bayesGARCH {bayesGARCH}

Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations

Description

Performs the Bayesian estimation of the GARCH(1,1) model with Student-t innovations.

Usage

  bayesGARCH(y, mu.alpha = c(0,0), Sigma.alpha = 1000 * diag(1,2), 
             mu.beta = 0, Sigma.beta = 1000,
             lambda = 0.01, delta = 2, control = list())

Arguments

y

vector of observations of size T. NA values are not allowed.

mu.alpha

hyper-parameter \mu_\alpha (prior mean) for the truncated Normal prior on parameter \alpha := (\alpha_0 \ \alpha_1)'. Default: a 2 \times 1 vector of zeros.

Sigma.alpha

hyper-parameter \Sigma_\alpha (prior covariance matrix) for the truncated Normal prior on parameter \alpha. Default: a 2 \times 2 diagonal matrix whose variances are set to 1'000, i.e., a diffuse prior. Note that the matrix must be symmetric positive definite.

mu.beta

hyper-parameter \mu_\beta (prior mean) for the truncated Normal prior on parameter \beta. Default: zero.

Sigma.beta

hyper-parameter \Sigma_\beta > 0 (prior variance) for the truncated Normal prior on parameter \beta. Default: 1'000, i.e., a diffuse prior.

lambda

hyper-parameter \lambda > 0 for the translated Exponential distribution on parameter \nu. Default: 0.01.

delta

hyper-parameter \delta \ge 2 for the translated Exponential distribution on parameter \nu. Default: 2 (to ensure the existence of the conditional variance).

control

list of control parameters (See *Details*).

Details

The function bayesGARCH performs the Bayesian estimation of the GARCH(1,1) model with Student-t innovations. The underlying algorithm is based on Nakatsuma (1998, 2000) for generating the parameters of the GARCH(1,1) scedastic function \alpha := (\alpha_0 \ \alpha_1)' and \beta and on Geweke (1993) and Deschamps (2006) for the generating the degrees of freedom parameter \nu. Further details and examples can be found in Ardia (2008) and Ardia and Hoogerheide (2010). Finally, we refer to Ardia (2009) for an extension of the algorithm to Markov-switching GARCH models.

The control argument is a list that can supply any of the following components:

n.chain

number of MCMC chain(s) to be generated. Default: n.chain=1.

l.chain

length of each MCMC chain. Default: l.chain=10000.

start.val

vector of starting values of chain(s). Default: start.val=c(0.01,0.1,0.7,20). A matrix of size n \times 4 containing starting values in rows can also be provided. This will generate n chains starting at the different row values.

addPriorConditions

function which allows the user to add constraints on the model parameters. Default: NULL, i.e. not additional constraints are imposed (see below).

refresh

frequency of reports. Default: refresh=10 iterations.

digits

number of printed digits in the reports. Default: digits=4.

Value

A list of class mcmc.list (R package coda).

Note

By using bayesGARCH you agree to the following rules:

• You must cite Ardia and Hoogerheide (2010) in working papers and published papers that use bayesGARCH. Use citation("bayesGARCH").

• You must place the following URL in a footnote to help others find bayesGARCH: https://CRAN.R-project.org/package=bayesGARCH.

• You assume all risk for the use of bayesGARCH.

The GARCH(1,1) model with Student-t innovations may be written as follows:

y_t = \epsilon_t (\varrho h_t)^{1/2}

for t=1,\ldots,T, where the conditional variance equation is defined as:

h_t := \alpha_0 + \alpha_1 y_{t-1}^2 + \beta h_{t-1}

where \alpha_0 > 0, \alpha_1 \ge 0, \beta \ge 0 to ensure a positive conditional variance. We set the initial variance to h_0 := 0 for convenience. The parameter \varrho := (\nu-2)/\nu is a scaling factor which ensures the conditional variance of y_t to be h_t. Finally, \epsilon_t follows a Student-t distribution with \nu degrees of freedom.

The prior distributions on \alpha is a bivariate truncated Normal distribution:

p(\alpha) \propto N_2(\alpha \mid \mu_\alpha, \Sigma_\alpha) I_{[\alpha > 0]}

where \mu_\alpha is the prior mean vector, \Sigma_\alpha is the prior covariance matrix and I_{[\bullet]} is the indicator function.

The prior distribution on \beta is a univariate truncated Normal distribution:

p(\beta) \propto N(\beta \mid \mu_\beta, \Sigma_\beta) I_{[\beta > 0]}

where \mu_\beta is the prior mean and \Sigma_\beta is the prior variance.

The prior distribution on \nu is a translated Exponential distribution:

p(\nu) = \lambda \exp [ - \lambda (\nu - \delta) ] I_{[\nu > \delta]}

where \lambda > 0 and \delta \ge 2. The prior mean for \nu is \delta + 1/\lambda.

The joint prior on parameter \psi := (\alpha, \beta, \nu) is obtained by assuming prior independence:

p(\psi) = p(\alpha) p(\beta) p(\nu).

The default hyperparameters \mu_\alpha, \Sigma_\alpha, \mu_\beta, \Sigma_\beta and \lambda define a rather vague prior. The hyper-parameter \delta \geq 2 ensures the existence of the conditional variance. The kth conditional moment for \epsilon_t is guaranteed by setting \delta \geq k.

