garchFit: Univariate or multivariate GARCH time series fitting

Description

Estimates the parameters of a univariate ARMA-GARCH/APARCH process, or --- experimentally --- of a multivariate GO-GARCH process model. The latter uses an algorithm based on fastICA(), inspired from Bernhard Pfaff's package gogarch.

Usage

garchFit(formula = ~ garch(1, 1), data,
	init.rec = c("mci", "uev"),
	delta = 2, skew = 1, shape = 4,
	cond.dist = c("norm", "snorm", "ged", "sged",
                      "std", "sstd", "snig", "QMLE"),
	include.mean = TRUE, include.delta = NULL, include.skew = NULL,
        include.shape = NULL,
        leverage = NULL, trace = TRUE,
	
	algorithm = c("nlminb", "lbfgsb", "nlminb+nm", "lbfgsb+nm"),
	hessian = c("ropt", "rcd"),
        control = list(),
        title = NULL, description = NULL, ...)
garchKappa(cond.dist = c("norm", "ged", "std", "snorm", "sged", "sstd", "snig"),
           gamma = 0, delta = 2, skew = NA, shape = NA)
.gogarchFit(formula = ~garch(1, 1), data, init.rec = c("mci", "uev"),
            delta = 2, skew = 1, shape = 4,
            cond.dist = c("norm", "snorm", "ged", "sged",
                          "std", "sstd", "snig", "QMLE"),
            include.mean = TRUE, include.delta = NULL, include.skew = NULL,
            include.shape = NULL,
            leverage = NULL, trace = TRUE,
            algorithm = c("nlminb", "lbfgsb", "nlminb+nm", "lbfgsb+nm"),
            hessian = c("ropt", "rcd"),
            control = list(),
            title = NULL, description = NULL, ...)

Value

for garchFit, an S4 object of class "fGARCH". Slot @fit contains the results from the optimization.

for .gogarchFit(): Similar definition for GO-GARCH modeling. Here, data must be multivariate. Still

“preliminary”, mostly undocumented, and untested(!). At least mentioned here...

Arguments

algorithm: a string parameter that determines the algorithm used for maximum likelihood estimation.
cond.dist: a character string naming the desired conditional distribution. Valid values are "dnorm", "dged", "dstd", "dsnorm", "dsged", "dsstd" and "QMLE". The default value is the normal distribution. See Details for more information.
control: control parameters, the same as used for the functions from nlminb, and 'bfgs' and 'Nelder-Mead' from optim.
data: an optional timeSeries or data frame object containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which armaFit is called. If data is an univariate series, then the series is converted into a numeric vector and the name of the response in the formula will be neglected.
delta: a numeric value, the exponent delta of the variance recursion. By default, this value will be fixed, otherwise the exponent will be estimated together with the other model parameters if include.delta=FALSE.
description: optional character string with a brief description.
formula: formula object describing the mean and variance equation of the ARMA-GARCH/APARCH model. A pure GARCH(1,1) model is selected e.g., for formula = ~garch(1,1). To specify an ARMA(2,1)-APARCH(1,1) process, use ~ arma(2,1) + aparch(1,1).
gamma: APARCH leverage parameter entering into the formula for calculating the expectation value.
hessian: a string denoting how the Hessian matrix should be evaluated, either hessian ="rcd", or "ropt". The default, "rcd" is a central difference approximation implemented in R and "ropt" uses the internal R function optimhess.
include.delta: a logical determining if the parameter for the recursion equation delta will be estimated or not. If false, the shape parameter will be kept fixed during the process of parameter optimization.
include.mean: this flag determines if the parameter for the mean will be estimated or not. If include.mean=TRUE this will be the case, otherwise the parameter will be kept fixed durcing the process of parameter optimization.
include.shape: a logical flag which determines if the parameter for the shape of the conditional distribution will be estimated or not. If include.shape=FALSE then the shape parameter will be kept fixed during the process of parameter optimization.
include.skew: a logical flag which determines if the parameter for the skewness of the conditional distribution will be estimated or not. If include.skew=FALSE then the skewness parameter will be kept fixed during the process of parameter optimization.
init.rec: a character string indicating the method how to initialize the mean and varaince recursion relation.
leverage: a logical flag for APARCH models. Should the model be leveraged? By default leverage=TRUE.
shape: a numeric value, the shape parameter of the conditional distribution.
skew: a numeric value, the skewness parameter of the conditional distribution.
title: a character string which allows for a project title.
trace: a logical flag. Should the optimization process of fitting the model parameters be printed? By default trace=TRUE.
...: additional arguments to be passed.

Author

Diethelm Wuertz for the Rmetrics R-port,
R Core Team for the 'optim' R-port,
Douglas Bates and Deepayan Sarkar for the 'nlminb' R-port,
Bell-Labs for the underlying PORT Library,
Ladislav Luksan for the underlying Fortran SQP Routine,
Zhu, Byrd, Lu-Chen and Nocedal for the underlying L-BFGS-B Routine.

Martin Maechler for cleaning up; mentioning .gogarchFit().

