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Multivariate Pareto Distributions: Inference and Financial Applications a
Emmanuel N. Papadakis & Efthymios G. Tsionas
a
a
Department of Economics , Athens University of Economics and Business , Athens, Greece Published online: 01 Mar 2010.
To cite this article: Emmanuel N. Papadakis & Efthymios G. Tsionas (2010) Multivariate Pareto Distributions: Inference and Financial Applications, Communications in Statistics - Theory and Methods, 39:6, 1013-1025, DOI: 10.1080/03610920902833560 To link to this article: http://dx.doi.org/10.1080/03610920902833560
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Communications in Statistics—Theory and Methods, 39: 1013–1025, 2010 Copyright © Taylor & Francis Group, LLC ISSN: 0361-0926 print/1532-415X online DOI: 10.1080/03610920902833560
Downloaded by [Athens University of Economics] at 04:27 30 March 2015
Multivariate Pareto Distributions: Inference and Financial Applications EMMANUEL N. PAPADAKIS AND EFTHYMIOS G. TSIONAS Department of Economics, Athens University of Economics and Business, Athens, Greece Univariate Pareto distributions are extensively studied. In this article, we propose a Bayesian inference methodology in the context of multivariate Pareto distributions of the second kind (Mardia’s type). Computational techniques organized around Gibbs sampling with data augmentation are proposed to implement Bayesian inference in practice. The new methods are shown to work well in artificial examples involving a trivariate distribution, and to an empirical application involving daily exchange rate data for four major currencies. Keywords Bayesian inference; Econometrics; Financial contagion; Financial crises; MCMC; Multivariate Pareto distribution. Mathematics Subject Classification 91G70; 62F15; 60J10; 60J22; 60E; 91G80.
1. Introduction Despite the fact that univariate Pareto distributions have been analyzed extensively using a variety of methods (Arnold and Press, 1989; Arnold et al., 1998), there is a lack of methods for analyzing multivariate Pareto distributions. For example, it is well known that for many financial data, heavy tails of the Pareto form are very reasonable approximations. However, there are also contagion and spillover effects in financial markets, so if we restrict attention to extreme tails we have also to account for the fact that the behavior of financial data in the tails may be correlated across financial markets, as we discuss in Sec. 4 in more detail. This naturally suggests multivariate Pareto models as reasonable candidates. Inference methods for multivariate Pareto distributions have been somewhat limited. (Mardia, 1962) discussed inference methods for the multivariate Pareto of the first kind; Arnold (1983) discussed some issues concerning Bayesian estimation for the Pareto distribution of the second kind, without deriving the posterior Received October 1, 2008; Accepted February 14, 2009 Address correspondence to Efthymios G. Tsionas, Department of Economics, Athens University of Economics and Business, 76 Patission St., Athens 104 34, Greece; E-mail:
[email protected]
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density. More recently, Hanagal (1996) studied maximum likelihood for certain multivariate Pareto distributions, setting the scale parameter equal to unity, which is the standard form of the multivariate Pareto distribution of the second type. The analysis of the later can be said that it is limited compared to ours because not only assumes scale homogeneity, he restricts even more his analysis setting scale parameter equal to a known fixed constant. For an introduction to the multivariate Pareto distributions, see Kotz et al. (2000, Ch. 52). In this article, we take up the general multivariate Pareto distribution of the second kind, and propose a complete scheme for exact, posterior inference based on Gibbs sampling with data augmentation. The new methods are applied to artificial data, as well as daily exchange rate data for four major currencies over the period 1996–2000.
