Type:	Package
Title:	Generalized Autoregressive Score Models
Version:	0.3.4.1
Description:	Simulate, estimate and forecast using univariate and multivariate GAS models as described in Ardia et al. (2019) <doi:10.18637/jss.v088.i06>.
License:	GPL-3
BugReports:	https://github.com/LeopoldoCatania/GAS/issues
URL:	https://github.com/LeopoldoCatania/GAS
LazyData:	TRUE
Imports:	Rcpp (≥ 0.12.2), Rsolnp, MASS, xts, numDeriv, zoo, cubature
LinkingTo:	Rcpp, RcppArmadillo
Depends:	R (≥ 3.4.0), methods
Suggests:	testthat
NeedsCompilation:	yes
Packaged:	2024-08-19 08:41:50 UTC; hornik
Repository:	CRAN
Date/Publication:	2024-08-19 09:00:52 UTC
Author:	Leopoldo Catania [aut, cre], David Ardia [ctb], Kris Boudt [ctb]
Maintainer:	Leopoldo Catania <leopoldo.catania@econ.au.dk>

Generalized Autoregressive Score models in R

Description

The GAS package allows us to simulate, estimate and forecast using univariate and multivariate Generalized Autoregressive Score (GAS) models (also known as Dynamic Conditional Score (DCS) models), see e.g., Creal et. al. (2013) and Harvey (2013). A detailed implementation of the package functionalities are reported in Ardia et. al. (2018, 2019).

Details

The authors acknowledge Google for financial support via the Google Summer of Code 2016 project "GAS"; see https://summerofcode.withgoogle.com/archive/2016/projects/4537082387103744/.

Current limitations:

• The multivariate GAS model for N>4 does not report the exact update for the correlation parameters since the Jacobian of the hyperspherical coordinates transformation needs to be coded for the case N>4. The Jacobian for N>4 is replaced by the identity matrix.

Note

By using GAS you agree to the following rules:

• You must cite Ardia et al. (2019) in working papers and published papers that use GAS. Use citation("GAS").

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

• You assume all risk for the use of GAS.

Author(s)

Leopoldo Catania [aut,cre], Kris Boudt [ctb], David Ardia [ctb]

Maintainer: Leopoldo Catania <leopoldo.catania@econ.au.dk>

References

Ardia D, Boudt K and Catania L (2018). "Downside Risk Evaluation with the R Package GAS." R Journal, 10(2), 410-421. doi:10.32614/RJ-2018-064.

Ardia D, Boudt K and Catania L (2019). "Generalized Autoregressive Score Models in R: The GAS Package." Journal of Statistical Software, 88(6), 1-28. doi:10.18637/jss.v088.i06.

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

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

Backtest a series of one-step ahead density predictions.

Description

The BacktestDensity() function accepts an object of the class uGASRoll, and returns a list with two elements: (i) the averages Negative Log Score (NLS) and weighted Continuous Ranked Probability Score (wCRPS) introduced by Gneiting and Ranjan (2012), and (ii) their values at each point in time. The wCRPS is evaluated using 5 weight functions, see Details.

Usage

BacktestDensity(Roll, lower, upper, K = 1000, a = NULL, b = NULL)

Arguments

Details

The average Negative Log Score (NLS) is computed as the negative of the average of the log scores evaluated during the out-of-sample period. The average weighted Continuous Ranked Probability Score (wCRPS) is computed as the average of the wCRPS evaluated during the out-of-sample period, see Gneiting and Ranjan (2012).
The wCRPS is evaluated using Equation 16 of Gneiting and Ranjan (2012). The weights functions implemented are:

• w(z) = 1: Uniform,

• w(z) = \phi_{a,b}(z): Center,

• w(z) = 1 - \phi_{a,b}(z): Tails,

• w(z) = \Phi_{a,b}(z): Right tail,

• w(z) = 1 - \Phi_{a,b}(z): Left tail,

where \phi_{a,b}(z) and \Phi_{a,b}(z) are the pdf and cdf of a Gaussian distribution with mean a and standard deviation b, respectively. The label "Uniform" represents the case where equal emphasis is given to all the parts of the distribution.

Value

A list with elements: average, series. The element "average" is a named vector with the averages NLS and wCRSP. The element "series" is a list: the fist element, LS, contains the out-of-sample Log Score (not with the negative sign), the second element, WCRPS, contains a matrix with the wCRPS series. The columns of this matrix are named: "uniform", "center", "tails", "tail_r", "tail_l", which are associated with the wCRSP with emphasis on: Uniform, Center, Tails, Right tail and Left tail, respectively.

Author(s)

Leopoldo Catania

References

Gneiting T, Ranjan R (2011). "Comparing Density Forecasts using Threshold -and Quantile-Weighted Scoring Rules." Journal of Business & Economic Statistics, 29(3), 411-422. doi:10.1198/jbes.2010.08110.

Examples

## Not run: 
data("cpichg")

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

Roll = UniGASRoll(cpichg, GASSpec, ForecastLength = 50,
                  RefitEvery = 10, RefitWindow = c("moving"))

BackTest = BacktestDensity(Roll, lower = -100, upper = 100)

BackTest$average

## End(Not run)

Backtest Value at Risk (VaR)

Description

This function implements several backtesting procedures for the Value at Risk (VaR). These are: (i) The statistical tests of Kupiec (1995), Christoffesen (1998) and Engle and Manganelli (2004), (ii) The tick loss function detailed in Gonzalez-Rivera et al. (2004), the mean and max absolute loss used by McAleer and Da Veiga (2008) and the actual over expected exceedance ratio.

Usage

BacktestVaR(data, VaR, alpha, Lags = 4)

Arguments

Details

This function implements several backtesting procedure for the Value at Risk. The implemented statistical tests are:

• LRuc The unconditional coverage test of Kupiec (1995).

• LRcc The conditional coverage test of Christoffesen (1998).

• DQ The Dynamic Quantile test of Engle and Manganelli (2004).

The implemented VaR backtesting quantities are:

• AD mean and maximum absolute deviation between the observations and the quantiles as in McAleer and Da Veiga (2008).

• Loss Average quantile loss and quantile loss series as in Gonzalez-Rivera et al. (2004).

• AE Actual over Expected exceedance ratio.

Value

A list with elements: LRuc, LRcc, DQ, AD, AE.

Author(s)

Leopoldo Catania

References

Christoffersen PF (1998). "Evaluating Interval Rorecasts." International Economic Review, 39(4), 841-862.

Engle RF and Manganelli S. (2004). "CAViaR: Conditional Autoregressive Value at Risk by Regression Quantiles." Journal of Business & Economic Statistics, 22(4), 367-381. doi:10.1198/073500104000000370.

Gonzalez-Rivera G, Lee TH, and Mishra, S (2004). "Forecasting Volatility: A Reality Check Based on Option Pricing, Utility Function, Value-at-Risk, and Predictive Likelihood." International Journal of Forecasting, 20(4), 629-645. doi:10.1016/j.ijforecast.2003.10.003.

Kupiec PH (1995). "Techniques for Verifying the Accuracy of Risk Measurement Models." The Journal of Derivatives, 3(2), 73-84. doi:10.3905/jod.1995.407942

McAleer M and Da Veiga B (2008). "Forecasting Value-at-Risk with a Parsimonious Portfolio Spillover GARCH (PS-GARCH) Model." Journal of Forecasting, 27(1), 1-19. doi:10.1002/for.1049.

