Estimating the Student's t degrees of freedom (dof) with a Gamma Prior over the dof

Description

BayesGA samples from the posterior distribution of the degrees of freedom (dof) with Gamma prior endowed upon the dof, using a random walk Metropolis (RMW) algorithm.

Usage

BayesGA(y, ini.nu = 1, S = 1000, delta = 0.001, a = 1, b = 0.1)

Arguments

Value

A vector of posterior sample estimates

References

Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462

Fernández, C., Steel, M. F. (1998). "On Bayesian modeling of fat tails and skewness", Journal of the American Statistical Association, doi:10.1080/01621459.1998.10474117

Juárez, M. A., Steel, M. F. (2010). "Model-Based Clustering of Non-Gaussian Panel Data Based on Skew-t Distributions", Journal of Business and Economic Statistics, doi:10.1198/jbes.2009.07145

Examples

# data from Student's t-distribution with dof = 0.1
y = rt(n = 100, df = 0.1)

# running the random walk Metropolis algorithm with default settings
nu = BayesGA(y)
# reporting the posterior mean estimate of the dof
mean(nu)

# application to log-return (daily index values) of United States (S&P500)
data(index_return)
# log-returns of United States
index_return_US <- dplyr::filter(index_return, Country == "United States")
y = index_return_US$log_return_rate

# running the random walk Metropolis algorithm with default settings
nu = BayesGA(y)
# reporting the posterior mean estimate of the dof from the log-return data of US
mean(nu)

Estimating the Student's t degrees of freedom (dof) with a Jeffreys Prior over the dof

Description

BayesJeffreys samples from the posterior distribution of the degrees of freedom (dof) with Jeffreys prior endowed upon the dof, using a random walk Metropolis (RMW) algorithm and Metropolis-adjusted Langevin algorithm (MALA).

Usage

BayesJeffreys(
  y,
  ini.nu = 1,
  S = 1000,
  delta = 0.001,
  sampling.alg = c("MH", "MALA")
)

Arguments

Value

A vector of posterior sample estimates

References

Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462

Gustafson, P. (1998). "A guided walk Metropolis algorithm", Statistics and Computing, doi:10.1023/A:1008880707168

Examples

# data from Student's t-distribution with dof = 0.1
y = rt(n = 100, df = 0.1)

# running the random walk Metropolis algorithm with default settings
nu1 = BayesJeffreys(y, sampling.alg = "MH")
# reporting the posterior mean estimate of the dof
mean(nu1)

# running MALA with default settings
nu2 = BayesJeffreys(y, sampling.alg = "MALA")
# reporting the posterior mean estimate of the dof
mean(nu2)

# application to log-return (daily index values) of United States (S&P500)
data(index_return)
# log-returns of United States
index_return_US <- dplyr::filter(index_return, Country == "United States")
y = index_return_US$log_return_rate

# running the random walk Metropolis algorithm with default settings
nu1 = BayesJeffreys(y, sampling.alg = "MH")
# reporting the posterior mean estimate of the dof from the log-return data of US
mean(nu1)

# running MALA with default settings
nu2 = BayesJeffreys(y, sampling.alg = "MALA")
# reporting the posterior mean estimate of the dof from the log-return data of US
mean(nu2)

Estimating the Student's t degrees of freedom (dof) with a Log-normal Prior over the dof

Description

BayesLNP samples from the posterior distribution of the degrees of freedom (dof) with Log-normal prior endowed upon the dof, using an Elliptical Slice Sampler (ESS).

Usage

BayesLNP(y, ini.nu = 1, S = 1000, mu = 1, sigma.sq = 1)

Arguments

Value

A vector of posterior sample estimates

References

Lee, S. Y. (2022). "The Use of a Log-Normal Prior for the Student t-Distribution", Axioms, doi:10.3390/axioms11090462

Murray, I., Prescott Adams, R., MacKay, D. J. (2010). "Elliptical slice sampling", Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics

Examples

# data from Student's t-distribution with dof = 0.1
y = rt(n = 100, df = 0.1)

# running the Elliptical Slice Sampler (ESS) with default settings
nu = BayesLNP(y)
# reporting the posterior mean estimate of the dof
mean(nu)

# application to log-return (daily index values) of United States (S&P500)
data(index_return)
# log-returns of United States
index_return_US <- dplyr::filter(index_return, Country == "United States")
y = index_return_US$log_return_rate

# running the Elliptical Slice Sampler (ESS) with default settings
nu = BayesLNP(y)
# reporting the posterior mean estimate of the dof from the log-return data of US
mean(nu)

Stock Market Index Return Data

Description

The stock market returns are recorded for four countries viz., United States (S&P500), Japan (NIKKEI225), Germany (DAX Index), and South Korea (KOSPI). Specifically log return rates (as computed in Section 5 of doi:10.3390/axioms11090462) are recorded for 5 months in the year 2009 for all the four countries, where these rates are considered to be Student's t-distributed and used for the purpose of estimating the corresponding degrees of freedom using a Bayesian model-based framework, developed in doi:10.3390/axioms11090462.

Usage

index_return

Format

A data frame with 4 columns:

Country

name of the country to which the log return rate corresponds to: 'United States', 'Japan', 'Germany', and 'South Korea'

log_return_rate

value of the log return rate

time_index

an index for the log return rate observations

date

the date on which the log return rate was recorded

Source

(Lee, 2022), doi:10.3390/axioms11090462.