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Original source:Tuo End number according to the tribe public number

 

 

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. Ideas are illustrated by fitting bi-variable time series data for weekly exchange rates, multi-variable SV models, including Granger causality in volatility, time-varying correlation, heavy-tailed error distributions, additive factor structures and multiplicative factor structures.

The single variable random volatility (SV) model provides an effective alternative to ARCH model 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:

  1. Asset income distribution is peak thick tail characteristics.
  2. Return on assets volatility cluster.
  3. Yields are cross-correlated.
  4. Volatility is cross-dependent.
  5. Granger fluctuations in one asset lead to fluctuations in another.
  6. There is usually a low dimensional factor structure that can explain most of the correlation.
  7. Correlation changes over time.

 

In addition to these seven 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 two-variable SV models (with bold acronyms). Furthermore, most of these models apply to multidimensional variables, with model 5 being the only exception.

 

Model 1 (basic MSV or MSV). The model is equivalent to combining two basic univariate SV models. Obviously, the model does not allow correlation between cross-returns or volatility, nor does it allow Granger causality. However, it allows peak thick-tailed characteristic yield distribution and volatility clustering.

Model 2 (constant correlation MSV or CC-MSV) in this model, yield shock correlation is allowed, so this 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, bidirectional Granger causality is allowed for fluctuations between the two assets. As far as we know, this model is an addition to the SV literature.

 

 

Use WinBUGS for Bayesian estimation

The model is established by calculating all unknown parameters a = (a 1… , a p) to complete the prior distribution Settings. For example, in model 1 (MSV), p = 6 and vector A with unknown parameters. 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 fit actual economic time series data. From January 1994 to December 2003, the data used were the weekly average adjusted logarithmic returns of the Australian and New Zealand exchange rates. The two sequences were chosen because the two economies are so closely linked to each other that the dependence between the two exchange rates was expected to be strong. The two series are plotted in a chart where the cross-dependence of yields and volatility does appear strong.

Time series chart of exchange rate returns.

Basis MSV

Basis of MSV

Due to non-standardized parameter Settings, simulation

Code snippet:

model volatility; {for (i in 1:N) { Yisigma2a[i] <- exp(-th[i,1]); Yisigma2b[i] <- exp(-th[i,2]); Y[i,1]~ dnorm(0,Yisigma2a[i]); Y [I, 2] ~ dnorm (0, Yisigma2b th [1, 1] ~ [I] dnorm (thmean [1, 1], itaua2); Th [1, 2] ~ dnorm (thmean [1, 2], itaub2 for (I in 2: N) {thmean [I, 1] < - mu1 + phi1 * (th [I - 1, 1] - mu1); Thmean [I,2] < -mu2 + phi2*(th[i-1,2]-mu2Copy the code

MSV  Granger Causality GC-MSV

Code snippet:

model volatility; { for (i in 1:N) { ysigmadet[i]<-exp(th[i,1]+th[i,2])*(1-rhoep*rhoep; Yisigma2[I,1,1] < -exp (th[I,2])/ysigmadet[I; Yisigma2[I,2,1] < -yisigma2 [I,1; for (I in 2:N) {thmean[I,1] < -mu1 + Phi1 * (th [I - 1, 1] - mu1) + phi12 * (th [I - 1, 2] - mu2); thmean [I, 2] < - mu2 + with phi2 * (th [I - 1, 2] - mu2)Copy the code

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.

Curves and density estimates of the edge 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 better understand the meaning of model definitions, 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. However, writing the first MCMC program for estimating multivariable SV models was not easy, and comparing alternative multivariable SV specifications was computationally complex. 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, it is very flexible to work with rich models. However, because WinBUGS provides the Gibbs sampling algorithm, we found that mixed sampling is usually slow and therefore takes a long time to sample.

 

 

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