Package 'gogarch' reference manual

Title:	Generalized Orthogonal GARCH (GO-GARCH) Models
Description:	Provision of classes and methods for estimating generalized orthogonal GARCH models. This is an alternative approach to CC-GARCH models in the context of multivariate volatility modeling.
Authors:	Bernhard Pfaff [aut, cre]
Maintainer:	Bernhard Pfaff <[email protected]>
License:	GPL (>= 2)
Version:	0.7-5
Built:	2024-11-01 11:15:14 UTC
Source:	https://github.com/cran/gogarch

Dow Jones Industrial Average and Nasdaq stock indices

Description

Levels of the Dow Jones Industrial Average and NASDAQ stock indices for the period 03/23/1990 until 03/23/2000.

Usage

data(BVDW)data(BVDW)

Format

A data frame with 2610 observations on the following 3 variables.

Date: Date in the format YYYYMMDD.
DJIA: Level of the DIJA.
NASDAQ: Level of the NASDAQ.

Details

This data set has been utilized in the source below and was kindly provided by Roy van der Weide.

Source

Boswijk, H. Peter and van der Weide, Roy (2006), Wake me up before you GO-GARCH, Tinbergen Institute Discussion Paper, TI 2006-079/4, University of Amsterdam and Tinbergen Institute.

Examples

data(BVDW)
str(BVDW)
data(BVDW)
str(BVDW)

Stock prices transportation sector, oil and kerosene prices

Description

This data frame contains the stock prices from American Airlines, South-West Airlines, Boeing and FedEx. In addition the spot prices for crude oil and kerosene are included. This data set was used in the article by Boswijk and van der Weide (2009). The data range is from July, 19 1993 until August, 12 2008.

Usage

data(BVDWAIR)data(BVDWAIR)

Format

A data frame with 3791 observations on the following 7 variables.

Date: POSIXt: The dates of observations.
CrudeOil: Crude oil price.
Kerosene: Kerosene price.
AmericanAir: Stock prices of American Airlines.
SouthWest: Stock prices of South-West Airlines.
Boeing: Stock prices of Boeing.
FedEx: Stock prices of Boeing.

Details

The stock price data was downloaded from Yahoo Finance and the price series for crude oil and kerosene were obtained from the U.S. Energy Information Administration (EIA).

Source

http://www.econstats.com

References

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Examples

data(BVDWAIR)
str(BVDWAIR)
data(BVDWAIR)
str(BVDWAIR)

Sector indices of the EURO STOXX 600

Description

The data frame contains the following sector indices of the EURO STOXX 600 index: Automobiles \& Parts, Banks, Basic Resources, Chemicals, Construction and Materials, Financial Services, Food \& Beverages, Health Care, Industrial Goods \& Services, Insurance, Media, Oil \& Gas, Technology, Telecommunications and Utilities. The data range is from 31th December 1986 until 21st November 2008.

Usage

data(BVDWSTOXX)data(BVDWSTOXX)

Format

A data frame with 5652 observations on the following 16 variables.

Date: POSIXt: The dates of observations.
AutoParts: Sector index Automobiles \& Parts
Banks: Sector index Banks
BasicRes: Sector index Basic Resources
Chemicals: Sector index Chemicals
ConstrMat: Sector index Construction and Materials
FoodBeverage: Sector index Food \& Beverages
FinService: Sector index Financial Services
HealthCare: Sector index Health Care
IndustrialGoods: Sector index Industrial Goods \& Services
Insurance: Sector index Insurance
Media: Sector index Media
OilGas: Sector index Oil \& Gas
Technology: Sector index Technology
Telecom: Sector index Telecommunications
Utilities: Sector index Utilities

Source

http://www.stoxx.com

References

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Examples

data(BVDWSTOXX)
str(BVDWSTOXX)
data(BVDWSTOXX)
str(BVDWSTOXX)

Autocorrelations of a Matrix Process

Description

This function computes the autocorrelation matrix for a given lag. For instance, it is used for estimating GO-GARCH models whence the method of moments is utilized.

Usage

cora(SSI, lag = 1, standardize = TRUE)
cora(SSI, lag = 1, standardize = TRUE)

Arguments

`SSI`	Array with dimension `dim = c(m, m, n)`
`lag`	Integer, the lag for which the autocorrelation is computed.
`standardize`	Logical, if `TRUE` (the default), the autocorrelation matrix is computed, otherwise the autocovariance matrix.