Examples

data("StockIndices")

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

FTSEMIB = StockIndices[, "FTSEMIB"]

InSampleData  = FTSEMIB[1:1500]
OutSampleData = FTSEMIB[1501:2404]

Fit = UniGASFit(GASSpec, InSampleData)

Forecast = UniGASFor(Fit, Roll = TRUE, out = OutSampleData)

alpha = 0.05

VaR = quantile(Forecast, alpha)

BackTest = BacktestVaR(OutSampleData, VaR, alpha)

Build confidence bands for the filtered parameters

Description

Build confidence bands for the filtered parameters sampling the coefficients from the asymptotic distribution as in Blasques et al. (2016).

Usage

ConfidenceBands(object, B = 10000, probs = c(0.01,0.1,0.9,0.99), ...)

Arguments

Details

This function implements the "In-Sample Simulation-Based Bands" Section 3.3 of Blasques et al. (2016).

Value

An object of the class array of dimension (T+1) x B x K, where T is the length of the time series, K is the number of parameters and B the number of draws. The first slice reports the estimated filtered parameters. The one step ahead prediction is also reported, this why T+1.

Author(s)

Leopoldo Catania

References

Blasques F, Koopman SJ, Lasak K, and Lucas, A (2016). "In-sample Confidence Bands and Out-of-Sample Forecast Bands for Time-Varying Parameters in Observation-Driven Models." International Journal of Forecasting, 32(3), 875-887. doi:10.1016/j.ijforecast.2016.04.002.

Examples

## Not run: 

# show the information of all the supported distributions
library("GAS")

data("cpichg")

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

Fit = UniGASFit(GASSpec, cpichg)

Bands = ConfidenceBands(Fit)

## End(Not run)

Information for the supported distributions

Description

Print the information regarding distributions supported in the GAS package.

Usage

DistInfo(DistLabel = NULL, N = 2, FULL = TRUE)

Arguments

. Run DistLabels() to see the labels of the currently implemented distributions.

Details

The information are printed in the console.

Author(s)

Leopoldo Catania

Examples

# show the information of all the supported distributions
library("GAS")

DistInfo()

Fissler and Ziegel (2016) (FZ) joint loss function for Value at Risk and Expected Shortfall.

Description

This function implements Fissler and Ziegel (2016) (FZ) joint loss function for Value at Risk and Expected Shortfall.

Usage

FZLoss(data, VaR, ES, alpha)

Arguments

Details

This function implements Fissler and Ziegel (2016) (FZ) joint loss function for Value at Risk and Expected Shortfall. The parameterization used is that of Patton et al. (2017) and is given by:

\frac{1}{\alpha ES_t^\alpha}I_t^\alpha(y_t - VaR_t^\alpha) + \frac{VaR_t^\alpha}{ES_t^\alpha} + \log{-ES_t^\alpha} - 1.

See also Fissler et al. (2015).

Value

A numeric vector containing the joing VaR and ES loss values.

Author(s)

Leopoldo Catania

References

Fissler, T., Ziegel, J.F., (2016). "Higher order elicitability and Osband's principle." The Annals of Statistics 44, 1680-1707.
Fissler, T., Ziegel, J.F., Tilmann, G. (2015). "Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting." arXiv preprint arXiv:1507.00244.
Patton, A. J., Ziegel, J.F., Chen, R. (2017). "Dynamic semiparametric models for expected shortfall (and Value-at-Risk)." arXiv preprint arXiv:1707.05108.

Examples

data("StockIndices")

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

FTSEMIB = StockIndices[, "FTSEMIB"]

InSampleData  = FTSEMIB[1:1500]
OutSampleData = FTSEMIB[1501:2404]

Fit = UniGASFit(GASSpec, InSampleData)

Forecast = UniGASFor(Fit, Roll = TRUE, out = OutSampleData)

alpha = 0.05

vVaR = quantile(Forecast, alpha)
vES  = ES(Forecast, alpha)

FZ = FZLoss(OutSampleData, vVaR, vES, alpha)

data: Goals scored by England against Scotland in international football matches.

Description

Number of goals scored by England against Scotland in international football matches. This is a 116 x 2 zoo object spanning the period 1872-1987 with a yearly frequency. The first column reports the number of goals scored by England against Scotland. The second column is a dummy variable equal 1 for matches played in England. This data set is taken from the Harvey (1989) pg 524.

Usage

data("Goals")

Format

A zoo object containing 116 x 2 observations.

Source

Harvey (1989) pg. 524

References

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge university press. https://cran.r-project.org/package=rugarch.

Estimate multivariate GAS models

Description

Estimate multivariate GAS models by Maximum Likelihood.

Usage

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

Arguments

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 indicates 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.

Value

An object of the class mGASFit.

Author(s)

Leopoldo Catania

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. doi:10.1002/jae.1279.

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

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

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

## Not run: 
# Specify an GAS model with multivariate Student-t
# conditional distribution and time-varying scales and correlations

library("GAS")

data("StockIndices")

GASSpec = MultiGASSpec(Dist = "mvt", ScalingType = "Identity",
                       GASPar = list(scale = TRUE, correlation = TRUE))

Fit = MultiGASFit(GASSpec, StockIndices)

Fit

## End(Not run)

Forecast with multivariate GAS models

Description

Forecast with multivariate GAS models. One-step ahead prediction of the conditional density is available in closed form. Multistep ahead prediction are performed by simulation as detailed in Blasques et al. (2016).

Usage

MultiGASFor(mGASFit, H = NULL, Roll = FALSE, out = NULL, B = 10000,
                     Bands = c(0.1, 0.15, 0.85, 0.9), ReturnDraws = FALSE)

Arguments

Value

An object of the class mGASFor

Author(s)

Leopoldo Catania

References

Blasques F, Koopman SJ, Lasak K, and Lucas, A (2016). "In-sample Confidence Bands and Out-of-Sample Forecast Bands for Time-Varying Parameters in Observation-Driven Models." International Journal of Forecasting, 32(3), 875-887. doi:10.1016/j.ijforecast.2016.04.002.

Examples

## Not run: 
# Specify a GAS model with multivatiate  Student-t conditional
# distribution and time-varying scales and correlations.

# Stock returns forecast

set.seed(123)

data("StockIndices")

mY = StockIndices[, 1:2]

# Specification mvt
GASSpec = MultiGASSpec(Dist = "mvt", ScalingType = "Identity",
                       GASPar = list(location = FALSE, scale = TRUE,
                                     correlation = TRUE, shape = FALSE))

# Perform H-step ahead forecast with confidence bands

# Estimation
Fit = MultiGASFit(GASSpec, mY)

# Forecast
Forecast  = MultiGASFor(Fit, H = 50)

Forecast

# Perform 1-Step ahead rolling forecast

InSampleData  = mY[1:1000, ]
OutSampleData = mY[1001:2404, ]

# Estimation
Fit = MultiGASFit(GASSpec, InSampleData)

Forecast  = MultiGASFor(Fit, Roll = TRUE, out = OutSampleData)

Forecast

## End(Not run)

Rolling forecast with multivariate GAS models

Description

One-step ahead rolling forecasts with model re-estimation. The function also reports several quantity for backtesting for point and density forecasts.

Usage

MultiGASRoll(data, GASSpec, ForecastLength = 500, Nstart = NULL,
             RefitEvery = 23, RefitWindow = c("moving", "recursive"),
             cluster = NULL, Compute.SE = FALSE, ...)