Details

"QMLE" stands for Quasi-Maximum Likelihood Estimation, which assumes normal distribution and uses robust standard errors for inference. Bollerslev and Wooldridge (1992) proved that if the mean and the volatility equations are correctly specified, the QML estimates are consistent and asymptotically normally distributed. However, the estimates are not efficient and “the efficiency loss can be marked under asymmetric ... distributions” (Bollerslev and Wooldridge (1992), p. 166). The robust variance-covariance matrix of the estimates equals the (Eicker-White) sandwich estimator, i.e.

$$V = H^{-1} G^{\prime} G H^{-1},$$

where $V$ denotes the variance-covariance matrix, $H$ stands for the Hessian and $G$ represents the matrix of contributions to the gradient, the elements of which are defined as

$$G_{t,i} = \frac{\partial l_{t}}{\partial \zeta_{i}},$$

where $t_{t}$ is the log likelihood of the t-th observation and $\zeta_{i}$ is the i-th estimated parameter. See sections 10.3 and 10.4 in Davidson and MacKinnon (2004) for a more detailed description of the robust variance-covariance matrix.

References

ATT (1984); PORT Library Documentation, http://netlib.bell-labs.com/netlib/port/.

Bera A.K., Higgins M.L. (1993); ARCH Models: Properties, Estimation and Testing, J. Economic Surveys 7, 305--362.

Bollerslev T. (1986); Generalized Autoregressive Conditional Heteroscedasticity, Journal of Econometrics 31, 307--327.

Bollerslev T., Wooldridge J.M. (1992); Quasi-Maximum Likelihood Estimation and Inference in Dynamic Models with Time-Varying Covariance, Econometric Reviews 11, 143--172.

Byrd R.H., Lu P., Nocedal J., Zhu C. (1995); A Limited Memory Algorithm for Bound Constrained Optimization, SIAM Journal of Scientific Computing 16, 1190--1208.

Davidson R., MacKinnon J.G. (2004); Econometric Theory and Methods, Oxford University Press, New York.

Engle R.F. (1982); Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation, Econometrica 50, 987--1008.

Nash J.C. (1990); Compact Numerical Methods for Computers, Linear Algebra and Function Minimisation, Adam Hilger.

Nelder J.A., Mead R. (1965); A Simplex Algorithm for Function Minimization, Computer Journal 7, 308--313.

Nocedal J., Wright S.J. (1999); Numerical Optimization, Springer, New York.

Examples

Run this code

## UNIVARIATE TIME SERIES INPUT:
   # In the univariate case the lhs formula has not to be specified ...

   # A numeric Vector from default GARCH(1,1) - fix the seed:
   N = 200
   x.vec = as.vector(garchSim(garchSpec(rseed = 1985), n = N)[,1])
   garchFit(~ garch(1,1), data = x.vec, trace = FALSE)

   # An univariate timeSeries object with dummy dates:
   stopifnot(require("timeSeries"))
   x.timeSeries = dummyDailySeries(matrix(x.vec), units = "GARCH11")
   garchFit(~ garch(1,1), data = x.timeSeries, trace = FALSE)

if (FALSE) {
   # An univariate zoo object:
   require("zoo")
   x.zoo = zoo(as.vector(x.vec), order.by = as.Date(rownames(x.timeSeries)))
   garchFit(~ garch(1,1), data = x.zoo, trace = FALSE)
}

   # An univariate "ts" object:
   x.ts = as.ts(x.vec)
   garchFit(~ garch(1,1), data = x.ts, trace = FALSE)

## MULTIVARIATE TIME SERIES INPUT:
   # For multivariate data inputs the lhs formula must be specified ...

   # A numeric matrix binded with dummy random normal variates:
   X.mat = cbind(GARCH11 = x.vec, R = rnorm(N))
   garchFit(GARCH11 ~ garch(1,1), data = X.mat)

   # A multivariate timeSeries object with dummy dates:
   X.timeSeries = dummyDailySeries(X.mat, units = c("GARCH11", "R"))
   garchFit(GARCH11 ~ garch(1,1), data = X.timeSeries)

if (FALSE) {
   # A multivariate zoo object:
   X.zoo = zoo(X.mat, order.by = as.Date(rownames(x.timeSeries)))
   garchFit(GARCH11 ~ garch(1,1), data = X.zoo)
}

   # A multivariate "mts" object:
   X.mts = as.ts(X.mat)
   garchFit(GARCH11 ~ garch(1,1), data = X.mts)

## MODELING THE PERCENTUAL SPI/SBI SPREAD FROM LPP BENCHMARK:

   stopifnot(require("timeSeries"))
   X.timeSeries = as.timeSeries(data(LPP2005REC))
   X.mat = as.matrix(X.timeSeries)
   if (FALSE) X.zoo = zoo(X.mat, order.by = as.Date(rownames(X.mat)))
   X.mts = ts(X.mat)
   garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.timeSeries)
   # The remaining are not yet supported ...
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.mat)
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.zoo)
   # garchFit(100*(SPI - SBI) ~ garch(1,1), data = X.mts)

## MODELING HIGH/LOW RETURN SPREADS FROM MSFT PRICE SERIES:

   X.timeSeries = MSFT
   garchFit(Open ~ garch(1,1), data = returns(X.timeSeries))
   garchFit(100*(High-Low) ~ garch(1,1), data = returns(X.timeSeries))

## GO-GARCH Modelling  (not yet!!) % FIXME

  ## data(DowJones30, package="fEcofin") # no longer exists
  ## X = returns(as.timeSeries(DowJones30)); head(X)
  ## N = 5; ans = .gogarchFit(data = X[, 1:N], trace = FALSE); ans
  ## ans@h.t

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