Fit univariate and multivariate GARCH-type models
garchFit.RdEstimates 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,
<!-- %recursion = c("internal", "filter", "testing"), -->
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, ...)Arguments
- formula
formulaobject describing the mean and variance equation of the ARMA-GARCH/APARCH model. A pure GARCH(1,1) model is selected e.g., forformula = ~garch(1,1). To specify an ARMA(2,1)-APARCH(1,1) process, use~ arma(2,1) + aparch(1,1).- data
-
an optional
"timeSeries"or"data.frame"object containing the variables in the model. If not found indata, the variables are taken fromenvironment(formula), typically the environment from whicharmaFitis called. Ifdatais an univariate series, then the series is converted into a numeric vector and the name of the response in the formula will be neglected. - init.rec
a character string indicating the method how to initialize the mean and varaince recursion relation.
- delta
a numeric value, the exponent
deltaof the variance recursion. By default, this value will be fixed, otherwise the exponent will be estimated together with the other model parameters ifinclude.delta=FALSE.- skew
a numeric value, the skewness parameter of the conditional distribution.
- shape
a numeric value, the shape parameter of the conditional distribution.
- 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.- include.mean
this flag determines if the parameter for the mean will be estimated or not. If
include.mean=TRUEthis will be the case, otherwise the parameter will be kept fixed durcing the process of parameter optimization.- include.delta
a
logicaldetermining if the parameter for the recursion equationdeltawill be estimated or not. If false, 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=FALSEthen the skewness parameter will be kept fixed during 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=FALSEthen the shape parameter will be kept fixed during the process of parameter optimization.- leverage
a logical flag for APARCH models. Should the model be leveraged? By default
leverage=TRUE.- trace
a logical flag. Should the optimization process of fitting the model parameters be printed? By default
trace=TRUE.- algorithm
a string parameter that determines the algorithm used for maximum likelihood estimation.
- 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 functionoptimhess.- control
control parameters, the same as used for the functions from
nlminb, and 'bfgs' and 'Nelder-Mead' fromoptim.- gamma
APARCH leverage parameter entering into the formula for calculating the expectation value.
- title
a character string which allows for a project title.
- description
optional character string with a brief description.
- ...
additional arguments to be passed.
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.
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...
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.
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().
See also
garchSpec,
garchFitControl,
class "fGARCH"
Examples
## 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)
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = ~garch(1, 1), data = x.vec, trace = FALSE)