Arguments

Value

An object of the class mGASRoll

Author(s)

Leopoldo Catania

Examples

## Not run: 
# Specify a GAS model with Multivariate Student-t conditional
# distribution and time-varying scale and correlation parameters

# stock returns Forecast

data("StockIndices")

mY = StockIndices[, 1:2]

# Specification mvt
GASSpec = MultiGASSpec(Dist = "mvt", ScalingType = "Identity",
                       GASPar = list(location = FALSE, scale = TRUE,
                                     correlation = TRUE, shape = FALSE))

# Perform 1-step ahead rolling forecast with refit
library(parallel)

Roll = MultiGASRoll(mY, GASSpec, ForecastLength = 250,
                       RefitEvery = 100, RefitWindow = c("moving"))

Roll

## End(Not run)

Simulate multivariate GAS processes

Description

Simulate multivariate GAS processes.

Usage

  MultiGASSim(fit = NULL, T.sim = 1000, N = NULL,
              kappa = NULL, A = NULL, B = NULL, Dist = NULL, ScalingType = NULL)

Arguments

Details

The function permits to simulate from an estimated mGASFit object. If fit is not provided, the user can specify a GAS model via the additional arguments kappa, A, B, Dist and ScalingType.

All the information regarding the supported multivariate conditional distributions can be investigated using the DistInfo function. The model is specified as:

y_{t}\sim p(y|\theta_{t})

where \theta_{t} is the vector of parameters for the density p(y|.). Note that, \theta_{t} includes also those parameters that are not time-varying. The GAS recursion for \theta_{t} is:

\theta_{t} = \Lambda(\tilde{\theta}_{t})

\tilde{\theta}_{t}=\kappa + A*s_{t-1} + B*\tilde{\theta}_{t-1}

where h(.) is the mapping function (see MultiMapParameters) and \tilde{\theta}_{t} is the vector of reparametrised parameters. The process is initialized at \theta_{1}=(I - B)^{-1}\kappa, where \kappa is the Kappa vector. The vector s_{t} is the scaled score of p(y|.) with respect to \tilde{\theta}_{t}. See Ardia et. al. (2016a) for further details.

Value

An object of the class mGASSim

Author(s)

Leopoldo Catania

References

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

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

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

Examples

# Simulate from a GAS process with Multivariate Student-t conditional
# distribution, time-varying locations, scales, correlations
# and fixed shape parameter.
library("GAS")

set.seed(786)

T.sim = 1000  # Number of observations to simulate.
N     = 3     # Trivariate series.
Dist  = "mvt" # Conditional Multivariate Studen-t distribution.

# Build unconditional vector of reparametrised parameters.

Mu  = c(0.1, 0.2, 0.3) # Vector of location parameters (this is not transformed).
Phi = c(1.0, 1.2, 0.3) # Vector of scale parameters for the firs, second and third variables.

Rho = c(0.1, 0.2, 0.3) # This represents vec(R), where R is the correlation matrix.
                       # Note that is up to the user to ensure that vec(R) implies a
                       # proper correlation matrix.

Theta = c(Mu, Phi, Rho, 7) # Vector of parameters such that the degrees of freedom are 7.

kappa = MultiUnmapParameters(Theta, Dist, N)

A = matrix(0, length(kappa), length(kappa))

# Update scales and correlations, do not update locations and shape parameters.
diag(A) = c(0, 0, 0, 0.05, 0.01, 0.09, 0.01, 0.04, 0.07, 0)

B = matrix(0, length(kappa), length(kappa))

# Update scales and correlations, do not update locations and shape parameters.
diag(B) = c(0, 0, 0, 0.7, 0.7, 0.5, 0.94, 0.97, 0.92, 0)

Sim = MultiGASSim(fit = NULL, T.sim, N, kappa, A, B, Dist, ScalingType = "Identity")

Sim

Multivariate GAS specification

Description

Specify the conditional distribution, scaling mechanism and time-varying parameters for multivariate GAS models.

Usage

MultiGASSpec(Dist = "mvnorm", ScalingType = "Identity",
             GASPar = list(location = FALSE, scale = TRUE,
                           correlation = FALSE, shape = FALSE),
             ScalarParameters = TRUE)

Arguments

.

Details

All the information regarding the supported multivariate conditional distributions can be investigated using the DistInfo function.

Value

An object of the class mGASSpec

Author(s)

Leopoldo Catania

References

Creal D, Koopman SJ, Lucas A (2011). "A Dynamic Multivariate Heavy-Tailed Model for Time-Varying Volatilities and Correlations." Journal of Business & Economic Statistics, 29(4), 552-563. doi:10.1198/jbes.2011.10070.

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

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

Examples

# Specify a GAS model with multivariate Student-t
# conditional distribution and time-varying locations,
# scales and correlations  parameters but constant shape parameter.

library("GAS")

GASSpec = MultiGASSpec(Dist = "mvt", ScalingType = "Identity",
                       GASPar = list(location = TRUE, scale = TRUE,
                                     correlation = TRUE, shape = FALSE))

GASSpec

Mapping function for univariate distributions

Description

Map unrestricted vector of parameters into the proper space. This function transforms the parameters updated using the GAS recursion into their proper space.

Usage

MultiMapParameters(Theta_tilde, Dist, N)

Arguments

Details

The order of the parameters is generally: locations, scales, correlations, shape. When the distribution defined by Dist does not have, say, the shape parameter, this should be simply omitted. See also DistInfo for specific distributions.

Value

A numeric vector of parameters.

Author(s)

Leopoldo Catania

Examples

# Map unrestricted parameters for the Multivariate Student-t distribution with N=3
library("GAS")

N = 3

Dist = "mvt"

# Vector of location parameters (this is not transformed).
Mu_tilde  = c(0.1,0.2,0.3)

# Vector of unrestricted scales parameters such that
# the scales will be equal to 1.0, 1.2 and 0.3, for the first, second and
# third variables, respectively.
Phi_tilde = c(log(1.0), log(1.2), log(0.3))

# The vector c(0.1,0.2,0.3) represents vec(R),
# where R is the correlation matrix.
# Note that is up to the user to ensure that
# vec(R) implies a proper correlation matrix
# The function UnMapR_C transforms vec(R) in a vector of unrestricted parameters. It is
# the inverse of the hyperspherical coordinates transformration.

Rho_tilde = UnMapR_C(c(0.1,0.2,0.3), N)

# Vector of unconditional reparametrised parameters such that the
# degrees of freedom are 7.
#
# LowerNu() prints the lower bound numerical parameter for the degree
# of freedom, see help(LowerNu)
#

Theta_tilde = c(Mu_tilde, Phi_tilde , Rho_tilde, log(7 - LowerNu()))

Theta = MultiMapParameters(Theta_tilde, Dist, N)

Theta

Inverse of MultiMapParameters

Description

Transform distribution parameters into the unrestricted parameters. The unrestricted vector of parameters is updated using the GAS recursion.

Usage

MultiUnmapParameters(Theta, Dist, N)

Arguments

Details

The order of the parameters is generally: locations, scales, correlations, shape. When the distribution defined by Dist does not have, say, the shape parameter, this should be simply omitted. See also DistInfo for specific distributions.

Value

A numeric vector of parameters.