Details

This function computes the autocorrelation matrix according to:

$\hat{\Gamma}_k (s) = \frac{1}{n} \sum_{t = k + 1}^n S_t S_{t-k}$

$\hat{\Phi}_k (s) = \hat{\Gamma}_0 (s)^{-1/2} \hat{\Gamma}_k (s) \hat{\Gamma}_0 (s)^{-1/2}$

It is computationally assured that $\hat{\Phi}_k (s)$ is symmetric by setting it equal to: $\hat{\Phi}_k (s) = \frac{1}{2}(\hat{\Phi}_k (s) + \hat{\Phi}_k (s)')$ . The standardization matrix $\hat{\Gamma}_0 (s)^{-1/2}$ is derived from the singular value decomposition of the co-variance matrix at lag zero.

Value

cora

Matrix with dimension dim = c(m, m).

Author(s)

Bernhard Pfaff

References

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Methods for Function goest

Description

These are methods for estimating GO-GARCH models. Currently only a method for estimating GO-GARCH models by Maximum-Likelihood is implemented.

Details

The declared estimation methods are called from function gogarch.

Methods

goest: signature(object = "Goestica")
goest: signature(object = "Goestmm")
goest: signature(object = "Goestml")
goest: signature(object = "Goestnls")

Author(s)

Bernhard Pfaff

Class "Goestica": GO-GARCH models estimated by fast ICA

Description

This class contains the GoGARCH class and has the mixing matrix $A$ as additional slot.

Objects from the Class

Objects can be created by calls of the form new("Goestmm", ...), or with the function gogarch whereby method = "ica" has been set.

Slots

ica:: Object of class "list": List object returned by fastICA.
Z:: Object of class "matrix": Transformation matrix.
U:: Object of class "matrix": Orthogonal matrix.
Y:: Object of class "matrix": Extracted component matrix.
H:: Object of class "list": List of conditional variance/covariance matrices.
models:: Object of class "list": List of univariate GARCH model fits.
estby:: Object of class "character": Estimation method.
X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Extends

Class "GoGARCH", directly. Class "Goinit", by class "GoGARCH", distance 2.

Methods

cvar: Returns the conditional variances as object with class attribute "mts" "ts".
ccov: Returns the conditional co-variances as object with class attribute "mts" "ts".
ccor: Returns the conditional correlationsas object with class attribute "mts" "ts".
coef: Returns the coeffiecients of the component GARCH models.
converged: Returns the convergence codes of the component GARCH models.
formula: Returns the formula for the component GARCH models.
goest: Fast ICA estimation of Go-GARCH models.
plot: Plotting of the conditional correlations.
predict: Returns the conditional covariances and mean forecasts and the forecasts of the component GARCH models, object is of class Gopredict.
residuals: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
resid: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
show: show-method for objects of class Goestmm.
summary: summary-method for objects of class Goestml, object is of class Gosum.
update: Updates an object of class Goestml.

Author(s)

Bernhard Pfaff

References

Broda, S.A. and Paolella, M.S. (2008): CHICAGO: A Fast and Accurate Method for Portfolio Risk Calculation, Swiss Finance Institute, Research Paper Series No. 08-08, Zuerich.

Class "Goestml": GO-GARCH models estimated by Maximum-Likelihood

Description

This class contains the GoGARCH class and has the outcome of nlminb as an additional slot.

Objects from the Class

Objects can be created by calls of the form new("Goestml", ...), or with the function gogarch whereby method = "ml" has been set.

Slots

opt:: Object of class "list": List returned by nlminb.
Z:: Object of class "matrix": Transformation matrix.
U:: Object of class "matrix": Orthogonal matrix.
Y:: Object of class "matrix": Extracted component matrix.
H:: Object of class "list": List of conditional variance/covariance matrices.
models:: Object of class "list": List of univariate GARCH model fits.
estby:: Object of class "character": Estimation method.
X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Extends

Class "GoGARCH", directly. Class "Goinit", by class "GoGARCH", distance 2.

Methods

angles: Returns the Eulerian angles.
cvar: Returns the conditional variances as object with class attribute "mts" "ts".
ccov: Returns the conditional co-variances as object with class attribute "mts" "ts".
ccor: Returns the conditional correlations as object with class attribute "mts" "ts".
coef: Returns the coeffiecients of the component GARCH models.
converged: Returns the convergence codes of the component GARCH models.
formula: Returns the formula for the component GARCH models.
goest: ML-Estimation of Go-GARCH models.
logLik: Returns the value of the log-Likelihood function.
plot: Plotting of the conditional correlations.
predict: Returns the conditional covariances and mean forecasts and the forecasts of the component GARCH models, object is of class Gopredict.
residuals: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
resid: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
show: show-method for objects of class Goestml.
summary: summary-method for objects of class Goestml, object is of class Gosum.
update: Updates an object of class Goestml.