#>
#> Mean and Variance Equation:
#> data ~ garch(1, 1)
#> <environment: 0x569a0c24a8c0>
#> [data = x.vec]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:49 2024 by user: georgi
#>
# An univariate timeSeries object with dummy dates:
stopifnot(require("timeSeries"))
#> Loading required package: timeSeries
#> Loading required package: timeDate
#>
#> Attaching package: ‘timeSeries’
#> The following objects are masked from ‘package:graphics’:
#>
#> lines, points
x.timeSeries = dummyDailySeries(matrix(x.vec), units = "GARCH11")
garchFit(~ garch(1,1), data = x.timeSeries, trace = FALSE)
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = ~garch(1, 1), data = x.timeSeries, trace = FALSE)
#>
#> Mean and Variance Equation:
#> data ~ garch(1, 1)
#> <environment: 0x569a0cfabfb8>
#> [data = x.timeSeries]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:49 2024 by user: georgi
#>
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)
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = ~garch(1, 1), data = x.ts, trace = FALSE)
#>
#> Mean and Variance Equation:
#> data ~ garch(1, 1)
#> <environment: 0x569a0e5f7118>
#> [data = x.ts]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:49 2024 by user: georgi
#>
## 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)
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 200
#> Recursion Init: mci
#> Series Scale: 0.003308171
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -0.05516776 0.05516776 -0.005516776 TRUE
#> omega 0.00000100 100.00000000 0.100000000 TRUE
#> alpha1 0.00000001 0.99999999 0.100000000 TRUE
#> gamma1 -0.99999999 0.99999999 0.100000000 FALSE
#> beta1 0.00000001 0.99999999 0.800000000 TRUE
#> delta 0.00000000 2.00000000 2.000000000 FALSE
#> skew 0.10000000 10.00000000 1.000000000 FALSE
#> shape 1.00000000 10.00000000 4.000000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 280.37902: -0.00551678 0.100000 0.100000 0.800000
#> 1: 280.37507: -0.00551675 0.101085 0.0997614 0.800840
#> 2: 280.37172: -0.00551671 0.100985 0.0984074 0.800529
#> 3: 280.36889: -0.00551668 0.102056 0.0981374 0.801378
#> 4: 280.36553: -0.00551664 0.101972 0.0967732 0.801110
#> 5: 280.36291: -0.00551659 0.102982 0.0962621 0.801921
#> 6: 280.36028: -0.00551652 0.102987 0.0948733 0.801815
#> 7: 280.35812: -0.00551642 0.103857 0.0941985 0.802668
#> 8: 280.35640: -0.00551626 0.103796 0.0928284 0.802909
#> 9: 280.35510: -0.00551594 0.104079 0.0921544 0.804093
#> 10: 280.35429: -0.00551543 0.103412 0.0912751 0.804938
#> 11: 280.35380: -0.00551470 0.102925 0.0909305 0.806189
#> 12: 280.35345: -0.00551337 0.102118 0.0903861 0.807156
#> 13: 280.35308: -0.00550880 0.101958 0.0898304 0.808116
#> 14: 280.35296: -0.00550326 0.101305 0.0896346 0.808798
#> 15: 280.35287: -0.00549602 0.101693 0.0894169 0.808659
#> 16: 280.35280: -0.00548956 0.101338 0.0890255 0.809195
#> 17: 280.35270: -0.00548306 0.100990 0.0889425 0.809846
#> 18: 280.35261: -0.00546712 0.100295 0.0886753 0.810712
#> 19: 280.35191: -0.00526092 0.100384 0.0880880 0.811016
#> 20: 280.34947: -0.00443606 0.0998012 0.0868493 0.812628
#> 21: 280.34213: -0.00121747 0.0993041 0.0846804 0.814752