Author(s)

Leopoldo Catania

Examples

# Unmap parameters for the Multivariate Student-t distribution with N=3
library(GAS)

N = 3

Dist = "mvt"

# Vector of location parameters (this is not transformed).
Mu  = c(0.1, 0.2, 0.3)

# Vector of scales parameters for the firs, second and third variables.
Phi = c(1.0, 1.2, 0.3)

# This represents vec(R), where R is the correlation matrix.
# Note that is up to the user to ensure that vec(R) implies a proper correlation matrix
Rho = c(0.1, 0.2, 0.3)

# Vector of parameters such that the degrees of freedom are 7.
Theta = c(Mu, Phi, Rho, 7)

Theta_tilde = MultiUnmapParameters(Theta, Dist, N)

Theta_tilde

# It works
all(abs(MultiMapParameters(Theta_tilde, Dist, N) - Theta) < 1e-16)

Numerical bounds imposed in parameters transformation.

Description

Prints the numerical bounds.

Usage

UpperNu()
LowerNu()
UpperA()
LowerA()
UpperB()
LowerB()

Details

UpperNu() and LowerNu() print the numerical upper and lower bounds for the degree of freedom parameter of the Student–t distribution, std. (including also sstd and mvt).
UpperA() and LowerA() print the numerical upper and lower bounds for the score parameter in the GAS recursion. These bounds are applied to each diagonal element of the matrix A that premultiplies the scaled score.
UpperB() and LowerB() print the numerical upper and lower bounds for the autoregressive parameter in the GAS recursion. These bounds are applied to each diagonal element of the matrix B that premultiplies the past value of the parameters.

Value

Prints the numerical bounds.

Author(s)

Leopoldo Catania

Examples

UpperNu()
LowerNu()
UpperA()
LowerA()
UpperB()
LowerB()

Goodness of fit for conditional densities

Description

This function implements density goodness of fit procedure of Diebold et al. (1998).

Usage

PIT_test(U, G = 20, alpha = 0.05, plot = FALSE)

Arguments

Details

This function implements density goodness of fit procedure of Diebold et al. (1998). The test relays on the result that, if the series of estimated conditional distributions is the true one, then the PIT series evaluated accordingly are iid Unif(0, 1) distributed. The test of the iid Uniform(0, 1) assumption consists of two parts. The first part concerns the independent assumption, and it tests if all the conditional moments of the data, up to the fourth one, have been accounted for by the model, while the second part checks if the conditional distribution assumption is reliable by testing if the PITs are Uniform over the interval (0, 1). See also Jondeau and Rockinger (2006) and Vlaar and Palm (1993).

Value

A list with elements: (i) Hist and (ii) IID. The fist element Hist concerns the test of the unconditional assumption of uniformity of the PIT, it is a list with elements:

• test Statistic test.

• crit The critical value of the test.

• pvalue The pvalue of the test.

• hist The histogram, evaluated using the hist function.

• confidence Approximated asymptotic confidence level.

The second element IID concerns the iid assumption, it is a list with elements:

• test A named numeric vector with elements: test1, test2, test3, test4 representing the Lagrange Multiplier test for the first four conditional moments of the PITs.

• crit The critical value of the test.

• pvalue A named numeric vector with elements: pvalue1, pvalue2, pvalue3, pvalue4 representing the pvalues of the Lagrange Multiplier test for the first four conditional moments of the PITs.

Author(s)

Leopoldo Catania

References

Diebold FX, Gunther TA and Tay AS (1998). "Evaluating Density Forecasts with Applications to Financial Risk Management." International Economic Review, 39(4), 863-883.

Jondeau E and Rockinger M (2006). "The Copula-Garch Model of Conditional Dependencies: An International Stock Market Application." Journal of International Money and Finance, 25(5), 827-853. doi:10.1016/j.jimonfin.2006.04.007.

Vlaar PJ and Palm FC (1993). "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps." Journal of Business & Economic Statistics, 11(3), 351-360. doi:10.1080/07350015.1993.10509963.

Examples

data("StockIndices")

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

FTSEMIB = StockIndices[, "FTSEMIB"]

Fit = UniGASFit(GASSpec, FTSEMIB)

U = pit(Fit)

Test = PIT_test(U, G = 20, alpha = 0.05, plot = TRUE)

Data: Daily logarithmic returns in percentage points of the DAX, FTSEMIB and CAC40 from 2007-01-03 to 2016-06-24

Description

Daily logarithmic returns in percentage points of the DAX, FTSEMIB and CAC40 from 2007-01-03 to 2016-06-24 obtained from Yahoo.

Usage

data("StockIndices")

Format

A matrix object of dimension 2,445 x 3 containing the daily logarithmic returns in percentage points from 2007-01-03 to 2016-06-24. Missing values are simply removed.

References

Yahoo finance.

Estimate univariate GAS models

Description

Estimate univariate GAS models by Maximum Likelihood.

Usage

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

Arguments

.

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.

Value

An object of the class uGASFit

Author(s)

Leopoldo Catania

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. 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

## Not run: 
# 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")


## End(Not run)

Forecast with univariate GAS models

Description

Forecast with univariate GAS models. The one-step ahead prediction of the conditional density is available in closed form. The multi-step ahead prediction is performed by simulation as detailed in Blasques et al. (2016).

Usage

UniGASFor(uGASFit, H = NULL, Roll = FALSE, out = NULL, B = 10000,
                     Bands = c(0.1, 0.15, 0.85, 0.9), ReturnDraws = FALSE)

Arguments

Value

An object of the class uGASFor.

Author(s)

Leopoldo Catania

References

Blasques F, Koopman SJ, Lasak K, and Lucas, A (2016). "In-sample Confidence Bands and Out-of-Sample Forecast Bands for Time-Varying Parameters in Observation-Driven Models." International Journal of Forecasting, 32(3), 875-887. doi:10.1016/j.ijforecast.2016.04.002.

Examples

# Specify an univariate GAS model with Student-t
# conditional distribution and time-varying location, scale and shape parameter

# Inflation Forecast

set.seed(123)

data("cpichg")

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

# Perform H-step ahead forecast with confidence bands

Fit = UniGASFit(GASSpec, cpichg)
Forecast = UniGASFor(Fit, H = 12)

Forecast

# Perform 1-Step ahead rolling forecast

InsampleData = cpichg[1:250]
OutSampleData = cpichg[251:276]

Fit = UniGASFit(GASSpec, InsampleData)

Forecast = UniGASFor(Fit, Roll = TRUE, out = OutSampleData)

Forecast

Rolling forecast with univariate GAS models

Description

One-step ahead rolling forecasts with model re-estimation. The function also reports several quantity for backtesting for point and density forecasts.

Usage

UniGASRoll(data, GASSpec, ForecastLength = 500, Nstart = NULL,
           RefitEvery = 23, RefitWindow = c("moving", "recursive"),
           cluster = NULL, Compute.SE = FALSE, ...)

Arguments

Value

An object of the class uGASRoll.

Author(s)

Leopoldo Catania

Examples

# Specify an univariate GAS model with Student-t
# conditional distribution and time-varying location, scale and shape parameter

# Inflation Forecast

data("cpichg")
help(cpichg)

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

# Perform 1-step ahead rolling forecast with refit
library("parallel")

Roll = UniGASRoll(cpichg, GASSpec, ForecastLength = 50,
                  RefitEvery = 12, RefitWindow = c("moving"))


Roll

Simulate Univariate GAS processes

Description

Simulate Univariate GAS processes.