Author(s)

Bernhard Pfaff

Class "Goestmm": Go-GARCH models estimated by Methods of Moments

Description

This class contains the GoGARCH class and has the weights vector and the matched orthogonal matrices $U$ as additional slots.

Objects from the Class

Objects can be created by calls of the form new("Goestmm", ...), or with the function gogarch whereby method = "mm" has been set.

Slots

weights:: Object of class "numeric": Weights for aggregating the matched orthogonal matrices $U$ .
Umatched:: Object of class "list": List of matched orthogonal matrices $U$ .
Z:: Object of class "matrix": Transformation matrix.
U:: Object of class "matrix": Orthogonal matrix.
Y:: Object of class "matrix": Extracted component matrix.
H:: Object of class "list": List of conditional variance/covariance matrices.
models:: Object of class "list": List of univariate GARCH model fits.
estby:: Object of class "character": Estimation method.
X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Extends

Class "GoGARCH", directly. Class "Goinit", by class "GoGARCH", distance 2.

Methods

cvar: Returns the conditional variances as object with class attribute "mts" "ts".
ccov: Returns the conditional co-variances as object with class attribute "mts" "ts".
ccor: Returns the conditional correlationsas object with class attribute "mts" "ts".
coef: Returns the coeffiecients of the component GARCH models.
converged: Returns the convergence codes of the component GARCH models.
formula: Returns the formula for the component GARCH models.
goest: Methods of moments estimation of Go-GARCH models.
plot: Plotting of the conditional correlations.
predict: Returns the conditional covariances and mean forecasts and the forecasts of the component GARCH models, object is of class Gopredict.
residuals: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
resid: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
show: show-method for objects of class Goestmm.
summary: summary-method for objects of class Goestml, object is of class Gosum.
update: Updates an object of class Goestml.

Author(s)

Bernhard Pfaff

References

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Class "Goestnls": GO-GARCH models estimated by Non-linear Least-Squares

Description

This class contains the GoGARCH class and has the outcome of optim as an additional slot.

Objects from the Class

Objects can be created by calls of the form new("Goestnls", ...), or with the function gogarch whereby method = "nls" has been set.

Slots

nls:: Object of class "list": List returned by optim.
Z:: Object of class "matrix": Transformation matrix.
U:: Object of class "matrix": Orthogonal matrix.
Y:: Object of class "matrix": Extracted component matrix.
H:: Object of class "list": List of conditional variance/covariance matrices.
models:: Object of class "list": List of univariate GARCH model fits.
estby:: Object of class "character": Estimation method.
X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Extends

Class "GoGARCH", directly. Class "Goinit", by class "GoGARCH", distance 2.

Methods

cvar: Returns the conditional variances as object with class attribute "mts" "ts".
ccov: Returns the conditional co-variances as object with class attribute "mts" "ts".
ccor: Returns the conditional correlationsas object with class attribute "mts" "ts".
coef: Returns the coeffiecients of the component GARCH models.
converged: Returns the convergence codes of the component GARCH models.
formula: Returns the formula for the component GARCH models.
goest: NLS-Estimation of Go-GARCH models.
plot: Plotting of the conditional correlations.
predict: Returns the conditional covariances and mean forecasts and the forecasts of the component GARCH models, object is of class Gopredict.
residuals: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
resid: Returns the residuals of the Go-GARCH model as object with class attribute "mts" "ts".
show: show-method for objects of class Goestnls.
summary: summary-method for objects of class GoGARCH, object is of class Gosum.
update: Updates an object of class GoGARCH.

Author(s)

Bernhard Pfaff

Specification and estimation of GO-GARCH models

Description

This function steers the specification and estimation of GO-GARCH models.