2. The Model and Methods Suppose Xt t = 1 T is a random sample of size T from a k-variate multivariate Pareto distribution of the second kind, where Xt = X1t Xkt . We exploit the representation (Mardia, 1962) Wit Xit = i + i i = 1 k t = 1 T (2.1) Zt where i > 0 is a scale parameter and i is a location parameter. Let the kdimensional vectors, = 1 2 k and = 1 2 k , respectively, to represent scale and location heterogeneity. The Wit ’s are independent random variables with the standard exponential distribution, that is p Wit = exp−Wit , Wit ≥ 0, and Zt follows independently a gamma distribution, namely p Zt = −1 Zt −1 exp−Zt , Zt > 0, > 0. It follows easily that the conditional distribution of Xt given Zt must be: k −1 k Xit − i Zt t = 1 T Xit > i i exp − p Xt Zt = Ztk i i=1 i=1 (2.2) Therefore, the joint distribution is k −1 k Xit − i p Xt Zt = −1 Ztk+ −1 Zt − Z t i exp − i i=1 i=1 t = 1 T Xit > i The likelihood function is L X =
T
p Xt
t=1
k −1 k Xit − i Ztk+ −1 Zt dZt i exp − 1 + i t=1 0 i=1 i=1 k −T T k Xit − i Ztk+ −1 exp − 1 + Zt dZt = −T i i i=1 t=1 0 i=1 (2.3) = −T
T
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After some algebraic manipulation we obtain: k −T T − +k
k T Xit − i 1+ i L X = + 1 · · · + k − 1
i i=1 t=1 i=1 (2.4) The marginal distribution of Xt has density given by: − +k
k −1 k Xit 1+ i p Xt 0 = + 1 · · · + k − 1
i i=1 i=1
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Xit > 0 t = 1 T
(2.5)
with the obvious generalization when = 0. In this latter case, we take the survival function: k xi − i − Xit ≥ i i > 0 > 0 t = 1 T P Xt ≥ xt = 1 + i i=1 (2.6) see Arnold (1983). From Arnold et al. (1994), the multivariate Pareto distribution of second kind with parameters a has multivariate Pareto of second kind marginals. For this distribution, provided that > 2, the following can be shown. The marginal expected values are: E Xit = i +
i for every i − 1
Regression remains linear with: xjt − j i 1+ j xjt − j 2 +1 Var Xit Xjt = xjt = i2 2 1+ − 1
j E Xit Xjt = xjt = i +
Given a prior distribution p a the posterior distribution is given by Bayes’ theorem as: p a X ∝ L a X p a One restriction that must be introduced among the parameters is the range constraint i ≤ min Xit t = 1 T i = 1 k
(2.7)
owing to the fact that Xit ≥ i i = 1 k t = 1 T . We choose the following prior distributions in Bayesian inference: p ∝ exp−
(2.8)
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k
(2.9)
exp−i i
i=1
p ∝
k
− i +1
i
exp−i i−1
(2.10)
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i=1
where the underscored letters stand for the so-called hyperparameters. In other words, we specify the priors but we assume implicitly a hierarchical representation of the form p . So, is the vector of parameters of interest and is the corresponding vector of parameters of the prior distribution. In our case the vector of hyperparameters is
, where the inside parenthesis group denotes the two hyperparameters for . The posterior is highly complicated to permit analytical expressions for marginal moments and densities, so numerical procedures must be used to make posterior inferences. To this end, we consider the joint posterior of X and Z, and examine Gibbs sampling with data augmentation as a candidate numerical approach to marginal inferences. Based on the posterior distribution we can derive the following conditionals: p a Z X ∝ −T
T
a Zt
exp−
(2.11)
t=1
T i Z X ˜ IG T + i Zt Xit − i + i i = 1 k
(2.12)
t=1
where G · · and IG · · indicates the Gamma and the Inverted Gamma distributions, respectively. If a random variable Y has a Gamma distribution with 1 parameters b by the transformation X = Y − 2 , the random variable X has an Inverted Gamma distribution with parameters b . The conditional posterior of the i ’s is given by T −1 p i Z X ∝ exp i + i Zt i = 1 k
(2.13)
t=1
subject to i ≤ min Xit t = 1 T i = 1 k Finally, k Xit − i Zt X ˜ G k + 1 + t = 1 T i i=1
(2.14)
Drawing from these distributions is straightforward. The distribution of is log-concave so specialized algorithms may be used. Here, we used the adaptive rejection technique proposed by Wild and Gilks (1993). A list of log-concave densities and log-concavity checks can be found in Wild and Gilks (1992, Table 2). The procedure is adaptive in the sense that as we add grid points, n → , we achieve better approximation of the target density due to the envelope density improvement. Drawing from the conditional distribution of i is accomplished by inverting directly the cumulative distribution function: the distribution is of the form f x ∝