Usage

UniGASSim(fit = NULL, T.sim = 1000,
          kappa = NULL, A = NULL, B = NULL, Dist = NULL, ScalingType = NULL)

Arguments

Details

The function permits to simulate from an estimated uGASFit object. If fit is not provided, the user can specify a GAS model via the additional arguments kappa, A, B, Dist and ScalingType.

All the information regarding the supported univariate conditional distributions can be investigated using the DistInfo function. The model is specified as

y_{t}\sim p(y|\theta_{t})

, where \theta_{t} is the vector of parameters for the density p(y|.). Note that, \theta_{t} includes also those parameters that are not time-varying. The GAS recursion for \theta_{t} is

\theta_{t} = \Lambda(\tilde{\theta}_{t})

,

\tilde{\theta}_{t}=\kappa + A*s_{t-1} + B*\tilde{\theta}_{t-1}

, where \Lambda(.) is the mapping function (see UniMapParameters) and \tilde{\theta}_{t} is the vector of reparametrised parameters. The process is initialized at \theta_{1}=(I - B)^{-1}\kappa, where \kappa is the vKappa vector. The vector s_{t} is the scaled score of p(y|.) with respect to \tilde{\theta}_{t}. See Ardia et. al. (2016a) for further details.

Value

An object of the class uGASSim.

Author(s)

Leopoldo Catania

References

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

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

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

Examples

# Simulate from a GAS process with Student-t conditional
# distribution, time-varying location, scale and fixed shape parameter.

library(GAS)

set.seed(786)

T.sim  = 1000 # number of observations to simulate
Dist   = "std" # conditional Studen-t distribution


# vector of unconditional reparametrised parameters such that, the unconditional level of
# \eqn{\theta}_{t} is (0, 1.5 ,7), i.e. location = 0, scale = 1.5,
# degrees of freedom = 7.

kappa = c(0.0, log(1.5), log(7-2.01))

# in this way we specify that the shape parameter is constant while the score
# coefficients for the location and the scale
# parameters are 0.001 and 0.01, respectively.

A = matrix(c(0.001 , 0.0  , 0.0 ,
             0.0   , 0.01 , 0.0 ,
             0.0   , 0.0  , 0.0 ), 3, byrow = TRUE)

B = matrix(c(0.7 , 0.0 , 0.0 ,
              0.0 , 0.98, 0.0 ,
              0.0 , 0.0 , 0.0),3,byrow = TRUE) # Matrix of autoregressive parameters.

Sim = UniGASSim(fit = NULL, T.sim, kappa, A, B, Dist, ScalingType = "Identity")

Sim

Univariate GAS specification

Description

Specify the conditional distribution, scaling mechanism and time–varying parameters for univariate GAS models.

Usage

UniGASSpec(Dist = "norm", ScalingType = "Identity",
           GASPar = list(location = FALSE, scale = TRUE,
                         skewness = FALSE, shape = FALSE, shape2 = FALSE))

Arguments

Details

All the information regarding the supported univariate conditional distributions can be investigated using the DistInfo function.

Value

An object of the class uGASSpec.

Author(s)

Leopoldo Catania

References

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

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

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

Examples

# Specify an univariate GAS model with Student-t
# conditional distribution and time-varying location, scale and shape parameter
library("GAS")

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

GASSpec

Mapping function for univariate distributions

Description

Map unrestricted vector of parameters into the proper space. This function transforms the parameters updated using the GAS recursion into their proper space.

Usage

UniMapParameters(Theta_tilde, Dist)

Arguments

Details

The order of the parameters is generally: location, scale, skewness, shape, shape2. When the distribution defined by Dist does not have, say, the shape parameter, this should be simply omitted. See also DistInfo for specific distributions.

Value

A numeric vector of parameters.

Author(s)

Leopoldo Catania

Examples

# Map unrestricted parameters for the Student-t distribution.
library("GAS")

Dist = "std"

# Vector of unconditional reparametrised parameters such that,
# Theta = c(0, 1.5 ,7), i.e., location = 0, scale = 1.5,
# degrees of freedom = 7.

# LowerNu() prints the lower bound numerical parameter for the degree
# of freedom, see help(LowerNu).

Theta_tilde = c(0.1, log(1.5), log(7 - LowerNu()))

Theta = UniMapParameters(Theta_tilde, Dist)

Theta

Unmapping function for univariate distributions, i.e. inverse of UniMapParameters

Description

Transform distribution parameters into the unrestricted parameters. The unrestricted vector of parameters is updated using the GAS recursion.

Usage

UniUnmapParameters(Theta, Dist)

Arguments

Details

The order of the parameters is generally: location, scale, skewness, shape, shape2. When the distribution defined by Dist does not have, say, the shape parameter, this should be simply omitted. See also DistInfo for specific distributions.

Value

A numeric vector of parameters.

Author(s)

Leopoldo Catania

Examples

# Unmap parameters for the Student-t distribution
library("GAS")

Dist = "std"

# Vector of parameters such that,
# Theta = c(0, 1.5 ,7), i.e., location = 0, scale = 1.5,
# degrees of freedom = 7.

Theta = c(0.1, 1.5, 7)

Theta_tilde = UniUnmapParameters(Theta, Dist)

Theta_tilde

# It works.
all(abs(UniMapParameters(Theta_tilde, Dist) - Theta) < 1e-16)

Data: Quarterly logarithmic change in percentage points of the Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) from 1947-04-01 to 2016-05-01

Description

Quarterly logarithmic change in percentage points of the Consumer Price Index for All Urban Consumers: All Items (CPIAUCSL) from 1947-04-01 to 2016-05-01 available at https://fred.stlouisfed.org/series/CPIAUCSL.

Usage

data("cpichg")

Format

A xts object containing 276 observations from 1947-04-01 to 2016-05-01.

References

US. Bureau of Labor Statistics, Consumer Price Index for All Urban Consumers: All Items [CPIAUCSL], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/CPIAUCSL, June 24, 2016.

Distributions of the GAS package

Description

Density, distribution function, quantile function, random generator, moments, scores and information matrix of univariate and multivariate distributions of the GAS package.

Usage

ddist_Uni(y, Theta, Dist, log = FALSE)
pdist_Uni(q, Theta, Dist)
qdist_Uni(p, Theta, Dist)
rdist_Uni(Theta, Dist)
mdist_Uni(Theta, Dist)
Score_Uni(y, Theta, Dist)
IM_Uni(Theta, Dist)

ddist_Multi(y, Theta, Dist, log = FALSE)
rdist_Multi(Theta, N, Dist)
Score_Multi(y, Theta, Dist)

Arguments

Details

The function mdist_Uni returns a vector with four elements: mean, variance, skewness and kurtosis coefficients. The functions Score_Uni and IM_Uni returns the score and the Fisher information matrix for univariate distributions. The function Score_Multi returns the score for multivariate distributions. See DistInfo for the lists of supported distributions. These functions are not vectorized. ddist_Uni and ddist_Multi give the density, pdist_Uni gives the distribution function, qdist_Uni gives the quantile function, and rdist_Uni and rdist_Multi generate random deviates.

Value

1. numeric scalar for: ddist_Uni, pdist_Uni, qdist_Uni, rdist_Uni,

2. numeric vector for: Score_Uni, Score_Multi and rdist_Multi,

3. matrix for IM_Uni.