Usage

gogarch(data, formula, scale = FALSE, estby = c("ica", "mm", "ml", "nls"),
  lag.max = 1, initial = NULL, garchlist = list(init.rec = "mci", delta
  = 2, skew = 1, shape = 4, cond.dist = "norm", include.mean = FALSE,
  include.delta = NULL, include.skew = NULL, include.shape = NULL,
  leverage = NULL, trace = FALSE, algorithm = "nlminb", hessian =
  "ropt", control = list(), title = NULL, description = NULL), ...) 
gogarch(data, formula, scale = FALSE, estby = c("ica", "mm", "ml", "nls"),
  lag.max = 1, initial = NULL, garchlist = list(init.rec = "mci", delta
  = 2, skew = 1, shape = 4, cond.dist = "norm", include.mean = FALSE,
  include.delta = NULL, include.skew = NULL, include.shape = NULL,
  leverage = NULL, trace = FALSE, algorithm = "nlminb", hessian =
  "ropt", control = list(), title = NULL, description = NULL), ...)

Arguments

`data`	Matrix: the original data set.
`formula`	Formula: valid formula for univariate GARCH models.
`scale`	Logical, if `TRUE` the data is scaled. The default is `scale = FALSE`.
`estby`	Character: by fast ICA `estby = "ica"` (the default), by Estbys of Moments `estby = "mm"` or by Maximum-Likelihood `estby = "ml"` or by non-linear Least-Squares `estby = "nls"`.
`initial`	Numeric: starting values for optimization (used if `estby = "ml"` or `estby = "nls"` has been chosen (see Details).
`lag.max`	Integer: The number of used lags for computing the matched orthogonal matrices $U$ (used if `estby = "mm"` has been chosen).
`garchlist`	List: Elements are passed to `garchFit`.
`...`	Ellipsis argument: is passed to the `goest` method (see details).

Details

The ellipsis argument is passed to the function fastICA if estby = "ica" has been set, or to optim if estby = "nls" is employed or to nlminb if the GO-GARCH model is estimated by maximum likelihood, i.e., estby = "ml". It is not employed if the methods of moments estimator is chosen.

If the argument initial is left NULL, the starting values are computed according seq(3.0, 0.1, length.out = l), whereby l is the length of initial for estby = "ml" and are set to rep(0.1, d, whereby for method = "nls". This length must be equal to $m * (m - 1)/2$ for estimation by Maximum-Likelihood and $m * (m + 1)/2$ for estimation by non-linear least-Squares, whereby $m$ is the number of columns of data.

Value

Dependent on the chosen estimation method either an object of class Goestica or, Goestmm or Goestml or Goestnls is returned. All of these classes extend the GoGARCH class.

Author(s)

Bernhard Pfaff

References

Van der Weide, Roy (2002), GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model, Journal of Applied Econometrics, 17(5), 549 – 564.

Boswijk, H. Peter and van der Weide, Roy (2006), Wake me up before you GO-GARCH, Tinbergen Institute Discussion Paper, TI 2006-079/4, University of Amsterdam and Tinbergen Institute.

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Broda, S.A. and Paolella, M.S. (2008): CHICAGO: A Fast and Accurate Method for Portfolio Risk Calculation, Swiss Finance Institute, Research Paper Series No. 08-08, Zuerich.

Examples

## Not run: 
library(vars)
## Boswijk / van der Weide (2009)
data(BVDWSTOXX)
BVDWSTOXX <- zoo(x = BVDWSTOXX[, -1], order.by = BVDWSTOXX[, 1])
BVDWSTOXX <- window(BVDWSTOXX, end = as.POSIXct("2007-12-31"))
BVDWSTOXX <- diff(log(BVDWSTOXX))
sectors <- BVDWSTOXX[, c("AutoParts", "Banks", "OilGas")]
sectors <- apply(sectors, 2, scale, scale = FALSE)
gogmm <- gogarch(sectors, formula = ~garch(1,1), estby = "mm",
         lag.max = 100)
gogmm
## Boswijk / van der Weide (2006)
data(BVDW)
BVDW <- zoo(x = BVDW[, -1], order.by = BVDW[, 1])
BVDW <- diff(log(BVDW)) * 100
gognls <- gogarch(BVDW, formula = ~garch(1,1), scale = TRUE,
          estby = "nls")
gognls
## van der Weide (2002)
data(VDW)
var1 <- VAR(scale(VDW), p = 1, type = "const")
resid <- residuals(var1)
gogml <- gogarch(resid, ~garch(1, 1), scale = TRUE,
         estby = "ml", control = list(iter.max = 1000))
gogml
solve(gogml@Z)