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exp Ax , subject to x ∈ 0 B , so a draw can be realized as x = A−1 logu exp AB − 1 + 1, where u is a standard uniform random number. Before proceeding to the next section where we use Markov Chain Monte Carlo techniques it is useful to mention some things about the well known Gibbs sampler. The Gibbs sampler (Casella and George, 1992; Gelfand and Smith, 1990) is a technique suitable for generating an irreducible aperiodic Markov chain that has as its stationary distribution a target distribution in a high-dimensional space but having some special structure. The Gibbs sampler can be considered as a special case of Metropolis–Hastings algorithm. In this case, full conditionals have sufficient information to uniquely determine a multivariate joint distribution by the Hammersley–Clifford theorem (Hammersley and Clifford, 1971; Robert and Casella, 1999) together with the required positivity condition (Besag, 1974) to state this result. Statements of the above can be found in Appendix. In many cases, it is not possible to draw from the probability density p X
but it is easy to draw from p X z where z can be interpreted as latent variables. Thus, a posterior simulation procedure which sequentially draws from p X z
and p z X is referred to as Gibbs sampling with data augmentation (Chib, 1995; Tanner and Wong, 1987). In Gibbs sampling with data augmentation we integrate out the latent variables vector z and the posterior is: p X =
p X z p z X dz ≈
N 1 p X z j N j=1
(2.15)
Equation (2.15) can be evaluated by simply calculating p X z j for each draw j = 1 N and averaging the result which converges by the weak law of large numbers to p X as N goes to infinity.
3. Artificial Data To ensure that the algorithm works well, it has been applied to artificial data from a trivariate distribution with i = i = 1, T = 50 and for = 05, = 15 and = 2. The prior parameters are = 1, and i = i = i = 0. Gibbs sampling has been implemented using 15,000 iterations, 5,000 of which are discarded to mitigate the impact of start-up effects. The posterior statistics are reported in Tables 1–2. For the case where = 05, marginal posterior distributions are presented in Fig. 1, and autocorrelation functions of the Gibbs draws are presented in Fig. 2 correspondingly. Also reported are numerical standard errors, indicating that numerical precision is acceptable, and Geweke’s convergence diagnostics (Geweke, 1992). Clearly, posterior means are reasonably close to their true values testifying to the usefulness of posterior inference and Gibbs sampling from multivariate Pareto distributions. From the shapes of marginal posterior distributions, it is to be noted that they are highly asymmetric, so a normal approximation implied by asymptotic properties of maximum likelihood would not be adequate even for samples of size 50. Gibbs sampling performs fairly well since autocorrelations die off sufficiently fast despite the fact that underlying tails are heavy. This implies that Gibbs sampling explores the posterior in a computationally efficient way.
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Parameters
Posterior mean
Posterior S.D.
Numerical standard error
Geweke’s convergence diagnostic
1 2 3 1 2 3
0.961 1.007 1.043 1.179 1.067 0.835 0.459
= 05 0.022 0.021 0.021 0.426 0.392 0.296 0.078
0.0002 0.0002 0.0002 0.0221 0.0202 0.0155 0.0032
1636 225 02341 −08289 −07792 −07292 −06676
1 2 3 1 2 3
0.959 0.972 0.969 1.6 1.405 1.392 1.703
= 15 0.022 0.022 0.022 0.595 0.507 0.514 0.418
0.0002 0.0002 0.0002 0.0358 0.0314 0.0311 0.0263
−02234 04068 06982 09161 0663 06237 09809
1 2 3 1 2 3
0.962 0.969 0.963 1.863 1.431 1.592 2.772
=2 0.021 0.021 0.021 0.733 0.575 0.653 0.82
0.0002 0.0002 0.0002 0.057 0.0424 0.0496 0.0665
06253 02055 1047 02456 04389 02924 04168
Table 2 Posterior statistics for exchange rate data Posterior mean
Posterior S.D.
Numerical standard error
Geweke’s convergence diagnostic
DM FF JY BP
0.227 0.626 2.055 0.0087
0.081 0.217 0.695 0.0039
0.0076 0.0216 0.0675 0.0002
−0262 −0228 −0247 −0247
DM FF JY BP
0.122 0.456 1.11 0.0022
0.0095 0.0096 0.0094 0.0011
93 · 10−5 0.0001 96 · 10−5 117 · 10−5
−0279 −1421 0576 −0538
2.861
0.774
0.0795
−0267
Parameters
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Figure 1. Marginal posterior distributions for artificial data = 05 .