Author(s)

Leopoldo Catania

Examples

# Skew Student-t distribution

# log density
Theta = c("location" = 0, "scales" = 1, "skewness" = 1.2, "shape" = 7)

ddist_Uni(y = 0.5, Theta, "sstd", TRUE)

# probability
pdist_Uni(q = -1.69, Theta, "sstd")

#quantile
qdist_Uni(p = 0.05, Theta, "sstd")

#random generator
rdist_Uni(Theta, "sstd")

#moments
mdist_Uni(Theta, "sstd")

data: Dow Jones 30 Constituents Closing Value Log Return in percentage points

Description

This dataset is taken from the rugarch package of Ghalanos (2015). Returns are in percentage points.

Dow Jones 30 Constituents closing value log returns from 1987-03-16 to 2009-02-03 from Yahoo Finance. Note that AIG was replaced by KFT (Kraft Foods) on September 22, 2008. This is not reflected in this data set as that would bring the starting date of the data to 2001.

Usage

data("dji30ret")

Format

A data.frame containing 5,521x30 observations.

Source

Yahoo Finance

References

Ghalanos A (2015). "rugarch: Univariate GARCH models." https://cran.r-project.org/package=rugarch.

A wrapper to the optim function.

Description

This function is a wrapper to the standard optim optimizer with method = "BFGS".

Usage

 fn.optim(par0, data, GASSpec, FUN)

Arguments

Details

The following control parameters are used for control:

• trace = 0

• abstol = 1e-8

See the documentation of optim.

Value

It returns a named list with four 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 minumum, iii) hessian, a numeric matrix containing the Hessian matrix evaluated at the minimum of the negative log likelihood, iv) convergence a numeric element indicating the convergence results of optim.

Author(s)

Leopoldo Catania

See Also

help(optim)

A wrapper to the solnp function of the Rsolnp package of Ghalanos and Theussl (2016).

Description

This function is a wrapper to the solnp function of the Rsolnp package of Ghalanos and Theussl (2016).

Usage

 fn.solnp(par0, data, GASSpec, FUN)

Arguments

Details

The following control parameters are used: trace = 0, rho = 1, outer.iter = 400, inner.iter = 1800, delta = 1e-08, tol = 1e-08. See the documentation of solnp.

Value

It returns a named list with four 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 minumum, iii) hessian, a numeric matrix containing the Hessian matrix evaluated at the minimum of the negative log likelihood, and iv) convergence a numeric element indicating the convergence results of solnp.

Author(s)

Leopoldo Catania

References

Alexios Ghalanos and Stefan Theussl (2015). "Rsolnp: General Non-linear Optimization Using Augmented Lagrange Multiplier Method". R package version 1.16.

See Also

help(solnp)

Class for the Multivariate GAS fitted object

Description

Class for the multivariate GAS fitted object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Data:

Object of class list. Contains the user's data.

Estimates:

Object of class list. Contains: lParList list of estimated parameters, optimiser object delivered from the optimization function, StaticFit ML estimates for the constant model, Inference inferential results for the estimated parameters.

GASDyn:

Object of class list. Contains: the series of filtered dynamic (GASDyn$mTheta) for the time–varying parameters, the series of scaled scores (GASDyn$mInnovation), the series of unrestricted filtered parameters (GASDyn$mTheta_tilde), the series of log densities (GASDyn$vLLK), the log likelihood evaluated at its optimum value (GASDyn$dLLK)

ModelInfo:

Object of class list. Contains information about the GAS specification:

• Spec Object of the class uGASSpec containing the GAS specification.

• iT numeric Number of observation.

• elapsedTime Numeric elapsed time in seconds.

Methods

• show signature(object = 'mGASFit'): print object information.

• summary signature(object = 'mGASFit'): Show summary.

• plot signature(x='mGASFit', y='missing'): Plot filtered dynamic and other estimated quantities.

• getFilteredParameters signature(object = "mGASFit"): Extract filtered parameters.

• getObs signature(object = "mGASFit"): Extract original observations.

• coef signature(object = 'uGASFit'): Returns a named vector of estimated coefficients. Also accepts the additional logical argument do.list. If do.list = TRUE, estimated coefficients are organized in a list with arguments: vKappa the intercept vector, mA the A system matrix, mB the B system matrix. By default, do.list = FALSE.

• getMoments signature(object = "mGASFit"): Extract conditional moments.

• residuals signature(object = 'mGASFit'): Extract the residuals. Also accepts the additional logical argument standardize. If standardize = TRUE, residuals are standardized by cholesky of the filtered covariance matrix. By default standardize = FALSE.

• convergence signature(object = 'mGASFit'): Extract convergence information.

Author(s)

Leopoldo Catania

Class for the Multivariate GAS Forecast object

Description

Class for the multivariate GAS forecast object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Forecast:

Object of class list. Contains forecasts:

• PointForecast: matrix with parameters forecasts.

• Moments: list with centered moments forecasts. The first element contains a matrix with the predicted conditional means. The second element contains an array with the predicted conditional covariances.

• vLS: numeric Log Score (Predictive Log Likelihood).

Bands:

array with confidence bands parameters forecasts. Available only if Roll = TRUE.

Draws:

If ReturnsDraws = TRUE it is a iH x iB matrix of draws from the predictive distribution.

Info:

list with forecast information.

Data:

list with original data.

Methods

• show signature(object = "uGASFor"): Show summary.

• plot signature(x='uGASFor',y='missing'): Plot forecasted quantities.

• getForecast signature(object = "uGASFor"): Extract parameters forecast.

• getObs signature(object = "uGASFor"): Extract original observations.

• getMoments signature(object = "uGASFor"): Extract moments forecasts.

• LogScore signature(object = "uGASFor"): Extract Log Scores.

Author(s)

Leopoldo Catania

Class for the Multivariate GAS Rolling object

Description

Class for the multivariate GAS rolling object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Forecast:

Object of class list. Contains forecasts:

• PointForecast: matrix with parameters forecasts.

• Moments: list with centered moments forecasts. The first element contains a matrix with the predicted conditional means. The second element contains an array with the predicted conditional covariances.

• vLS: numeric Log Score (Predictive Log Likelihood).

Info:

list with forecast information.

Data:

list with original data.

Methods

• show signature(object = 'mGASRoll'): Show summary.

• plot signature(x = 'mGASRoll',y = 'missing'): Plot forecasted quantities.

• getForecast signature(object = 'mGASRoll'): Extract parameters forecast.

• getObs signature(object = 'mGASRoll'): Extract original observations.

• getMoments signature(object = 'mGASRoll'): Extract moments forecasts.

• LogScore signature(object = 'mGASRoll'): Extract Log Scores.

• residuals signature(object = 'mGASRoll'): Extract the forecast errors. Also accepts the additional logical argument standardize. If standardize = TRUE, forecast errors are standardized by cholesky of the forecast covariance matrix. By default standardize = FALSE.

• coef signature(object = 'mGASFit'): Returns a matrix of estimated coefficients. Each row of the matrix corresponds to a refit of the model during the forecast period according to the RefitEvery argument provided in the MultiGASRoll function. Also accepts the additional logical argument do.list. If do.list = TRUE, estimated coefficients are organized in a list of lists according according to the RefitEvery argument provided in the MultiGASRoll function. Each list is populated by three arguments: vKappa the intercept vector, mA the A system matrix, mB the B system matrix. By default, do.list = FALSE.