The Bayesian estimation of the GARCH(1,1) model with Normal innovations is obtained as a special case by setting lambda=100 and delta=500. In this case, the generated values for \nu are centered around 500 which ensure approximate Normality for the innovations.

The function addPriorConditions allows to add prior conditions on the model parameters \psi := (\alpha_0 \ \alpha_1 \ \beta \ \nu)'. The function must return TRUE if the constraint holds and FALSE otherwise.

By default, the function is:

    addPriorConditions <- function(psi)
    {
      TRUE
    }
  

and therefore does not add any other constraint than the positivity of the parameters which are obtained through the prior distribution for \psi.

You simply need to modify addPriorConditions in order to add constraints on the model parameters \psi. For instance, to impose the covariance-stationary conditions to hold, i.e. \alpha_1 + \beta < 1, just define the function addPriorConditions as follows:

    addPriorConditions <- function(psi)
    {
      psi[2] + psi[3] < 1
    }
  

Note that adding prior constraints on the model parameters can diminish the acceptance rate and therefore lead to a very inefficient sampler. This would however indicate that the condition is not supported by the data.

The estimation strategy implemented in bayesGARCH is fully automatic and does not require any tuning of the MCMC sampler. The generation of the Markov chains is however time consuming and estimating the model over several datasets on a daily basis can therefore take a significant amount of time. In this case, the algorithm can be easily parallelized, by running a single chain on several processors. Also, when the estimation is repeated over updated time series (i.e. time series with more recent observations), it is wise to start the algorithm using the posterior mean or median of the parameters obtained at the previous estimation step. The impact of the starting values (burn-in phase) is likely to be smaller and thus the convergence faster.

Finally, note that as any MH algorithm, the sampler can get stuck to a given value, so that the chain does not move anymore. However, the sampler uses Taylor-made candidate densities that are especially ‘constructed’ at each step, so it is almost impossible for this MCMC sampler to get stuck at a given value for many subsequent draws. In the unlikely case that such ill behavior would occur, one could scale the data (to have standard deviation 1), or run the algorithm with different initial values or a different random seed.

Author(s)

David Ardia [e-mail]

References

Ardia, D. (2009) Bayesian Estimation of a Markov-Switching Threshold Asymmetric GARCH Model with Student-t Innovations. Econometrics Journal 12(1), pp. 105-126. doi: 10.1111/j.1368-423X.2008.00253.x

Ardia, D., Hoogerheide, L.F. (2010) Bayesian Estimation of the GARCH(1,1) Model with Student-t Innovations. R Journal 2(2), pp.41-47. doi: 10.32614/RJ-2010-014

Ardia, D. (2008) Financial Risk Management with Bayesian Estimation of GARCH Models. Lecture Notes in Economics and Mathematical Systems 612. Springer-Verlag, Berlin, Germany. ISBN 978-3-540-78656-6, e-ISBN 978-3-540-78657-3, doi: 10.1007/978-3-540-78657-3

Deschamps, P.J. (2006) A Flexible Prior Distribution for Markov Switching Autoregressions with Student-t Errors. Journal of Econometrics 133, pp.153-190.

Geweke, J.F. (1993) Bayesian Treatment of the Independent Student-t Linear Model. Journal of Applied Econometrics 8, pp.19-40.

Nakatsuma, T. (2000) Bayesian Analysis of ARMA-GARCH Models: A Markov Chain Sampling Approach. Journal of Econometrics 95(1), pp.57-69.

Nakatsuma, T. (1998) A Markov-Chain Sampling Algorithm for GARCH Models. Studies in Nonlinear Dynamics and Econometrics 3(2), pp.107-117.

See Also

garchFit (R package fGarch) for the classical Maximum Likelihood estimation of GARCH models.

Examples

  ## !!! INCREASE THE NUMBER OF MCMC ITERATIONS !!!

  ## LOAD DATA
  data(dem2gbp)
  y <- dem2gbp[1:750]

  ## RUN THE SAMPLER (2 chains)
  MCMC <- bayesGARCH(y, control = list(n.chain = 2, l.chain = 200))

  ## MCMC ANALYSIS (using coda)
  plot(MCMC)
  
  ## FORM THE POSTERIOR SAMPLE
  smpl <- formSmpl(MCMC, l.bi = 50)

  ## POSTERIOR STATISTICS
  summary(smpl)
  smpl <- as.matrix(smpl)
  pairs(smpl)

  ## GARCH(1,1) WITH NORMAL INNOVATIONS
  MCMC <- bayesGARCH(y, lambda = 100, delta = 500,
                     control = list(n.chain = 2, l.chain = 200))

  ## GARCH(1,1) WITH NORMAL INNOVATIONS AND 
  ## WITH COVARIANCE STATIONARITY CONDITION
  addPriorConditions <- function(psi){psi[2] + psi[3] < 1}
  MCMC <- bayesGARCH(y, lambda = 100, delta = 500,
                     control = list(n.chain = 2, l.chain = 200, 
                     addPriorConditions = addPriorConditions))