#> 22: 280.33288: 0.00415509 0.0983925 0.0835405 0.816652
#> 23: 280.32556: 0.00933509 0.0984673 0.0851411 0.815292
#> 24: 280.32295: 0.0112564 0.0984683 0.0875808 0.813418
#> 25: 280.32262: 0.0109708 0.0988384 0.0887005 0.812137
#> 26: 280.32261: 0.0107322 0.0988356 0.0888526 0.812038
#> 27: 280.32261: 0.0107066 0.0988719 0.0888596 0.811988
#> 28: 280.32261: 0.0107061 0.0988616 0.0888549 0.812004
#>
#> Final Estimate of the Negative LLH:
#> LLH: -861.9494 norm LLH: -4.309747
#> mu omega alpha1 beta1
#> 3.541769e-05 1.081941e-06 8.885493e-02 8.120038e-01
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -2.110947e+07 -4.842148e+08 2.852385e+03 -1.458198e+03
#> omega -4.842148e+08 -2.671144e+13 -2.328641e+08 -2.747076e+08
#> alpha1 2.852385e+03 -2.328641e+08 -2.618403e+03 -2.563703e+03
#> beta1 -1.458198e+03 -2.747076e+08 -2.563703e+03 -2.938561e+03
#> attr(,"time")
#> Time difference of 0.004027128 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.02163911 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = GARCH11 ~ garch(1, 1), data = X.mat)
#>
#> Mean and Variance Equation:
#> GARCH11 ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = X.mat]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:49 2024 by user: georgi
#>
# A multivariate timeSeries object with dummy dates:
X.timeSeries = dummyDailySeries(X.mat, units = c("GARCH11", "R"))
garchFit(GARCH11 ~ garch(1,1), data = X.timeSeries)
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 200
#> Recursion Init: mci
#> Series Scale: 0.003308171
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -0.05516776 0.05516776 -0.005516776 TRUE
#> omega 0.00000100 100.00000000 0.100000000 TRUE
#> alpha1 0.00000001 0.99999999 0.100000000 TRUE
#> gamma1 -0.99999999 0.99999999 0.100000000 FALSE
#> beta1 0.00000001 0.99999999 0.800000000 TRUE
#> delta 0.00000000 2.00000000 2.000000000 FALSE
#> skew 0.10000000 10.00000000 1.000000000 FALSE
#> shape 1.00000000 10.00000000 4.000000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 280.37902: -0.00551678 0.100000 0.100000 0.800000
#> 1: 280.37507: -0.00551675 0.101085 0.0997614 0.800840
#> 2: 280.37172: -0.00551671 0.100985 0.0984074 0.800529
#> 3: 280.36889: -0.00551668 0.102056 0.0981374 0.801378
#> 4: 280.36553: -0.00551664 0.101972 0.0967732 0.801110
#> 5: 280.36291: -0.00551659 0.102982 0.0962621 0.801921
#> 6: 280.36028: -0.00551652 0.102987 0.0948733 0.801815
#> 7: 280.35812: -0.00551642 0.103857 0.0941985 0.802668
#> 8: 280.35640: -0.00551626 0.103796 0.0928284 0.802909
#> 9: 280.35510: -0.00551594 0.104079 0.0921544 0.804093
#> 10: 280.35429: -0.00551543 0.103412 0.0912751 0.804938
#> 11: 280.35380: -0.00551470 0.102925 0.0909305 0.806189
#> 12: 280.35345: -0.00551337 0.102118 0.0903861 0.807156
#> 13: 280.35308: -0.00550880 0.101958 0.0898304 0.808116
#> 14: 280.35296: -0.00550326 0.101305 0.0896346 0.808798
#> 15: 280.35287: -0.00549602 0.101693 0.0894169 0.808659
#> 16: 280.35280: -0.00548956 0.101338 0.0890255 0.809195
#> 17: 280.35270: -0.00548306 0.100990 0.0889425 0.809846
#> 18: 280.35261: -0.00546712 0.100295 0.0886753 0.810712
#> 19: 280.35191: -0.00526092 0.100384 0.0880880 0.811016
#> 20: 280.34947: -0.00443606 0.0998012 0.0868493 0.812628