Author(s)

Leopoldo Catania

Class for Multivariate GAS Simulation

Description

Class for multivariate GAS model simulation.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

ModelInfo:

Object of class list. Contains information about the multivariate GAS specification:

• iT: numeric Time length of simulated observations.

• iN: numeric Cross sectional dimension.

• iK : numeric number of (possibly) time–varying parameters implied by the distributional assumption.

• vKappa numeric vector of unconditional level for the reparametrized vector of parameters.

• mA matrix of coefficients of dimension iK x iK that premultiply the conditional score in the GAS updating recursion.

• mB matrix of autoregressive coefficients of dimension iK x iK.

• Dist character label of the conditional distribution, see DistInfo

• ScalingType character representing the scaling mechanism for the conditional score, see DistInfo

GASDyn:

Object of class list. Contains: the series of simulated parameters (GASDyn$mTheta), the series of scaled scores (GASDyn$mInnovation), the series of unrestricted simulated parameters (GASDyn$mTheta_tilde), the series of log densities (GASDyn$vLLK), the log likelihood evaluated at its optimum value (GASDyn$dLLK)

Data:

Object of class matrix. Matrix of dimension iN x iT of simulated data

Methods

• show signature(object = 'mGASSim'): Show summary.

• plot signature(x = 'mGASSim',y = 'missing'): Plot simulated data and parameters.

• getFilteredParameters signature(object = 'mGASSim'): Extract simulated parameters.

• getObs signature(object = 'mGASSim'): Extract simulated observations

• coef signature(object = 'mGASSim'): Extract delivered coefficients

• getMoments signature(object = 'uGASFor'): Extract simulated moments.

Author(s)

Leopoldo Catania

Class for the Multivariate GAS model specification

Description

Class for the Multivariate GAS model specification.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Spec:

Object of class list. Contains information abount the multivariate GAS specification:

• Dist: character Containing the conditional distribution assumption.

• ScalingType: character indicating the scaling mechanism for the conditional score.

• GASPar: list with elements: location, scale, correlation, shape.

• ScalarParameters: logical indicates if the parameters of the locations, scales and correlations dynamic have to be scalars or a diagonal matrices.

Methods

• show signature(object = 'mGASSpec'): Show summary.

Author(s)

Leopoldo Catania

Plot output from an object of the from the GAS package.

Description

This method provides plot functionalities for the uGASFit, mGASFit, uGASSim, mGASSim, uGASFor, mGASFor, uGASRoll and mGASRoll objects defined in the GAS package.

Usage

  plot(x, y, ...)
  PlotMenu(x)

Arguments

Details

plot accepts the additional argument numeric argument which. By default which = NULL. If which is provided, plot() does not show the interactive menu and plot the corresponding option. The available options for each object class is printed by the function PlotMenu(x). By default which = NULL, that is, plot() display an interactive menu.

Value

Displays a plot of an object of class uGASFit, mGASFit, uGASSim, mGASSim, uGASFor, mGASFor, uGASRoll, mGASRoll.

Author(s)

Leopoldo Catania

Examples

## Not run: 
## Plot filtered estimates of a GAS model estimated on the
## Quarterly logarithmic change in percentage points of the Consumer Price Index data set (cpichg)
library("GAS")

data("cpichg")

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

Fit = UniGASFit(GASSpec, cpichg)

plot(Fit, which = 1)

## End(Not run)

Data: Daily logarithmic returns in percentage points of the S&P500 index from 1950-01-04 to 2016-06-24

Description

Daily logarithmic returns in percentage points of the S&P500 index from 1950-01-04 to 2016-06-24 obtained from yahoo finance.

Usage

data("sp500ret")

Format

A xts object of dimension 16,727 x 1 containing the daily logarithmic returns in percentage points from 1950-01-04 to 2016-06-24.

Source

Yahoo Finance

Data: SP500 Daily 5 minutes Realized Volatility from 2000-01-03 to 2000-01-10

Description

Oxford-Man Institute Daily 5 minutes Realized Volatility from 2000-01-03 to 2000-01-10 for the SP500 Index available at https://realized.oxford-man.ox.ac.uk/data.

Usage

data("sp500rv")

Format

A xts object containing 4,310 observations from 2000-01-03 to 2000-01-10.

References

https://realized.oxford-man.ox.ac.uk/data

Data from Bien et al (2011).

Description

From the readme.bnp.txt file in the JAE Data Archive available at http://qed.econ.queensu.ca/jae/2011-v26.4/bien-nolte-pohlmeier/:

The high-frequency data used in the paper come from the Trades and Quotation (TAQ) database. The data contains time-stamped quotations of Citicorp stock traded at the NYSE over the period from 20th February to 23rd February 2001.

In the study, 30-second bid and ask quote changes are constructed from the irregularly-spaced quote data. The study covers observations recorded from 9:35 EST until 16:00 EST.

The data contains 3080 rows and eight columns - in order:

1. year 2. month 3. day 4. time in number of seconds after the 9:35 EST 5. best ask quote 6. best bid quote 7. 30-second change of the ask quote in number of ticks 8. 30-second change of the bid quote in number of ticks.

Usage

data("tqdata")

Format

A data.frame object containing 3,080 observat. ions.

References

Bien K, Nolte, I, Pohlmeier W (2011). "An Inflated Multivariate Integer Count Hurdle Model: An Application to Bid and Ask Quote Dynamics". Journal of Applied Econometrics, 26(4), 669-707. doi:10.1002/jae.1122

Class for the univariate GAS fitted object

Description

Class for the univariate GAS fitted object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

ModelInfo:

Object of class list. Contains information about the GAS specification:

• Spec: object of the class uGASSpec containing the GAS specification.

• iT: numeric number of observation.

• elapsedTime: numeric elapsed Time in seconds.

GASDyn:

Object of class list. Contains: the series of filtered dynamic (GASDyn$mTheta) for the time–varying parameters, the series of scaled scores (GASDyn$mInnovation), the series of unrestricted filtered parameters (GASDyn$mTheta_tilde), the series of log densities (GASDyn$vLLK), the log likelihood evaluated at its optimum value (GASDyn$dLLK)

.

Estimates:

Object of class list. Contains: lParList list of estimated parameters, optimiser object delivered from the optimization function, StaticFit ML estimates for the constant model, Inference inferential results for the estimated parameters.

Data:

The user's data.

Testing:

Statistical tests results.

Methods

• show signature(object = 'uGASFit'): print object information.

• summary signature(object = 'uGASFit'): Show summary.

• plot signature(x = 'uGASFit',y = 'missing'): Plot filtered dynamic and other estimated quantities.

• getFilteredParameters signature(object = 'uGASFit'): Extract filtered parameters.

• getObs signature(object = 'uGASFit'): Extract original observations.

• coef signature(object = 'uGASFit'): Returns a named vector of estimated coefficients. Also accepts the additional logical argument do.list. If do.list = TRUE, estimated coefficients are organized in a list with arguments: vKappa the intercept vector, mA the A system matrix, mB the B system matrix. By default, do.list = FALSE.

• pit signature(object = 'uGASFit'): Extract Probability Integral Transformation.

• getMoments signature(object = 'uGASFit'): Extract conditional moments.

• residuals signature(object = 'uGASFit'): Extract the residuals. Also accepts the additional logical argument standardize. If standardize = TRUE, residuals are standardized by the filtered standard deviation. By default standardize = FALSE.