## End(Not run)
## Not run: 
library(vars)
## Boswijk / van der Weide (2009)
data(BVDWSTOXX)
BVDWSTOXX <- zoo(x = BVDWSTOXX[, -1], order.by = BVDWSTOXX[, 1])
BVDWSTOXX <- window(BVDWSTOXX, end = as.POSIXct("2007-12-31"))
BVDWSTOXX <- diff(log(BVDWSTOXX))
sectors <- BVDWSTOXX[, c("AutoParts", "Banks", "OilGas")]
sectors <- apply(sectors, 2, scale, scale = FALSE)
gogmm <- gogarch(sectors, formula = ~garch(1,1), estby = "mm",
         lag.max = 100)
gogmm
## Boswijk / van der Weide (2006)
data(BVDW)
BVDW <- zoo(x = BVDW[, -1], order.by = BVDW[, 1])
BVDW <- diff(log(BVDW)) * 100
gognls <- gogarch(BVDW, formula = ~garch(1,1), scale = TRUE,
          estby = "nls")
gognls
## van der Weide (2002)
data(VDW)
var1 <- VAR(scale(VDW), p = 1, type = "const")
resid <- residuals(var1)
gogml <- gogarch(resid, ~garch(1, 1), scale = TRUE,
         estby = "ml", control = list(iter.max = 1000))
gogml
solve(gogml@Z)

## End(Not run)

Class "GoGARCH": Estimated GO-GARCH Models

Description

This class defines the slots for estimated GO-GARCH models. It contains the class Goinit.

Objects from the Class

Objects can be created by calls of the form new("GoGARCH", ...).

Slots

Z:: Object of class "matrix": Transformation matrix.
U:: Object of class "Orthom": Orthonormal matrix.
Y:: Object of class "matrix": Extracted component matrix.
H:: Object of class "list": List of conditional variance/covariance matrices.
models:: Object of class "list": List of univariate GARCH model fits.
estby:: Object of class "character": Estimation method.
CALL:: Object of class "call": Result of match.call in generating function.
X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Extends

Class "Goinit", directly.

Methods

cvar: Returns the conditional variances as object with class attribute "mts" "ts".
ccov: Returns the conditional co-variances as object with class attribute "mts" "ts".
ccor: Returns the conditional correlationsas object with class attribute "mts" "ts".
coef: Returns the coeffiecients of the component GARCH models.
converged: Returns the convergence codes of the component GARCH models.
formula: Returns the formula for the component GARCH models.
plot: Plotting of the conditional correlations.
predict: Returns the conditional covariances and mean forecasts and the forecasts of the component GARCH models, object is of class Gopredict.
residuals: Returns the residuals of the GO-GARCH model.
show: show-method for objects of class GoGARCH.
summary: summary-method for objects of class GoGARCH, object is of class Gosum.
update: Updates an object of class GoGARCH.

Author(s)

Bernhard Pfaff

Constructor function for objects of class "Goinit"

Description

This function can be utilized to create objects of class Goinit. These objects are the starting point for estimating GO-GARCH models.

Usage

goinit(X, garchf = ~garch(1, 1), scale = FALSE)
goinit(X, garchf = ~garch(1, 1), scale = FALSE)

Arguments

`X`	Matrix: the data matrix.
`garchf`	Formula: A formula object that will be used in the GARCH models of the uncorrelated components.
`scale`	Logical, if `TRUE` the data `X` will be scaled, the default value is `FALSE` for no scaling of the data.

Details

This function computes the variance/covariance matrix of X. Next the singular value decomposition is applied and the projection matrix as well as the diagonal matrix with the square roots of the eigen values are computed.

Value

An object of class Goinit.

Author(s)

Bernhard Pfaff

Examples

## Not run: 
library(vars)
data(VDW)
var1 <- VAR(VDW, p = 1, type = "const")
resid <- resid(var1)
goinit(resid, scale = TRUE)

## End(Not run)
## Not run: 
library(vars)
data(VDW)
var1 <- VAR(VDW, p = 1, type = "const")
resid <- resid(var1)
goinit(resid, scale = TRUE)

## End(Not run)

Class "Goinit": Initialisation of GO-GARCH models

Description

This class defines the required slots for estimating GO-GARCH models.

Objects from the Class

Objects can be created by calls of the form new("Goinit", ...), or more conveniently by goinit().

Slots

X:: Object of class "matrix": The data matrix.
V:: Object of class "matrix": Covariance matrix of X.
P:: Object of class "matrix": Left singular values of Var/Cov matrix of X.
Dsqr:: Object of class "matrix": Square roots of eigenvalues on diagonal, else zero.
garchf:: Object of class "formula": Garch formula used for uncorrelated component GARCH models.
name:: Object of class "character": The name of the original data object.