#> 21: 280.34213: -0.00121747 0.0993041 0.0846804 0.814752
#> 22: 280.33288: 0.00415509 0.0983925 0.0835405 0.816652
#> 23: 280.32556: 0.00933509 0.0984673 0.0851411 0.815292
#> 24: 280.32295: 0.0112564 0.0984683 0.0875808 0.813418
#> 25: 280.32262: 0.0109708 0.0988384 0.0887005 0.812137
#> 26: 280.32261: 0.0107322 0.0988356 0.0888526 0.812038
#> 27: 280.32261: 0.0107066 0.0988719 0.0888596 0.811988
#> 28: 280.32261: 0.0107061 0.0988616 0.0888549 0.812004
#>
#> Final Estimate of the Negative LLH:
#> LLH: -861.9494 norm LLH: -4.309747
#> mu omega alpha1 beta1
#> 3.541769e-05 1.081941e-06 8.885493e-02 8.120038e-01
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -2.110947e+07 -4.842148e+08 2.852385e+03 -1.458198e+03
#> omega -4.842148e+08 -2.671144e+13 -2.328641e+08 -2.747076e+08
#> alpha1 2.852385e+03 -2.328641e+08 -2.618403e+03 -2.563703e+03
#> beta1 -1.458198e+03 -2.747076e+08 -2.563703e+03 -2.938561e+03
#> attr(,"time")
#> Time difference of 0.003959179 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.02163386 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = GARCH11 ~ garch(1, 1), data = X.timeSeries)
#>
#> Mean and Variance Equation:
#> GARCH11 ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = X.timeSeries]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:49 2024 by user: georgi
#>
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)
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 200
#> Recursion Init: mci
#> Series Scale: 0.003308171
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -0.05516776 0.05516776 -0.005516776 TRUE
#> omega 0.00000100 100.00000000 0.100000000 TRUE
#> alpha1 0.00000001 0.99999999 0.100000000 TRUE
#> gamma1 -0.99999999 0.99999999 0.100000000 FALSE
#> beta1 0.00000001 0.99999999 0.800000000 TRUE
#> delta 0.00000000 2.00000000 2.000000000 FALSE
#> skew 0.10000000 10.00000000 1.000000000 FALSE
#> shape 1.00000000 10.00000000 4.000000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 280.37902: -0.00551678 0.100000 0.100000 0.800000
#> 1: 280.37507: -0.00551675 0.101085 0.0997614 0.800840
#> 2: 280.37172: -0.00551671 0.100985 0.0984074 0.800529
#> 3: 280.36889: -0.00551668 0.102056 0.0981374 0.801378
#> 4: 280.36553: -0.00551664 0.101972 0.0967732 0.801110
#> 5: 280.36291: -0.00551659 0.102982 0.0962621 0.801921
#> 6: 280.36028: -0.00551652 0.102987 0.0948733 0.801815
#> 7: 280.35812: -0.00551642 0.103857 0.0941985 0.802668
#> 8: 280.35640: -0.00551626 0.103796 0.0928284 0.802909
#> 9: 280.35510: -0.00551594 0.104079 0.0921544 0.804093
#> 10: 280.35429: -0.00551543 0.103412 0.0912751 0.804938
#> 11: 280.35380: -0.00551470 0.102925 0.0909305 0.806189
#> 12: 280.35345: -0.00551337 0.102118 0.0903861 0.807156
#> 13: 280.35308: -0.00550880 0.101958 0.0898304 0.808116
#> 14: 280.35296: -0.00550326 0.101305 0.0896346 0.808798
#> 15: 280.35287: -0.00549602 0.101693 0.0894169 0.808659
#> 16: 280.35280: -0.00548956 0.101338 0.0890255 0.809195
#> 17: 280.35270: -0.00548306 0.100990 0.0889425 0.809846
#> 18: 280.35261: -0.00546712 0.100295 0.0886753 0.810712
#> 19: 280.35191: -0.00526092 0.100384 0.0880880 0.811016
#> 20: 280.34947: -0.00443606 0.0998012 0.0868493 0.812628