An important issue in MCMC methods is to assess if the draw has converged after say N draws and for this task several convergence criteria have been proposed in the literature. Geweke (1992, 1999) proposed the comparison of the estimated posterior mean g 1 from the first N1 draws with the estimated posterior mean g 2
from the last N2 draws.
Figure 2. Autocorrelation functions for artificial data = 05 .
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Assume that g is some function of interest and we employ Gibbs sampling to approximate the expectation E g
. We assume that we have only two parameters j j = 1 2 and we use the notation j = 1 2 to represent the jth draw, for each j = 1 N . Generalization to the case of more parameters is straightforward. j
j
The problem is to estimate the mean of G j = g 1 , g 2
, where G j is, in our case, a 2 × 1 stochastic process or in general an s × 1 stochastic process. The first problem we have to deal with is the dependence of successive draws while assume that the sequence j is identically distributed. The asymptotically efficient, in terms of N , estimator of this mean is simply the sample average of all the
g s j , s = 1 2. In our notation we take g sN = N −1 Nj=1 G j , with corresponding numerical standard errors NSE g sN t , s = 1 2. Given the sequence G j Nj=1 , comparison of first N1 values in the sequence with the last N2 in the sequence
N1 is probable to uncover failure of convergence. Define g 1N = N1 −1 j=1 G j ,
g 2N = N2 −1 Nj=N ∗ G j , where N ∗ = N − N2 + 1 and the corresponding numerical 1
and NSE g 2
standard errors NSE g 1
and NSE g 2
. Let NSE g N
N
N
N
N
1 denote consistent estimated numerical standard errors for G j j=1 and G j Nj=N ∗ , respectively. The statistic
g 1N − g 2N
D G= → N 0 1
2 2 2
1
+ NSE g NSE g N N
(3.1)
if the chain has converged. The above expression is Geweke’s convergence diagnostic. In our cases, it appears that Gibbs sampling has converged based on the values of this statistic. Numerical standard errors and convergence diagnostics are computed based on the spectral density at zero of Gibbs draws. This is estimated using an autoregressive approximation of order ten.
4. Contagion Before proceeding to the empirical section we should give the general framework on which it is based. This is the notion of contagion and we present two definitions according to the literature. Broadly, contagion is the cross-country transmission of shocks. Contagion does not need to be related to crises although it assumes more importance during crisis periods. Another, more restrictive definition, treats contagion as the transmission of shocks to other countries or cross-country correlation, beyond any fundamental link among the countries and among the shocks. The latter definition is explained by what is called herding behavior or mutual mimetic contagion (Topol, 1991). Among the different mechanisms that explain the herding behavior by international investors, the literature has emphasized the role of asymmetric information. Due to the fact that information is costly, investors remain uninformed about the countries in which they invest. The relatively uninformed investors follow the supposedly informed investors so the markets move jointly. Moreover, investors try to infer future price ups and downs based on how the rest of the market is reacting. These types of reactions lead to herding behavior and panics. Kiyotaki and Moore (2002) explored the reasons why some countries appear to have more contagion than others and they examine two different mechanisms
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which may generate contagion. King and Wadhwani (1990) studied the case where mistakes or idiosyncratic changes in one market may be transmitted to other markets causing increases to volatility. An implication of their arguments is that increases in volatility may be self reinforcing and persistent. Previous research has used two general frameworks of analysis. The first is based on the theory of efficient markets and the second turns attention to several non economic factors like psychological and other non rational reasons. This family of models is broadly known as “fads” models. Lux (1995) adopted a probabilistic approach to model observed heterogeneity and the rather complex micro-structure of speculative markets. There are also a lot of other empirical or theoretical approaches, like the so-called “wake up call hypothesis” and propagation (Kaminsky et al., 2003).