Methods

show: Prints the slots, whereby for X only the head is displayed.

Author(s)

Bernhard Pfaff

Examples

showClass("Goinit")
showClass("Goinit")

Log-Likelihood function of GO-GARCH models

Description

This function returns the negative of the log-Likelihood function for GO-GARCH models.

Usage

gollh(params, object, garchlist)
gollh(params, object, garchlist)

Arguments

`params`	Vector of initial values for `theta`.
`object`	An object of class `Goinit` or an extension thereof.
`garchlist`	List, elements are passed to `garchFit`.

Details

The log-Likelihood function of GO-GARCH models is given as:

$L_{\theta, \alpha, \beta} = - \frac{1}{2} \sum_{t=1}^T m \log(2\pi) + \log|Z_\theta Z_\theta '| + \log|H_t| + y' H_t^{-1}y_t$

whereby $Z = P \Delta^{\frac{1}{2}} U_0$ , $y_t = Z^{-1}x_t$ and $H_t$ is the conditional variance matrix of the independent components.

Value

negll

Scalar, the negative value of the log-Likelihood function.

Author(s)

Bernhard Pfaff

References

Van der Weide, Roy (2002), GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model, Journal of Applied Econometrics, 17(5), 549 – 564.

Non-linear least-squares estimation of matrix B

Description

This is the target function for estimating the matrix $B$ by non-linear least-squares. It is used in the estimation method goest if method = "nls" is chosen.

Usage

gonls(params, SSI)
gonls(params, SSI)

Arguments

`params`	The initial values of the $vech(B)$ .
`SSI`	A list with two elements, each a list itself, containing $S_t = s_t s_t' - I_m$ and $S_{t-1} = s_{t-1} s_{t-1}' - I_m$ , respectively.

Details

Boswijk and van der Weiden (2006) proposed the following criterion function:

$S(A) = \frac{1}{n} \sum_{t = 1}^n tr([s_t s_t' - I_m - B(s_{t-1} s_{t-1}' - I_m)B]^2) = S^*(B)$

for retrieving the matrix $U$ . This matrix is the eigen vector matrix of $B$ . The linear map $Z = P \Delta^{1/2} U$ and its inverse can then be computed for calculating the component matrix $Y = X Z^{-1}$ .

Value

`f`	`numeric`: The value of the target function.

Author(s)

Bernhard Pfaff

References

Boswijk, H. Peter and van der Weide, Roy (2006), Wake me up before you GO-GARCH, Tinbergen Institute Discussion Paper, TI 2006-079/4, University of Amsterdam and Tinbergen Institute.

Class "Gopredict": Prediction of GO-GARCH Models

Description

This class defines the slots for forecasts from a GO-GARCH model.

Objects from the Class

Objects can be created by calls of the form new("Gopredict", ...), or with the method predict of formal class objects GoGARCH and Goestml.

Slots

Hf:: Object of class "list": The forecasted conditional covariances.
Xf:: Object of class "matrix": The transformed forecasts of the component GARCH mean models.
CGARCHF:: Object of class "list": The original forecasts of the component GARCH models.

Methods

ccor: Returns the forecasted conditional correlations.
ccov: Returns the forecasted conditional co-variances.
cvar: Returns the forecasted conditional variances.
show: show-method for objects of class Gopredict.

Note

In case more than 10 forecasts steps are computed, the show-method displays only the head of the returned objects. Furthermore, the show-method displays the forecasted conditional variances only. The forecasted conditional co-variances and/or the forecasted conditional correlations can be retrieved with the methods ccov or ccor, respectively.

Author(s)

Bernhard Pfaff

Class "Gosum": Summary object of GO-GARCH model

Description

The formal summary class of GoGARCH objects or objects that extend this class.

Objects from the Class

Objects can be created by calls of the form new("Gosum", ...) or are set by the summary-method.

Slots

name:: character: the name of the original data object.
method:: character: the estimation method.
model:: formula: The GARCH model formula for the component GARCH models.
garchc:: list: The elements are matcoef matrices generated by garchFit for the components.
Zinv:: matrix: The inverse of the linear map $X = Y Z$ .

Methods

show: show-method for objects of class Gosum.

Author(s)

Bernhard Pfaff

Creates an object of class GoGARCH based on Euler angles

Description

This function returns an object of class GoGARCH based on an input vector of Euler angles.

