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In this article, we make it easy to estimate and compare likelihood based multivariate random volatility (SV) models using a free Bayesian software called WinBUGS. Illustrate ideas by fitting bivariate time series data for weekly exchange rates, multivariable SV models, including Granger causality in volatility, time-varying correlation, heavy-tailed error distributions, additive factor structures and multiplicative factor structures.
The univariate random volatility (SV) model provides a powerful alternative to ARCH type models and can explain the conditional and unconditional properties of volatility.
Multivariate SV model
The stylized reality of returns on financial assets
Considering that the multivariable SV model is most useful for describing the dynamics of returns on financial assets, we first summarize some well-documented stylized facts about returns on financial assets:
- Asset income distribution is peak thick tail characteristics.
- Return on assets volatility cluster.
- Returns are cross-correlated.
- Volatility is cross-dependent.
- Granger fluctuations in one asset lead to fluctuations in another.
- There is usually a low dimensional factor structure that can explain most of the correlation.
- Correlation changes over time.
In addition to these seven stylized facts, questions such as the dimension of a parameter space and the positive semi-deterministic covariance matrix are of practical importance. As we review existing models and introduce our new models, we will comment on their appropriateness to deal with stylized facts and the two issues mentioned above.
To illustrate the differences and connections between alternative multivariable SV models, we focus on the bivariate case in this paper. In particular, we considered nine different bivariate SV models (with bold acronyms), two of which were new to the literature. In addition, most of these specifications apply to multidimensional generalizations, with Model 5 being the only exception.
Model 1 (basic MSV or MSV). This model is equivalent to stacking two basic univariate SV models together. Obviously, the specification does not allow cross-correlation between returns or volatility, nor does it allow Granger causality. However, it does allow for peak thick-tailed characteristic yield distributions and volatility clustering.
Model 2 (constant correlation MSV or CC-MSV) in this model, return impact correlation is allowed, so the model is similar to Bollerslev’s constant conditional correlation (CCC) ARCH model. So yields are interdependent.
Model 3 (MSV with Granger causality or GC-MSV). Since φ 21 can be different from zero, the second asset’s fluctuation is allowed to be Granger by the first asset’s fluctuation. So both yields and volatility are interdependent. However, the cross-dependence of volatility is realized through granger causality and volatility clustering. In addition, when two φ 12 and φ 21 are non-zero, the bilateral Granger causality is allowed to fluctuate between the two assets. As far as we know, this specification is an addition to the SV literature.
Use WinBUGS for Bayesian estimation
The model in Section 2.2 is established by calculating all unknown parameters a = (a 1… , a p) to complete the specification of prior distribution. For example, in model 1 (MSV), p = 6 and vector A with unknown parameters is. Bayesian inference is based on the joint posterior distribution of all unobserved quantities θ in the model. The vector θ includes a vector of unknown parameters and potential logarithmic volatility, i.e., θ = (a, h 1… , h T).
The empirical description
data
In this section, we will introduce models that match actual financial time series data. From January 1994 to December 2003, the data used were the average corrected logarithmic returns of 519 Australian and New Zealand dollars per week. The two series were chosen because the two economies are so closely linked to each other that strong dependence between the two exchange rates was anticipated. The two series are plotted in a chart where the cross-dependence of yields and volatility does appear strong.
Time series plot of the Australian and New Zealand dollar/US dollar exchange rates.
The results of
We report the mean, standard error, and 95% confidence interval posterior distributions for the first six models and the posterior distributions for the last three models, as well as the computation time to generate 100 iterations for each of the nine.
Graph and density estimates of the marginal distributions of D, μ and φ in the model (Afactor-T-MSV).
The density of σ edge distributions η, σ ε1, and σ ε2 are estimated in the model (AFactor MSV).
The density of the edge distribution of ν is estimated at 1, ν 2, and ω in the model (AFactor MSV).
DIC for all models
To understand the implications of better specifications, we obtain smooth estimates of volatility and correlation for models (Afactor-T-MSV) and models (DC-MSV).
conclusion
In this paper, we propose estimating and comparing multivariable SV models using Bayesian MCMC techniques through WinBUGS. MCMC is a powerful approach with many advantages over other approaches. Unfortunately, writing the first MCMC program for estimating multivariable SV models was not easy, and comparing alternative multivariable SV specifications was computationally expensive. WinBUGS imposes a short and sharp learning curve. In the two-variable setup, we show that it is simple to implement and fairly fast to compute. In addition, handling rich specifications is very flexible. However, because WinBUGS provides the single-motion Gibbs sampling algorithm, as one would expect, we found that mixing is usually slow and therefore requires long sampling.
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