GAS (version 0.3.4.1)

UniGASFit: Estimate univariate GAS models

Description

Estimate univariate GAS models by Maximum Likelihood.

Usage

UniGASFit(GASSpec, data, fn.optimizer = fn.optim, Compute.SE = TRUE)

Value

An object of the class uGASFit

Arguments

GASSpec

An object of the class uGASSpec created using the function UniGASSpec

.

data

numeric vector of length Tx1 containing the time series of observations. It can also be an object of the class ts, xts or zoo.

fn.optimizer

function. This is a generic optimization function that can be provided by the user. By default fn.optimizer = fn.optim where fn.optim is a wrapper to the optim function. See Details for user defined optimization routines.

Compute.SE

logical. Should asymptotic Standard Errors be computed? By default Compute.SE = TRUE

Author

Leopoldo Catania

Details

Maximum Likelihood estimation of GAS models is an on-going research topic. General results are reported by Blasques et al. (2014b), Blasques et al. (2014a) and Harvey (2013), while results for specific models have been derived by Blasques et al. (2014c) and Andres (2014).

Starting values for the optimizer are chosen in the following way: (i) estimate the static version of the model (i.e., with A = 0 and B = 0) and set the initial value of the intercept parameter accordingly, and (ii) perform a grid search for the coefficients contained in A and B. Further technical details are presented in Section 3.2 of Ardia et. al. (2016a).

The user is free to employ his/her own optimization routine via the fn.optimizer argument. fn.optimizer accepts a function object. The user provided optimizer has to satisfy strict requirements. The arguments of the fn.optimizer are : i) par0 a vector of starting values, ii) data the data provided, iii) GASSpec an object of the class uGASSpec, and iv) FUN the likelihood function. The output of fn.optimizer has to be an object of the class list with four named elements: i) pars: a numeric vector where the estimated parameters are stored, ii) value: a numeric containing the value of the negative log likelihood evaluated at its minimum, iii) hessian, a numeric matrix containing the Hessian matrix evaluated at the minimum of the negative log likelihood, this is used for inferential purposes, and iv) convergence a numeric variable reporting information about the convergence of the optimization. This quantity is printed by the show() and summary() methods. convergence = 0 has to indicate successful completion.

The user is allowed to not include the last two elements of the output of the fn.optimizer function, that is, the values hessian = NULL and convergence = NULL are admissible. In the case of hessian = NULL, the Hessian matrix is evaluated numerically using the hessian function in the numDeriv package of Gilbert and Varadhan (2016). If the provided hessian is not positive definite, a try with the hessian evaluation used by the BFGS quasi-Newton implementation in the function optim is made.

By default, the optim optimizer with method = "BFGS" is employed.

References

Ardia D, Boudt K and Catania L (2016a). "Generalized Autoregressive Score Models in R: The GAS Package." https://www.ssrn.com/abstract=2825380.

Blasques F, Koopman SJ, Lucas A (2014a). "Maximum Likelihood Estimation for Correctly Specified Generalized Autoregressive Score Models: Feedback Effects, Contraction Conditions and Asymptotic Properties." techreport TI 14-074/III, Tinbergen Institute. https://www.tinbergen.nl/discussionpaper/?paper=2332.

Blasques F, Koopman SJ, Lucas A (2014b). "Maximum Likelihood Estimation for Generalized Autoregressive Score Models." techreport TI 2014-029/III, Tinbergen Institute. https://www.tinbergen.nl/discussionpaper/?paper=2286.

Blasques F, Koopman SJ, Lucas A, Schaumburg J (2014c). "Spillover Dynamics for Systemic Risk Measurement using Spatial Financial Time Series Models." techreport TI 2014-103/III, Tinbergen Institute. https://www.tinbergen.nl/discussionpaper/?paper=2369.

Creal D, Koopman SJ, Lucas A (2013). "Generalized Autoregressive Score Models with Applications." Journal of Applied Econometrics, 28(5), 777-795. tools:::Rd_expr_doi("10.1002/jae.1279").

Gilbert P, Varadhan R (2016). numDeriv: Accurate Numerical Derivatives. R package 2016.8-1, https://CRAN.R-project.org/package=numDeriv.

Ghalanos A, Theussl S (2016). "Rsolnp: General Non-Linear Optimization using Augmented Lagrange Multiplier Method." https://cran.r-project.org/package=Rsolnp.

Harvey AC (2013). Dynamic Models for Volatility and Heavy Tails: With Applications to Financial and Economic Time Series. Cambridge University Press.

Ye Y (1988). Interior Algorithms for Linear, Quadratic, and Linearly Constrained Convex Programming. Ph.D. thesis, Stanford University.

Examples

if (FALSE) {
# Specify an univariate GAS model with Student-t
# conditional distribution and time-varying scale.
library("GAS")

data("sp500ret")

GASSpec = UniGASSpec(Dist = "std", ScalingType = "Identity",
                     GASPar = list(location = FALSE, scale = TRUE,
                                   shape = FALSE))

Fit = UniGASFit(GASSpec, sp500ret)

Fit

# Estimate the model with a different optimizer.
# Assume we want to use the Nelder and Mead optimization provided by
# the optim() function, we create
# the wrapper fn.NM.optim in this way

fn.NM.optim <- function(par0, data, GASSpec, FUN) {

  optimizer = optim(par0, FUN, data = data, GASSpec = GASSpec, method = "Nelder-Mead",
                    control = list(trace = 0), hessian = TRUE)

  out = list(pars = optimizer$par,
             value = optimizer$value,
             hessian = optimizer$hessian,
             convergence = optimizer$convergence)

  return(out)

}

Fit.NM.optim = UniGASFit(GASSpec, sp500ret, fn.optimizer = fn.NM.optim )

Fit.NM.optim


# Estimate time-varying Negative Binomial distribution for the Goals dataset.
# Let's use the gosolnp() optimizer for the time-varying model estimation and
# the solnp() optimizer for estimation of the static model for the choice of
# the starting values. The logical is(GASSpec, "list") is TRUE when the function
# is evaluated for the choice of starting values, and FALSE when the function
# is evaluated for the time-varying model.
# We can also make use of parallel computation calling a cluster object defined
# in the Global environment.

library("Rsolnp")
fn.gosolnp <- function(par0, data, GASSpec, FUN) {

  if (is(GASSpec, "list")) {

    optimiser = suppressWarnings(solnp(par0, FUN, data = data,
                                       GASSpec = GASSpec,
                                       control = list(trace = 0)))

  } else {

    cluster = get("cluster", envir = globalenv())

    optimiser = suppressWarnings(gosolnp(
      pars = NULL,
      fun = FUN, data = data, cluster = cluster,
      GASSpec = GASSpec,
      n.sim = 100000,
      n.restarts = 10,
      LB = c(-5, -2, -2, -2),
      UB = c(5, 8, 3.0, 5.0))

    )
  }

  out = list(pars = optimiser$pars,
             value = tail(optimiser$values, 1),
             hessian = optimiser$hessian,
             convergence = optimiser$convergence)

  return(out)

}

data("Goals")

library("parallel")

cluster = makeCluster(2)

GASSpec = UniGASSpec(Dist = "negbin", ScalingType = "Inv",
                     GASPar = list(location = TRUE, scale = FALSE))


vY = na.omit(Goals[, 1])

Fit = UniGASFit(GASSpec, vY, fn.optimizer = fn.gosolnp)

Fit

stopCluster(cluster)
rm("cluster")

}