Winbugs Multivariate stochastic volatility model: Bayesian estimation and model comparison

In this article, we use a free Bayesian software called Winbugs, which makes it easy to estimate and compare the likelihood-based multivariable stochastic volatility (SV) model. By fitting the two-variable time series data of the weekly exchange rate, nine multivariate SV models, including Granger causality in volatility, time-varying correlation, heavy-tailed error distribution, additive factor structure and multiplication factor structure, illustrate the idea.

The univariate stochastic volatility (SV) model provides a powerful alternative to the arch type model, which explains volatility conditions and unconditional attributes.

The stylized facts of the financial assets income of the multivariate SV model

Considering that the multivariate SV model is most useful for describing the dynamics of financial asset returns, we first summarize some of the stylized facts about the well-documented benefits of financial assets:

1. Asset income distribution is leptokurtic.

2. Asset return rate volatility cluster.

3. Returns are cross-related.

4. Volatility is cross-dependent.

5. Sometimes, one asset Granger's fluctuations lead to the volatility of another asset (that is, volatility spreads from one market to another).

6. There is usually a lower dimensional factor structure that can interpret most of the correlations.

7. The correlation is changed over time.

In addition to these seven stylized facts, questions such as the dimension of the parameter space and the positive semi-deterministic matrix of the covariance matrices are of practical importance. As we review existing models and present our new models, we will comment on their handling of the stylized facts and the appropriateness of the two above.

To illustrate the differences and linkages between alternative multivariable SV models, we focus on the two variables in this article. In particular, we have considered nine different two-variable SV models (acronyms with bold), two of which are new to the literature. In addition, most of these specifications apply to multidimensional generalizations, and model 5 is the only exception.

For t = 1, ...,t, the logarithmic return (median) observed at time T is represented as y t = (y 1t,y 2 t) '. Let ε ton = (ε1 ton , ε2 ton ) ', η- ton = (η1 ton , η2 ton ) ', μ= (μ1,μ2) ',? ton = (? 1 tons ,huh? 2,t ) ', Ω ton = DIAG (EXP (? ton /2)), and

Model 1 (Basic MSV or msv). And? 0 =μ. The model is equivalent to stacking two basic single-variable SV models together. Clearly, the specification does not allow for cross-correlation between earnings or volatility, nor does it allow Granger causality. However, it does allow leptokurtic return distributions and volatility clustering.

Model 2 (constant dependent MSV or CC-MSV). And? 0 =μ. In this model, it is permissible to return impact correlation, so the model is similar to Bollerslev's constant condition-dependent (CCC) arch model. Therefore, the returns are interdependent.

Model 3 (MSV with Granger causality or GC-MSV). And? 0 =μ and φ12 = 0. Since φ21 can be different from 0, the volatility of the second asset allows Granger to fluctuate from the first asset. Therefore, the rate of return and volatility are interdependent. However, the cross-dependence of volatility is realized by Granger causality and volatility clustering. Furthermore, when two φ12 and φ21 are nonzero, the fluctuation of bilateral Granger causality between the two assets is permissible. As far as we know, this specification is the new content of SV literature.

Bayesian estimation using Winbugs

The model in section 2.2 is accomplished by the specification of a priori distribution of all unknown parameters a = (a 1, ...,a p). For example, in Model 1 (MSV),p = 6 and Vector a of an unknown parameter is. Bayesian inference is based on the combined posterior distribution of all the observed θ in the model. Vector θ includes a vector of unknown parameters and potential logarithmic volatility, i.e. θ= (a,h 1, ...,h T).

Experience Statement data

In this section, we will describe the model that matches the actual financial time series data. From January 1994 to December 2003, the data used was the average corrected logarithmic return of AUD and NZD 519 times a week. The choice of these two series is because the two economies are closely connected to each other , thus anticipating a strong dependency between the two currencies. These two series are plotted in the diagram, where the cross-dependence of returns and volatility is really strong.

Time series Chart of the AUD and NZD/USD exchange rate returns.

Results

We report the average of the posterior distributions of the first six models, the standard error and the 95% confidence interval, and the posterior distribution of the last three models, and the calculation time for generating 100 iterations for each of the nine.

The graph and density estimates of the marginal distributions of D, μ and φ in Model 8 (AFACTOR-T-MSV).

The density of the edge distribution of σ is estimated η,σε1, and σε2 in Model 8 (afactor tert-msv).

The density of the ν edge distribution is estimated 1,ν2, and Ω in Model 8 (afactor tert-msv).

All models of DIC

To understand the meaning of a better specification, we obtained a smoothing estimate of the volatility and correlation of Model 8 (AFACTOR-T-MSV) and Model 5 (DC-MSV).

Smoothing estimation of exchange rate volatility and time-varying correlation of model 5 (DC-MSV).

The volatility estimation of smoothing factor and time-varying correlation of model 8 (factor-t-msv).

Conclusion

In this paper, we propose to estimate and compare multivariable SV models by using Bayesian MCMC technique in Winbugs. MCMC is a powerful method that has many advantages over other methods. Unfortunately, it is not easy to write the first MCMC program used to estimate the multivariate SV model, and the more alternative Multivariable SV specification is computationally expensive. Winbugs imposed a short and sharp learning curve. In the two-variable setting, we show that the implementation is simple and the computation speed is fairly fast. In addition, the processing of rich specifications is also very flexible. However, since Winbugs provides a single-action Gibbs sampling algorithm, as one might expect, we find that the blending is usually slow and therefore requires long sampling.

Winbugs Multivariate stochastic volatility model: Bayesian estimation and model comparison