association Vol 61 No 313 199 203. Philipov A and Glickman M E 2006 Multivariate stochastic volatility via Wishart pro. cesses Journal of Business and Economic Statistics 24 313 328. Shera D 1998 A Library of Markov Chain Monte Carlo routines for Matlab. http www mathworks com matlabcentral fileexchange 198. Srivastava M S 2003 Singular Wishart and multivariate beta distributions Annals of. Statistics 31 5 1537 1560, Uhlig H 1997 Bayesian vector autoregressions with stochastic volatility Econometrica.
65 1 59 73, Figure 1 Trace plots of k and A from the simulation of A M s W. Generating Random Wishart Matrices with Fractional Degrees of Freedom in OX Yu Cheng Ku and Peter Bloom eld Department of Statistics North Carolina State University February 17 2010 Abstract Several recently proposed Multivariate Stochastic Volatility MSV models suggest using Wishart processes to capture the dynamic correlation structure of asset returns When estimating these models with