## An adaptive Markov Chain Monte Carlo method for GARCH model (2009)

Citations: | 4 - 4 self |

### BibTeX

@MISC{Takaishi09anadaptive,

author = {Tetsuya Takaishi},

title = {An adaptive Markov Chain Monte Carlo method for GARCH model},

year = {2009}

}

### OpenURL

### Abstract

Abstract. We propose a method to construct a proposal density for the Metropolis-Hastings algorithm in Markov Chain Monte Carlo (MCMC) simulations of the GARCH model. The proposal density is constructed adaptively by using the data sampled by the MCMC method itself. It turns out that autocorrelations between the data generated with our adaptive proposal density are greatly reduced. Thus it is concluded that the adaptive construction method is very efficient and works well for the MCMC simulations of the GARCH model.

### Citations

2493 |
E.: Equation of state calculations by fast computing machines
- Metropolis, Rosenbluth, et al.
- 1953
(Show Context)
Citation Context ...se an MCMC method generating data with a small τ. i=1 3.2 Metropolis-Hastings algorithm The most general and simple method to draw values from a given probability distribution is the Metropolis method=-=[12]-=- or its generalized version, MetropolisHastings method[13]. Let P(x) is a probability distribution from which data x4 Tetsuya Takaishi shall be sampled. First starting from x, we propose a candidate ... |

1443 | Generalized Autoregressive Conditional Heteroskedasticity
- Bollerslev
(Show Context)
Citation Context ... proposed. In 1982 Engle[10] proposed Autoregressive Conditional Heteroskedasticity (ARCH) model where the present volatility is assumed to depend on squares of the past observations. Later Bollerslev=-=[11]-=- proposed Generalized ARCH (GARCH) model which includes additional past volatility terms to the present volatility estimate. A conventional approach to infer GARCH model parameters is the Maximum Like... |

1347 |
Monte carlo sampling methods using Markov chains and their applications
- Hastings
- 1970
(Show Context)
Citation Context ...Metropolis-Hastings algorithm The most general and simple method to draw values from a given probability distribution is the Metropolis method[12] or its generalized version, MetropolisHastings method=-=[13]-=-. Let P(x) is a probability distribution from which data x4 Tetsuya Takaishi shall be sampled. First starting from x, we propose a candidate x ′ which is drawn from a certain probability distribution... |

1047 |
Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of the United Kingdom
- Engle
- 1982
(Show Context)
Citation Context ...ized facts. In finance volatility is an important quantity to measure risk. To forecast volatility, various empirical models to mimic the properties of the volatility have been proposed. In 1982 Engle=-=[10]-=- proposed Autoregressive Conditional Heteroskedasticity (ARCH) model where the present volatility is assumed to depend on squares of the past observations. Later Bollerslev[11] proposed Generalized AR... |

416 | Stochastic volatility: likelihood inference and comparison with ARCH models
- Kim, Shephard, et al.
- 1998
(Show Context)
Citation Context ...data for analysis. In this study the Metropolis algorithm is implemented as follows. We draw a candidate θ ′ by adding a small random value δθ to the present value θ:6 Tetsuya Takaishi θ ′ = θ + δθ, =-=(15)-=- where δθ = d(r −0.5). r is a uniform random number in [0, 1] and d is a constant to tune the Metropolis acceptance. We choose d so that the acceptance becomes greater than 50%. 0.2 Adaptive 0.2 Metro... |

184 | Empirical properties of asset returns: stylized facts and statistical issues. quantitative finance
- Cont
- 2001
(Show Context)
Citation Context ...es of asset returns show various interesting properties which can not be explained from the assumption that the time series obeys the Brownian motion. Those properties are classified as stylized facts=-=[1]-=-. Some examples of the stylized facts are (i) fat-tailed distribution of return (ii) volatility clustering (iii) slow decay of the autocorrelation time of the absolute returns. The true dynamics behin... |

52 |
Bayesian Inference on GARCH models using the Gibbs Sampler
- Bauwens, Lubrano
- 1998
(Show Context)
Citation Context ...an inference. Usually the Bayesian inference procedure is performed by MCMC methods. There is no unique way to implement MCMC methods. So far a variety of methods to MCMC procedure have been developed=-=[14]-=--[19]. In a recent survey[18] it is shown that Acceptance-Rejection/Metropolis-Hastings (AR/MH) algorithm works better than other algorithms. In the AR/MH algorithm the proposal density is assumed to ... |

20 | Bayesian Analysis of ARMA-GARCH Models: A Markov Chain Sampling Approach
- Nakatsuma
- 2000
(Show Context)
Citation Context ...ee similar behavior. To quantify the correlation we measure the autocorrelation function (ACF). The ACF of certain successive data x is defined by ACF(t) = 1 ∑N N j=1 (x(j)− < x >)(x(j + t)− < x >) , =-=(16)-=- where < x > and σ 2 x are the average value and the variance of x respectively. Fig. 2 shows the ACF for the adaptive construction method and the Metropolis algorithm. The ACF of the adaptive constru... |

8 |
Avalanche dynamics and trading friction effects on stock market returns
- Iori
- 1999
(Show Context)
Citation Context ...w decay of the autocorrelation time of the absolute returns. The true dynamics behind the stylized facts is not fully understood. There are some attempts to make physical models based on spin dynamics=-=[2]-=--[9] and they are able to capture some of the stylized facts. In finance volatility is an important quantity to measure risk. To forecast volatility, various empirical models to mimic the properties o... |