Usage

gotheta(theta, object, garchlist = list(init.rec = "mci", delta = 2,
skew = 1, shape = 4, cond.dist = "norm", include.mean = FALSE,
include.delta = NULL, include.skew = NULL, include.shape = NULL,
leverage = NULL, trace = FALSE, algorithm = "nlminb", hessian = "ropt",
control = list(), title = NULL, description = NULL))
gotheta(theta, object, garchlist = list(init.rec = "mci", delta = 2,
skew = 1, shape = 4, cond.dist = "norm", include.mean = FALSE,
include.delta = NULL, include.skew = NULL, include.shape = NULL,
leverage = NULL, trace = FALSE, algorithm = "nlminb", hessian = "ropt",
control = list(), title = NULL, description = NULL))

Arguments

`theta`	Vector of Euler angles.
`object`	An object of formal class `Goinit` or an extension thereof.
`garchlist`	List with optional elements passed to `garchFit`.

Details

In a first step the orthogonal matrix $U$ is computed as the product of rotation matrices given the vector theta of Euler angles with the function UprodR. The linear map $Z$ is computed next as $Z = P D^{\frac{1}{2}} U'$ . The unobserved components $Y$ are calculated as $Y = X Z^{-1}$ . These are then utilized in the estimation of the univariate GARCH models according to object@garchf. The conditional variance/covariance matrices are calculated according to $V_t = Z H_t Z'$ whereby $H_t$ signifies a matrix with the conditional variances of the unvariate GARCH models on its diagonal.

Value

Returns an object of class GoGARCH.

Author(s)

Bernhard Pfaff

References

Van der Weide, Roy (2002), GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model, Journal of Applied Econometrics, 17(5), 549 – 564.

Examples

## Not run: 
library(vars)
data(VDW)
var1 <- VAR(VDW, p = 1, type = "const")
resid <- resid(var1)
gin <- goinit(resid, scale = TRUE)
gotheta(0.5, gin)

## End(Not run)
## Not run: 
library(vars)
data(VDW)
var1 <- VAR(VDW, p = 1, type = "const")
resid <- resid(var1)
gin <- goinit(resid, scale = TRUE)
gotheta(0.5, gin)

## End(Not run)

Class "Orthom": Orthogonal matrices

Description

This class defines an orthogonal matrix, which is characterized by $det(M) = 1$ and $M M' = I$ .

Objects from the Class

Objects can be created by calls of the form new("Orthom", ...). In addition the function UprodR returns an object of formal class Orthom.

Slots

M:: Object of class "matrix".

Methods

M: Returns the slot M of class Orthom.
print: print-method for objects of class Orthom.
show: show-method for objects of class Orthom.
t: Transpose of object@M.

Note

Objects are validated by validOrthomObject(). This function is utilised by validObject().

Author(s)

Bernhard Pfaff

Examples

showClass("Orthom")
showClass("Orthom")

Rotation matrix, 2-dimensional

Description

Given an angle $\theta$ whereby $\theta \in [0, \pi/2)$ the function Rd2 returns a 2-dimensional rotation matrix of Euler angles.

Usage

Rd2(theta)
Rd2(theta)

Arguments

theta

Numeric, angle in the interval $[0, \pi/2)$ .

Value

`R`	A 2-dimensional rotation matrix.

Author(s)

Bernhard Pfaff

Examples

Rd2(pi/3)
Rd2(pi/3)

Matching of Orthogonal Matrices for Cayley transforms

Description

This function matches an orthogonal matrix to the importance of the columns of the matrix to which it should be matched.

Usage

Umatch(from, to)
Umatch(from, to)

Arguments

`from`	Matrix: orthogonal
`to`	Matrix: orthogonal

Value

mat

Matched matrix.

Author(s)

Bernhard Pfaff

References

Boswijk, H. Peter and van der Weide, Roy (2009), Method of Moments Estimation of GO-GARCH Models, Working Paper, University of Amsterdam, Tinbergen Institute and World Bank.

Liebeck, H. and Osborne, A. (1991), The Generation of All Rational Orthogonal Matrices, The American Mathematical Monthly, 98 (2) (Feb. 1991), 131 – 133.

Returns a symmetric matrix from a vector

Description

This function returns the symmetric matrix $X$ from a vector that resulted from $v = vech(X)$ .

Usage

unvech(v)
unvech(v)

Arguments

`v`	Vector, numeric.

Details

The vector v must have length equal to $m * (m + 1) / 2$ , whereby $m$ is a dimension of the symmetric matrix $X_{m \times m}$ .

Value

`X`	Matrix, symmetric of order $m \times m$ .