#> 21: 280.34213: -0.00121747 0.0993041 0.0846804 0.814752
#> 22: 280.33288: 0.00415509 0.0983925 0.0835405 0.816652
#> 23: 280.32556: 0.00933509 0.0984673 0.0851411 0.815292
#> 24: 280.32295: 0.0112564 0.0984683 0.0875808 0.813418
#> 25: 280.32262: 0.0109708 0.0988384 0.0887005 0.812137
#> 26: 280.32261: 0.0107322 0.0988356 0.0888526 0.812038
#> 27: 280.32261: 0.0107066 0.0988719 0.0888596 0.811988
#> 28: 280.32261: 0.0107061 0.0988616 0.0888549 0.812004
#>
#> Final Estimate of the Negative LLH:
#> LLH: -861.9494 norm LLH: -4.309747
#> mu omega alpha1 beta1
#> 3.541769e-05 1.081941e-06 8.885493e-02 8.120038e-01
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -2.110947e+07 -4.842148e+08 2.852385e+03 -1.458198e+03
#> omega -4.842148e+08 -2.671144e+13 -2.328641e+08 -2.747076e+08
#> alpha1 2.852385e+03 -2.328641e+08 -2.618403e+03 -2.563703e+03
#> beta1 -1.458198e+03 -2.747076e+08 -2.563703e+03 -2.938561e+03
#> attr(,"time")
#> Time difference of 0.003915787 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.02603769 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = GARCH11 ~ garch(1, 1), data = X.mts)
#>
#> Mean and Variance Equation:
#> GARCH11 ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = X.mts]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 3.5418e-05 1.0819e-06 8.8855e-02 8.1200e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 3.542e-05 2.183e-04 0.162 0.871
#> omega 1.082e-06 1.051e-06 1.030 0.303
#> alpha1 8.885e-02 5.450e-02 1.630 0.103
#> beta1 8.120e-01 1.242e-01 6.538 6.25e-11 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 861.9494 normalized: 4.309747
#>
#> Description:
#> Tue Apr 30 14:51:50 2024 by user: georgi
#>
## 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)
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 377
#> Recursion Init: mci
#> Series Scale: 0.7911981
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -1.06338438 1.063384 0.1063384 TRUE
#> omega 0.00000100 100.000000 0.1000000 TRUE
#> alpha1 0.00000001 1.000000 0.1000000 TRUE
#> gamma1 -0.99999999 1.000000 0.1000000 FALSE
#> beta1 0.00000001 1.000000 0.8000000 TRUE
#> delta 0.00000000 2.000000 2.0000000 FALSE
#> skew 0.10000000 10.000000 1.0000000 FALSE
#> shape 1.00000000 10.000000 4.0000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 509.82704: 0.106338 0.100000 0.100000 0.800000
#> 1: 509.51991: 0.106356 0.0941456 0.0992929 0.796649
#> 2: 509.34861: 0.106401 0.0914068 0.105150 0.798661
#> 3: 509.31324: 0.106450 0.0850532 0.106653 0.796875
#> 4: 509.18616: 0.106664 0.0845871 0.109854 0.802489
#> 5: 509.13752: 0.107048 0.0796272 0.109273 0.805331
#> 6: 509.11455: 0.107636 0.0785584 0.109980 0.809041
#> 7: 509.09156: 0.108344 0.0773904 0.109886 0.808412
#> 8: 508.86157: 0.130216 0.0904027 0.123041 0.783222
#> 9: 508.76023: 0.151709 0.0685017 0.102923 0.822562
#> 10: 508.74002: 0.151708 0.0695354 0.103784 0.823131
#> 11: 508.73297: 0.151702 0.0694221 0.104546 0.821891
#> 12: 508.72844: 0.151608 0.0697581 0.105658 0.821808
#> 13: 508.72336: 0.151493 0.0696262 0.105848 0.820859
#> 14: 508.71847: 0.151405 0.0705829 0.106223 0.820227
#> 15: 508.71404: 0.151289 0.0705200 0.106456 0.819285
#> 16: 508.71023: 0.151174 0.0708824 0.107305 0.818955