6 |
A simple model of price formation
- Sznajd-Weron, Weron
- 2002
(Show Context)
Citation Context ...d ω are the parameters to be estimated. Let θ = (ω, α, β) be a parameter set of the GARCH model. The likelihood function of the GARCH model is written as L(y|θ) = Π n i=1 1 √ 2πσ2 t exp (− y2 t σ2 ). =-=(5)-=- t This function plays a central role in ML estimations and also for the Bayesian inference.An Adaptive Markov Chain Monte Carlo Method for GARCH Model 3 3 Bayesian inference In this section we brief... |

6 | Dynamics of price and trading volume in a spin model of stock markets with heterogeneous agents
- Kaizoji, Bornholdt, et al.
(Show Context)
Citation Context ...robability distribution of θ when the data y are given. With this π(θ|y) values of the parameters are inferred as the expectation values of θ given by 〈θ〉 = 1 ∫ θπ(θ|y)dθ, (7) Z where ∫ Z = π(θ|y)dθ. =-=(8)-=- Z is a normalization constant irrelevant to MCMC estimations. 3.1 MCMC In general the integral of eq.(7) can not be performed analytically. The MCMC technique gives a method to estimate eq.(7) numeri... |

6 | Comparison of MCMC methods for estimating GARCH models, COE discussion paper series
- Asai, Watanabe
- 2004
(Show Context)
Citation Context ...esian inference procedure is performed by MCMC methods. There is no unique way to implement MCMC methods. So far a variety of methods to MCMC procedure have been developed[14]-[19]. In a recent survey=-=[18]-=- it is shown that Acceptance-Rejection/Metropolis-Hastings (AR/MH) algorithm works better than other algorithms. In the AR/MH algorithm the proposal density is assumed to be a multivariate Student’s t... |

6 |
Bayesian Estimation of GARCH model by Hybrid Monte Carlo
- Takaishi
(Show Context)
Citation Context ...ference. Usually the Bayesian inference procedure is performed by MCMC methods. There is no unique way to implement MCMC methods. So far a variety of methods to MCMC procedure have been developed[14]-=-=[19]-=-. In a recent survey[18] it is shown that Acceptance-Rejection/Metropolis-Hastings (AR/MH) algorithm works better than other algorithms. In the AR/MH algorithm the proposal density is assumed to be a ... |

5 |
Bayesian analysis of GARCH option pricing models
- Mitsui, Watanabe
- 2003
(Show Context)
Citation Context ...od can be high. The posterior density of GARCH parameters often resembles to a Gaussian-like shape. Thus one may choose a density similar to a Gaussian distribution as the proposal density. Following =-=[17, 18]-=-, in order to cover the tails of the posterior density we use a (p-dimensional) multivariate Student’s t-distribution given by g(θ) = Γ((ν + p)/2)/Γ(ν/2) detΣ 1/2 (νπ) p/2 where θ and M are column vec... |

3 |
Bornholdt’s spin model of a market dynamics in high dimensions
- Yamano
- 2002
(Show Context)
Citation Context ...roperties of the financial time series volatility. Thus in this study we use GARCH(1,1) model for our testbed. The volatility σ2 t of the GARCH model is now written as σ 2 t = ω + αy2 t−1 + βσ2 t−1 , =-=(4)-=- where α, β and ω are the parameters to be estimated. Let θ = (ω, α, β) be a parameter set of the GARCH model. The likelihood function of the GARCH model is written as L(y|θ) = Π n i=1 1 √ 2πσ2 t exp ... |

2 |
A simple model for stocks markets
- Sanchez
- 2002
(Show Context)
Citation Context ...nce which estimates the GARCH parameters numerically by using the MCMC method. From the Bayes’ theorem the posterior density π(θ|y) with data y = (y1, y2, . . . , yn) is given by π(θ|y) ∝ L(y|θ)π(θ), =-=(6)-=- where L(y|θ) is the likelihood function. π(θ) is the prior density for θ. The functional form of π(θ) is not known a priori. Here we assume that the prior density π(θ) is constant. π(θ|y) gives a pro... |

2 | Simulations of financial markets in a Potts-like model
- Takaishi
- 2005
(Show Context)
Citation Context ...cay of the autocorrelation time of the absolute returns. The true dynamics behind the stylized facts is not fully understood. There are some attempts to make physical models based on spin dynamics[2]-=-=[9]-=- and they are able to capture some of the stylized facts. In finance volatility is an important quantity to measure risk. To forecast volatility, various empirical models to mimic the properties of th... |

1 |
Expectation bubbles in a spin model of markets
- Bornholdt
- 2001
(Show Context)
Citation Context ... 2 t = ω + yt = σtǫt, (1) q∑ i=1 αiy 2 t−i + p∑ i=1 βiσ 2 t−i , (2) where ω > 0, αi > 0 and βi > 0 to ensure a positive volatility. Furthermore the stationary condition given by q∑ αi + i=1 p∑ βi < 1 =-=(3)-=- i=1 is also required. ǫt is an independent normal error ∼ N(0, 1). In many empirical studies it is shown that (p = q = 1) GARCH model well captures the properties of the financial time series volatil... |

1 |
A spin model of market dynamics with random nearest neighbor coupling
- Yamano
- 2002
(Show Context)
Citation Context ...s constant. π(θ|y) gives a probability distribution of θ when the data y are given. With this π(θ|y) values of the parameters are inferred as the expectation values of θ given by 〈θ〉 = 1 ∫ θπ(θ|y)dθ, =-=(7)-=- Z where ∫ Z = π(θ|y)dθ. (8) Z is a normalization constant irrelevant to MCMC estimations. 3.1 MCMC In general the integral of eq.(7) can not be performed analytically. The MCMC technique gives a meth... |