5. Empirical Application For empirical application, we consider daily data for exchange rates over the period 1996–2000 for the German mark, the French franc, the Japanese yen, and the British pound (1,001 observations in total). All rates are taken against the U.S. dollar. Since the issue is to make inferences about the extent of heavy tails, we consider only the observations for which absolute exchange rate changes exceed twice the median for each series for a given date. This restricts attention to periods of high volatility in all currencies simultaneously. This provides a total of 108 observations. Posterior statistics are reported in Table 2 along with numerical standard errors (indicating acceptable precision) and Geweke’s convergence diagnostics (indicating that Gibbs sampler has converged to the posterior distribution). The posterior mean of is 2.861 indicating heavy tails. Marginal posteriors are reported in Fig. 3, and they exhibit the same characteristic shape as in the case of the artificial example, indicating that inferences based on asymptotic normal approximations are inadequate. Notably, the marginal posterior of is nearly uniform around its posterior mean (0.0087) and around a small range indicating fairly high confidence to this estimate (notice also the small posterior standard deviation). Autocorrelation functions for Gibbs draws, reported in Fig. 4, indicate the absence of significant serial correlation with the exception of the parameters: at lag 50, autocorrelations for these parameters are typically near 0.40, which is relatively high but not destructive for making computationally efficient and reliable posterior inferences. Another important observation is that the joint posterior is highly non-normal as evidenced by the graphs in Fig. 5. Due to the range constraint among the ’s, bivariate distributions are highly asymmetric, so one would expect that the performance of Gibbs sampling could be impaired. On the contrary, the Gibbs sampler converges fairly fast, and autocorrelations are not destructive. 5.1. Measuring Contagion The model offers a natural way to define contagion in the mth market as:
Cm x∗ = P Xmt ≥ xm∗ ∩ Xit ≥ xi∗ i = m x∗ ∈ k m = 1 k
(5.1)
which gives the probability that volatility is high (in the sense that absolute changes in returns exceed a given value) conditional on the fact that in all other markets
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Figure 3. Marginal posterior distributions for exchange rate data. P∩ X ≥x∗ i=1k
it i volatility is also high. Clearly, Cm x∗ = P∩ X . For each drawn value ∗ it ≥xi i=m of the parameters, we can simulate a multivariate Pareto model based on the representation in (2.1) and estimate the contagion measure by approximating the probabilities involved using a frequency approach. This can be repeated for each
Figure 4. Autocorrelation functions for exchange rate data.
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Figure 5. Aspects of posterior distributions for exchange rate data.
parameter draw, and the results can be averaged to yield the final contagion measure. We have estimated the contagion measure for two alternative definitions of x∗ , namely the median vector of the data (Case 1), and five times the median (Case 2) to obtain the following results. 5.2. Contagion Measures for Currencies Currency DM FF JY BP
Case 1
Case 2
.904 .901 .897 .892
.878 .883 .892 .892
These results provide evidence in favor of substantial contagion effects. Indeed, the correlations of the original data (in absolute changes exceeding the original median) are extremely high-indicating substantial co-movement in the heavy tails of exchange rate changes. In terms of the original data, the kind of tail behavior we examine here is in the probability range of 1% but this does not imply that measuring co-movement of rare events is not worthy of consideration. In fact, portfolio managers, forecasters, and policy makers would have been interested in exactly this kind of information for rational decision making. Of course, with the introduction of the euro this information has historical interest only, unless we
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are willing to interpret the above results in the context of some existing major tendency in the euro-yen trend. More realistically, multivariate Pareto models could be applied to other currencies involving the euro as a whole (for which, however, we do not presently have a sizeable number of observations) and draw relevant conclusions for financial decision making, and policy analysis.
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Appendix Definition A.1 (Positivity Condition). Let p y1 y2 yk be the joint density of a k dimensional random vector Y and let p i yi denote the marginal density of Yi , i = 1 k If p i yi > 0 for every i = 1 k implies that p y1 y2 yk > 0, then the joint density p is said to satisfy the positivity condition. Let pi yi y1 yi−1 yi+1 yk denote the conditional density of Yi Y−i = y−i . Theorem A.1 (Hammersley–Clifford). Under the positivity condition, the joint density p satisfies p y1 y2 yk ∝
k pj yj y1 yi−1 yi+1 yk
p y y1 yi−1 yi+1 yk
j=1 j j
for every y and y in the support of p (Hammersley and Clifford, 1971). Under the positivity condition, the Gibbs sampler generates an irreducible Markov Chain, thus providing the necessary convergence properties.
Acknowledgment This research project is co-financed by E.U.-European Social Fund (75%) and the Greek Ministry of Development – GSRT (25%). Research support is gratefully acknowledged.
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