#> 17: 508.70624: 0.151057 0.0709395 0.107483 0.818015
#> 18: 508.70241: 0.150956 0.0717984 0.107803 0.817387
#> 19: 508.69901: 0.150836 0.0716372 0.108257 0.816599
#> 20: 508.69619: 0.150718 0.0720058 0.109044 0.816221
#> 21: 508.68815: 0.150061 0.0750247 0.109122 0.812611
#> 22: 508.67779: 0.149315 0.0751932 0.111191 0.809976
#> 23: 508.67153: 0.148688 0.0766777 0.114346 0.806316
#> 24: 508.67090: 0.149243 0.0770851 0.115532 0.804599
#> 25: 508.67086: 0.148699 0.0770358 0.115443 0.804762
#> 26: 508.67085: 0.148859 0.0770391 0.115425 0.804785
#> 27: 508.67085: 0.148857 0.0770402 0.115430 0.804778
#>
#> Final Estimate of the Negative LLH:
#> LLH: 420.3749 norm LLH: 1.115053
#> mu omega alpha1 beta1
#> 0.11777514 0.04822674 0.11543003 0.80477759
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -822.50417 -66.39566 44.87188 -46.12927
#> omega -66.39566 -22459.56936 -7066.22458 -10357.90647
#> alpha1 44.87188 -7066.22458 -4084.55043 -4274.92985
#> beta1 -46.12927 -10357.90647 -4274.92985 -5674.92502
#> attr(,"time")
#> Time difference of 0.004879951 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.02381349 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = 100 * (SPI - SBI) ~ garch(1, 1), data = X.timeSeries)
#>
#> Mean and Variance Equation:
#> 100 * (SPI - SBI) ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = X.timeSeries]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 0.117775 0.048227 0.115430 0.804778
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 0.11778 0.03509 3.357 0.000789 ***
#> omega 0.04823 0.01851 2.605 0.009186 **
#> alpha1 0.11543 0.03771 3.061 0.002206 **
#> beta1 0.80478 0.05422 14.842 < 2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> -420.3749 normalized: -1.115053
#>
#> Description:
#> Tue Apr 30 14:51:50 2024 by user: georgi
#>
# 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))
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 248
#> Recursion Init: mci
#> Series Scale: 0.03467707
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -0.27446113 0.2744611 -0.02744611 TRUE
#> omega 0.00000100 100.0000000 0.10000000 TRUE
#> alpha1 0.00000001 1.0000000 0.10000000 TRUE
#> gamma1 -0.99999999 1.0000000 0.10000000 FALSE
#> beta1 0.00000001 1.0000000 0.80000000 TRUE
#> delta 0.00000000 2.0000000 2.00000000 FALSE
#> skew 0.10000000 10.0000000 1.00000000 FALSE
#> shape 1.00000000 10.0000000 4.00000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 342.27498: -0.0274461 0.100000 0.100000 0.800000
#> 1: 340.94425: -0.0274245 0.0415953 0.171831 0.810282
#> 2: 340.93440: -0.0274205 0.0422745 0.170657 0.808213
#> 3: 340.92348: -0.0274161 0.0447136 0.170835 0.807844
#> 4: 340.91370: -0.0274100 0.0453986 0.169929 0.805652
#> 5: 340.90869: -0.0274003 0.0476051 0.170026 0.804585
#> 6: 340.90521: -0.0273777 0.0482184 0.169865 0.802336
#> 7: 340.90248: -0.0273321 0.0489754 0.171281 0.801438
#> 8: 340.89717: -0.0270041 0.0480428 0.177299 0.797774
#> 9: 340.86736: -0.0231806 0.0537295 0.174808 0.789982
#> 10: 340.73004: -0.0101016 0.0439580 0.164309 0.806907
#> 11: 340.56766: 0.0160790 0.0439679 0.183551 0.801336
#> 12: 340.52900: 0.0197153 0.0459647 0.190266 0.789344