• convergence signature(object = 'uGASFit'): Extract convergence information.

• quantile signature(object = 'uGASSim'): Compute quantiles of the filtered estimated density at each point in time. It accepts the additional argument probs representing the vector of probabilities.

• ES signature(object = 'uGASSim'): Compute Expected Shortfall of the filtered estimated density at each point in time. It accepts the additional argument probs representing the vector of probabilities.

Author(s)

Leopoldo Catania

Class for the univariate GAS forecast object

Description

Class for the univariate GAS forecast object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Forecast:

Object of class list. Contains forecasts:

• PointForecast: matrix with parameters forecasts.

• Moments: matrix with centered moments forecasts.

• vLS: numeric Log Score (Predictive Log Likelihood).

• vU: numeric Out-of-sample Probability Integral Transformation (PIT).

Bands:

array with confidence bands parameters forecasts. Available only if Roll = TRUE.

Draws:

If ReturnsDraws = TRUE it is a iH x iB matrix of draws from the predictive distribution.

Info:

list with forecast information.

Data:

list with original data.

Methods

• show signature(object = 'uGASFor'): Show summary.

• plot signature(x = 'uGASFor', y = 'missing'): Plot forecasted quantities.

• getForecast signature(object = 'uGASFor'): Extract parameters forecast.

• getObs signature(object = 'uGASFor'): Extract original observations.

• pit signature(object = 'uGASFor'): Extract Probability Integral Transformation, only if Roll = TRUE.

• quantile signature(object = 'uGASFor'): Extract quantile forecasts. For multistep ahead prediction ES is computed by simulation and ReturnsDraws = TRUE should have been selected. It accepts the additional argument probs representing the vector of probabilities.

• ES signature(object = 'uGASFor'): Extract Expected Shortfall forecasts. For multistep ahead prediction ES is computed by simulation and ReturnsDraws = TRUE should have been selected. It accepts the additional argument probs representing the vector of probabilities.

• getMoments signature(object = 'uGASFor'): Extract moments forecasts.

• LogScore signature(object = 'uGASFor'): Extract Log Scores.

Author(s)

Leopoldo Catania

Class for the univariate GAS rolling object

Description

Class for the univariate GAS rolling object.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Forecast:

Object of class list. Contains forecasts:

• PointForecast: matrix with parameters forecasts.

• Moments: matrix with centered moments forecasts.

• vLS: numeric Log Score (Predictive Log Likelihood).

• vU: numeric Out-of-sample Probability Integral Transformation (PIT).

Info:

list with forecast information.

Data:

list with original data.

Testing:

Statistical tests results.

Methods

• show signature(object = 'uGASRoll'): Show summary.

• plot signature(x = 'uGASRoll', y = 'missing'): Plot forecasted quantities.

• getForecast signature(object = 'uGASRoll'): Extract parameters forecast.

• getObs signature(object = 'uGASRoll'): Extract original observations.

• pit signature(object = 'uGASRoll'): Extract Probability Integral Transformation, only if Roll = TRUE

• quantile signature(object = 'uGASRoll'): Extract quantile forecasts. It accepts the addional argument probs representing the vector of probabilities.

• ES signature(object = 'uGASRoll'): Extract Expected Shortfall forecasts. It accepts the addional argument probs representing the vector of probabilities.

• getMoments signature(object = 'uGASRoll'): Extract moments forecasts.

• LogScore signature(object = 'uGASRoll'): Extract Log Scores.

• residuals signature(object = 'uGASRoll'): Extract the forecast errors. Also accepts the additional logical argument standardize. If standardize = TRUE, forecast errors are standardized by the forecast standard deviation. By default standardize = FALSE.

• coef signature(object = 'uGASFit'): Returns a matrix of estimated coefficients. Each row of the matrix corresponds to a refit of the model during the forecast period according to the RefitEvery argument provided in the UniGASRoll function. Also accepts the additional logical argument do.list. If do.list = TRUE, estimated coefficients are organized in a list of lists according according to the RefitEvery argument provided in the UniGASRoll function. Each list is populated by three arguments: vKappa the intercept vector, mA the A system matrix, mB the B system matrix. By default, do.list = FALSE.

Author(s)

Leopoldo Catania

Class for Univariate GAS Simulation

Description

Class for Univariate GAS model Simulation.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

ModelInfo:

Object of class list. Contains information about the univariate GAS specification:

• iT numeric Time length of simulated observations.

• iK numeric Number of (possibly) time-varying parameters implied by the distributional assumption.

• vKappa numeric Vector of unconditional level for the reparametrised vector of parameters.

• mA matrix Of coefficients of dimension iK x iK that premultiply the conditional score in the GAS updating recursion.

• mB matrix Of autoregressive coefficients of dimension iK x iK.

• Dist character Label of the conditional distribution, see DistInfo

• ScalingType character Representing the scaling mechanism for the conditional score, see DistInfo.

GASDyn:

Object of class list. Contains: the series of simulated parameters (GASDyn$mTheta), the series of scaled scores (GASDyn$mInnovation), the series of unrestricted simulated parameters (GASDyn$mTheta_tilde), the series of log densities (GASDyn$vLLK), the log likelihood evaluated at its optimum value (GASDyn$dLLK).

Data:

Object of class numeric. Vector of length iT of simulated data.

Methods

• show signature(object = 'uGASSim'): Show summary.

• plot signature(x = 'uGASSim', y = 'missing'): Plot simulated data and parameters.

• getFilteredParameters signature(object = 'uGASSim'): Extract simulated parameters.

• getObs signature(object = 'uGASSim'): Extract simulated observations.

• coef signature(object = 'uGASSim'): Extract delivered coefficients.

• quantile signature(object = 'uGASSim'): Compute quantiles of the filtered simulated density at each point in time. It accepts the additional argument probs representing the vector of probabilities.

• ES signature(object = 'uGASSim'): Compute the Expected Shortfall of the filtered simulated density at each point in time. It accepts the additional argument probs representing the vector of probabilities.

Author(s)

Leopoldo Catania

Class for the univariate GAS model specification

Description

Class for the univariate GAS model specification.

Objects from the Class

A virtual Class: No objects may be created from it.

Slots

Spec:

Object of class list. Contains information about the univariate GAS specification:

• Dist: character containing the conditional distribution assumption.

• ScalingType: character indicating the scaling mechanism for the conditional score.

• iK: numeric representing the number of (possibly) time-varying parameters implied by the distributional assumption.

• GASPar list with elements: location, scale, skewness, shape, shape2.

Methods

• show signature(object = 'uGASSpec'): Show summary.

Author(s)

Leopoldo Catania

US Monthly Civilian Unemployment Rate (UNRATE) from 1948-01-01 to 2016-05-01

Description

From https://fred.stlouisfed.org/series/UNRATE: The unemployment rate represents the number of unemployed as a percentage of the labor force. Labor force data are restricted to people 16 years of age and older, who currently reside in 1 of the 50 states or the District of Columbia, who do not reside in institutions (e.g., penal and mental facilities, homes for the aged), and who are not on active duty in the Armed Forces.

Usage

data("usunp")

Format

A xts object containing 821 observations from 1948-01-01 to 2016-05-01.

References

US. Bureau of Labor Statistics, Civilian Unemployment Rate [UNRATE], retrieved from FRED, Federal Reserve Bank of St. Louis; https://fred.stlouisfed.org/series/UNRATE, July 2, 2016.