Author(s)

Bernhard Pfaff

Examples

v <- c(1, 2, 3, 4, 5, 6)
unvech(v)
v <- c(1, 2, 3, 4, 5, 6)
unvech(v)

Creation of an orthogonal matrix

Description

This function returns an orthogonal matrix which results of the matrix products of rotation matrices.

Usage

UprodR(theta)
UprodR(theta)

Arguments

theta

Vector, of angles of the rotation matrices.

Details

The length of theta must be equal to $m * (m - 1) / 2$ , where $m$ is the dimension of the orthogonal matrix. The elements of theta must lie in the interval $[0, \pi/2)$ .

Value

result

Object of class Orthom.

Author(s)

Bernhard Pfaff

References

Vilenkin, N. Ja. (1968), Special Functions and the Theory of Group Representations, Translations of Mathematical Monographs, 22, American Math. Soc., Providence, Rhode Island, USA.

Examples

theta <- c(pi/3, pi/5, pi/7)
U <- UprodR(theta)
U
theta <- c(pi/3, pi/5, pi/7)
U <- UprodR(theta)
U

Validation function for objects of class Goinit

Description

This function validates objects of class Goinit.

Usage

validGoinitObject(object)
validGoinitObject(object)

Arguments

object

Object of class Goinit.

Details

This function is utilized by validObject(). It is tested whether object@V, object@P, object@Dsqr are square matrices; object@V coincides with the singular value decomposition.

Value

TRUE

Logical, TRUE if the object passes the validation, otherwise an informative error message is returned.

Author(s)

Bernhard Pfaff

Examples

data(VDW)
go <- goinit(VDW)
validObject(go)
data(VDW)
go <- goinit(VDW)
validObject(go)

Validation function for objects of class Orthom

Description

This function validates objects of class Orthom.

Usage

validOrthomObject(object)
validOrthomObject(object)

Arguments

object

Object of class Orthom.

Details

This function is utilized by validObject(). It is tested whether object@M is a square matrix, has $det(M) = 1$ and $MM' = I$ .

Value

TRUE

Logical, TRUE if the object passes the validation, otherwise an informative error message is returned.

Author(s)

Bernhard Pfaff

Examples

theta <- c(pi/3, pi/5, pi/7)
U <- UprodR(theta)
validObject(U)
theta <- c(pi/3, pi/5, pi/7)
U <- UprodR(theta)
validObject(U)

Dow Jones Industrial Average and Nasdaq stock indices

Description

The daily (log) returns of the Dow Jones Industrial Average and the NASDAQ composite, respectively. The daily observations start at the first of January, 1990, and end in October 2001.

Usage

data(VDW)data(VDW)

Format

A data frame with 3082 observations on the following 2 variables.

DJIA: Log-return of Dow Jones Industrial Average.
NASDAQ: Log-return of NASDAQ.

Details

This data set has been utilized in the source below and can be downloaded from the web-site of the Journal of Applied Econometrics (see link below).

Source

Van der Weide, Roy (2002), GO-GARCH: A Multivariate Generalized Orthogonal GARCH Model, Journal of Applied Econometrics, 17(5), 549 – 564.

References

http://qed.econ.queensu.ca/jae/2002-v17.5/van_der_weide/

Examples

data(VDW)
str(VDW)
data(VDW)
str(VDW)

Package 'gogarch'

Help Index

Dow Jones Industrial Average and Nasdaq stock indices

Description

Usage

Format

Details

Source

See Also

Examples

Stock prices transportation sector, oil and kerosene prices

Description

Usage

Format

Details

Source

References

Examples

Sector indices of the EURO STOXX 600

Description

Usage

Format

Source

References

Examples

Autocorrelations of a Matrix Process

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Methods for Function goest

Description

Details

Methods

Author(s)

See Also

Class "Goestica": GO-GARCH models estimated by fast ICA

Description

Objects from the Class

Slots

Extends

Methods

Author(s)

References

See Also

Class "Goestml": GO-GARCH models estimated by Maximum-Likelihood

Description

Objects from the Class

Slots

Extends

Methods

Author(s)

See Also

Class "Goestmm": Go-GARCH models estimated by Methods of Moments

Description

Objects from the Class

Slots

Extends

Methods

Author(s)

References

See Also

Class "Goestnls": GO-GARCH models estimated by Non-linear Least-Squares

Description

Objects from the Class

Slots

Extends

Methods

Author(s)

See Also

Specification and estimation of GO-GARCH models

Description

Usage

Arguments

Details

Value