#> 13: 340.52544: 0.0172714 0.0457522 0.185566 0.793380
#> 14: 340.52539: 0.0175126 0.0455331 0.185311 0.793841
#> 15: 340.52539: 0.0175241 0.0455418 0.185276 0.793848
#> 16: 340.52539: 0.0175254 0.0455424 0.185274 0.793849
#>
#> Final Estimate of the Negative LLH:
#> LLH: -493.1704 norm LLH: -1.98859
#> mu omega alpha1 beta1
#> 6.077311e-04 5.476465e-05 1.852739e-01 7.938490e-01
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -323896.988 -3942880 -1274.877 -2692.172
#> omega -3942880.327 -3716487012 -2283988.677 -3321697.132
#> alpha1 -1274.877 -2283989 -1841.165 -2492.527
#> beta1 -2692.172 -3321697 -2492.527 -3686.317
#> attr(,"time")
#> Time difference of 0.004188299 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.01663399 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = Open ~ garch(1, 1), data = returns(X.timeSeries))
#>
#> Mean and Variance Equation:
#> Open ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = returns(X.timeSeries)]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 6.0773e-04 5.4765e-05 1.8527e-01 7.9385e-01
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 6.077e-04 1.778e-03 0.342 0.7325
#> omega 5.476e-05 3.765e-05 1.455 0.1458
#> alpha1 1.853e-01 8.095e-02 2.289 0.0221 *
#> beta1 7.938e-01 6.301e-02 12.599 <2e-16 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> 493.1704 normalized: 1.98859
#>
#> Description:
#> Tue Apr 30 14:51:50 2024 by user: georgi
#>
garchFit(100*(High-Low) ~ garch(1,1), data = returns(X.timeSeries))
#>
#> Series Initialization:
#> ARMA Model: arma
#> Formula Mean: ~ arma(0, 0)
#> GARCH Model: garch
#> Formula Variance: ~ garch(1, 1)
#> ARMA Order: 0 0
#> Max ARMA Order: 0
#> GARCH Order: 1 1
#> Max GARCH Order: 1
#> Maximum Order: 1
#> Conditional Dist: norm
#> h.start: 2
#> llh.start: 1
#> Length of Series: 248
#> Recursion Init: mci
#> Series Scale: 1.759168
#>
#> Parameter Initialization:
#> Initial Parameters: $params
#> Limits of Transformations: $U, $V
#> Which Parameters are Fixed? $includes
#> Parameter Matrix:
#> U V params includes
#> mu -0.01484999 0.01484999 -0.001484999 TRUE
#> omega 0.00000100 100.00000000 0.100000000 TRUE
#> alpha1 0.00000001 0.99999999 0.100000000 TRUE
#> gamma1 -0.99999999 0.99999999 0.100000000 FALSE
#> beta1 0.00000001 0.99999999 0.800000000 TRUE
#> delta 0.00000000 2.00000000 2.000000000 FALSE
#> skew 0.10000000 10.00000000 1.000000000 FALSE
#> shape 1.00000000 10.00000000 4.000000000 FALSE
#> Index List of Parameters to be Optimized:
#> mu omega alpha1 beta1
#> 1 2 3 5
#> Persistence: 0.9
#>
#>
#> --- START OF TRACE ---
#> Selected Algorithm: nlminb
#>
#> R coded nlminb Solver:
#>
#> 0: 338.79385: -0.00148500 0.100000 0.100000 0.800000
#> 1: 338.46090: -0.00148500 0.0930914 0.100064 0.793760
#> 2: 338.24428: -0.00148500 0.0927929 0.109355 0.793249
#> 3: 338.13232: -0.00148499 0.0865650 0.112004 0.786856
#> 4: 337.95165: -0.00148499 0.0903645 0.120451 0.785916
#> 5: 337.83363: -0.00148498 0.0880066 0.123980 0.777629
#> 6: 337.71984: -0.00148497 0.0934008 0.130363 0.773527
#> 7: 337.62836: -0.00148495 0.0936956 0.132882 0.764569
#> 8: 337.54525: -0.00148493 0.0995383 0.138251 0.759700
#> 9: 337.47146: -0.00148490 0.100375 0.140485 0.750701
