Bayes Estimation Of GARCH-SN Models By MCMC

Background:Lots of time series contain lots of information. Reasonable and adequate use of the information is very meaningful to reveal the law of development and forecast trends. Classical regression methods are modeled by variables’means, with the random disturbance hypothesis of zero expectations and Independent identical normal distribution. The forecast error varies rather than a constant, and depends on the change of the regression error, suggesting that the mean square error may be autocorrelation. Autoregressive conditional heteroskedasticity (ARCH) model’s conditional mean and variance is an observed values function. Generalized autoregressive conditional heteroskedasticity (GARCH) model is an important development of ARCH model. It’s the leap from linear ARCH to nonlinear GARCH.Skew-normal (SN) distribution is a new class of distribution which is obtaind by introducing skewness parameter into normal distribution. GARCH-SN model is good at fitting non-normal random disturbance data. GARCH model is nonlinear and its parameter estimation is complex. Classical methods are usually powerless. Least squares estimate is inapplicable in heteroscedasticity case. Maximum likelihood estimates (MLE) has lots of calculation while slow convergence rate, and may be unlimited. SN distribution’s density function is obtained by fast Fourier transform (FFT) of the characteristic function in quasi maximum likelihood estimation (QMLE) of GARCH-SN model, so the result is approximate too. The traditional frequencies methods might be limited for parameters estimation of time series models with SN distribution. There would be important theoretical and actual research value to solve the parameters estimation of GARCH-SN model.Objective:In the view of Bayesian theoretical, we intend to study the GARCH-SN model’s Bayesian estimation. We could use the Monte Carlo Markov chain (MCMC) method to solve complex high-dimensional operator in Bayesian estimation. MCMC method could construct a discrete time Markov chain, and its stationary distribution would be the posterior distribution of the estimated parameters. Then we could do statistical analysis by the samples from the Markov chain, such as the calculation of posterior mean and its standard deviation, posterior median, etc. Bayesian estimation could avoid defects of the classic maximum lilelihood estimation method, for example, complex computations, no finite solution, and approximate solution, etc. We would like to establish a simple and reliable method for parameter estimationMethods:First, we study the estimation of the skewness parameter λ in SN distribution. Supposed Y～SN (λ), Y could be expressed as the following random form Y=√1-δ2V+δU, with δ=λ/√1+λ2, V～N(0,1) and U～TN(0,1;[0,∞)). V and U would be independent of each other. Supposed the prior δ～U[-1,1], the skewness parameter would be following the distribution t(0,1/2;2).Then, supposed Y-{yt} and yt=μ+δt|t-1εt, set α=?{α0,α1,…,αq) and β=(β1,…,βp), the parameter vector Φ could be expressed as Φ={α,β,μ,λ).The sampling iterative steps would be as follows.(1) Extract the auxiliary variables from full conditional distribution using ARS. (2) Extract the skewness parameter λ from posterior distribution using M-H algorithm.(3) Extract the parameter μ. of stationary time series from posterior distribution using M-H algorithm.(4) Extract the parameters α={αi}i=0q and β={βj}j=1p using Griddy-Gibbs sampling.We obtain all the parameters estimates of the GARCH(p,q)-SN(λ) model by MCMC sampling.Simulation:The statistical simulation was calculated using R2.15.0. The simulated GARCH(1,1)-SN(λ) models in the study were as follows, θ=(0.05,0.20,0.50,0.20),θ=(0.05,0.20,0.50,0.80),θ=(0.05,0.20,0.70,0.20) θ=(0.05,0.20,0.70,0.80),θ=(0.05,0.50,0.20,0.20),θ=(0.05,0.50,0.20,0.80) θ=(0.05,0.70,0.20,0.20),θ=(0.05,0.70,0.20,0.80). We repeated sampling10000times, the sample size were200,500, and1000, respectively. The simulated results of parameters (α0,α1,β,λ) in different input sets were as follows.(1) All Markov chains are smooth convergence, and the density curves of posterior distributions are smooth, roughly bell-shaped characteristics.1-rejectionRate is closer to1and the simulation process is more stable while input set size is larger.(2) The correlation between parameters was not significant.(3) Parameter estimates satisfy the stationary conditions α1+β<1.(4) α0is smaller, and approximates to α0as the sample size increasing. The effect is good when the sample size is above500.(5) When α1+β<1but not very close to1, λ is small, the model is stable. α1is smaller, and approximates to α1as the sample size increasing,β is larger, and approximates to β as the sample size increasing. The effect is good when the sample size is above500.(6) When α1+β<1but not very close to1, λ is large, the model is stable,α1, is smaller, and approximates to α1, as the sample size increasing.β is larger, and approximates to β as the sample size increasing when β is small. β is smaller, and approximates to β as the sample size increasing when β is large. The effect is good when the sample size is above500.(7) When α1+β<1and close to1, λ is small, the model is stable. α1, and β are both increased as the sample size increasing. The effect is good when the sample size is above500.(8) When α1+β<1and close to1, λ is lager, the model is stable.α1, approximates to α1as the sample size increasing. β is larger, and approximates to β as the sample size increasing when β is small. β approximates to β as the sample size increasing when β is large.(9) If the disturbance distribution hypothesis was mistaken and the sample size was large enough, the estimation of α0would be stability. The estimation of α1, would be biased when α1+β is less than1but close to1, and λ is large. The estimation of β would be biased when α1+β is less than1but not close to1, and λ is large.Case:We study Guangzhou monthly death toll time series model application and comparation by R2.15.0and GTM3.0software. There’re48months data (mean±Std. Deviation:1939.5±304.9, median:1814.5, mix:1544, max:2881). The data does not meet the assumptions of normality (Shapiro-Wilk value=0.8635, P<0.001). Seasonal autocorrelation at lag12month,24month,36month promptes to model the seasonal difference sequence. It’s not relevant after First-order difference and seasonal difference. It’s suggested to model autocorrelation with lag1and12. The residual of AR(1)12model does not obey the standard normal distribution which promptes to model the time series model with non-normal distribution assumption. Then, we model seasonal AR(1)12-SN(λ) and compute12-step forecast values. Seasonal AR (1)12-SN(λ) model’s short-and-medium-term forecast effect is good.Conclusion:Firstly, we construct the Bayessian estimation of GARCH(p,q)-SN(λ) model. MCMC mothod is effective to simulate the marginal posterior distribution of all parameters. The simulation process is stable and convergence. It’s one of the simple and reliable mothod for GARCH (p,q)-SN(λ) model’s parameters estimation.Secondly, Parameter estimates may be biased if ignoring the random disturbance distribution identification, so it’s necessary to identife the distribution of the random disturbance.Thirdly, time series models with SN distribution could be better fitting the series fluctuation characteristics when lots of series disturbance were not satisfied the standard normal distribution hypothesis. The class of models would be used for predictions in epidemic cases, hospital outpatients and investment rate of return, etc. It would be a meaningful application extension for time series models.Shortcomings:Bayes estimation of the model requires great sample. The estimation error would be too large if the sample size is not great enough. So it would be not applied to small sample of time series forecasting applications. The calculation is large, so it would be needed to improve processing speed for the real-time time series forecasting applications.