#> 10: 337.40214: -0.00148486 0.106092 0.145870 0.745701
#> 11: 337.33811: -0.00148478 0.106731 0.148700 0.736855
#> 12: 337.27971: -0.00148466 0.112207 0.154164 0.731676
#> 13: 337.22953: -0.00148446 0.113576 0.156827 0.722862
#> 14: 337.17847: -0.00148403 0.118016 0.163018 0.717519
#> 15: 337.15010: -0.00148366 0.121418 0.163737 0.708886
#> 16: 337.12658: -0.00148293 0.117033 0.171770 0.710523
#> 17: 337.08704: -0.00148284 0.122915 0.178235 0.693051
#> 18: 337.03054: -0.00148210 0.136576 0.184775 0.680722
#> 19: 337.00277: -0.00148078 0.144688 0.188565 0.663381
#> 20: 336.98756: -0.00147267 0.154426 0.195482 0.648918
#> 21: 336.97675: -0.00144968 0.154594 0.204920 0.641641
#> 22: 336.97583: -0.00142340 0.157318 0.206707 0.637693
#> 23: 336.97552: -0.00139450 0.158213 0.207352 0.636343
#> 24: 336.97467: -0.00128837 0.159968 0.208616 0.633682
#> 25: 336.97264: -0.000975485 0.163087 0.210866 0.628926
#> 26: 336.96825: -0.000197740 0.167982 0.214399 0.621399
#> 27: 336.95854: 0.00166593 0.175778 0.220016 0.609263
#> 28: 336.93800: 0.00599522 0.188577 0.229193 0.588989
#> 29: 336.90456: 0.0144044 0.207806 0.242786 0.557755
#> 30: 336.89568: 0.0148500 0.204967 0.240403 0.561330
#> 31: 336.89184: 0.0148500 0.157893 0.205616 0.629260
#> 32: 336.86883: 0.0148500 0.184188 0.226104 0.592041
#> 33: 336.85962: 0.0148500 0.173554 0.219500 0.608290
#> 34: 336.85725: 0.0148500 0.155984 0.209335 0.635860
#> 35: 336.85498: 0.0148500 0.164112 0.214240 0.623640
#> 36: 336.85486: 0.0148500 0.163014 0.213762 0.625457
#> 37: 336.85485: 0.0148500 0.162753 0.213693 0.625907
#> 38: 336.85485: 0.0148500 0.162780 0.213714 0.625865
#> 39: 336.85485: 0.0148500 0.162785 0.213717 0.625858
#>
#> Final Estimate of the Negative LLH:
#> LLH: 476.9354 norm LLH: 1.923127
#> mu omega alpha1 beta1
#> 0.02612363 0.50376670 0.21371740 0.62585764
#>
#> R-optimhess Difference Approximated Hessian Matrix:
#> mu omega alpha1 beta1
#> mu -132.010667 -11.71868 6.265931 -31.66371
#> omega -11.718680 -165.10094 -231.511196 -402.35533
#> alpha1 6.265931 -231.51120 -664.732441 -708.75592
#> beta1 -31.663715 -402.35533 -708.755923 -1073.12968
#> attr(,"time")
#> Time difference of 0.004318714 secs
#>
#> --- END OF TRACE ---
#>
#>
#> Time to Estimate Parameters:
#> Time difference of 0.02533388 secs
#>
#> Title:
#> GARCH Modelling
#>
#> Call:
#> garchFit(formula = 100 * (High - Low) ~ garch(1, 1), data = returns(X.timeSeries))
#>
#> Mean and Variance Equation:
#> 100 * (High - Low) ~ garch(1, 1)
#> <environment: 0x569a0b6b9d80>
#> [data = returns(X.timeSeries)]
#>
#> Conditional Distribution:
#> norm
#>
#> Coefficient(s):
#> mu omega alpha1 beta1
#> 0.026124 0.503767 0.213717 0.625858
#>
#> Std. Errors:
#> based on Hessian
#>
#> Error Analysis:
#> Estimate Std. Error t value Pr(>|t|)
#> mu 0.02612 0.08964 0.291 0.770733
#> omega 0.50377 0.35141 1.434 0.151694
#> alpha1 0.21372 0.09596 2.227 0.025941 *
#> beta1 0.62586 0.18294 3.421 0.000623 ***
#> ---
#> Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#>
#> Log Likelihood:
#> -476.9354 normalized: -1.923127
#>
#> Description:
#> Tue Apr 30 14:51:50 2024 by user: georgi
#>
## 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