Package {tvGarchKF}

Automatic selection of tuning parameters for non-parametric tv-GARCH estimation

Description

This function provides an automatic procedure for selecting tuning parameters for non-parametric estimation of the tv-GARCH parameter curves. Generalized additive models (GAMs) are used as reference smoothers to estimate the effective degrees of freedom (EDF) associated with each time-varying parameter curve. The resulting EDF values are then used as a common complexity criterion to calibrate comparable tuning parameters for smoothing spline, local polynomial and B-spline estimators.

Usage

auto.tvGarchKF(
  object,
  data,
  k_grid = 3:8,
  bs_gam = "ps",
  gam_method = "REML",
  spar_grid = seq(0.3, 1.3, by = 0.01),
  span_grid = seq(0.5, 2.5, by = 0.05),
  loess_degree = c(1, 2, 2),
  bs_degree_grid = c(2, 2, 2),
  tol = 0.001
)

Arguments

Details

The automatic selection procedure first fits a GAM reference model to each initial time-varying parameter curve and extracts its effective degrees of freedom. These EDF values provide a common measure of curve complexity. For each non-parametric estimator, the function then searches over the corresponding tuning-parameter grid and selects the specification whose effective degrees of freedom most closely match those of the GAM reference model.

Consequently, the smoothing spline, local polynomial and B-spline estimators are calibrated to have approximately comparable levels of smoothness, despite being bases on different smoothing mechanisms.

Value

An object of class "auto.tvGarchKF" containing the fitted GAM reference models and the tuning parameters automatically selected for smoothing spline, local polynomial, and B-spline estimators.

The returned object includes the estimated effective degrees of freedom (edf), smoothing parameters (spar), local polynomial spans (span), polynomial and spline degrees (degree), basis dimensions (k), numbers of knots (nknots), and measures of agreement between the effective degrees of freedom of each non-parametric estimator and those of the corresponding GAM reference model.

The selected tuning parameters are organised by estimation method and time-varying parameter curve, c(u), \alpha(u) and \beta(u), and can subsequently be supplied to tvGarchNonParamFit() for non-parametric estimation of the time-varying GARCH parameter curves.

Examples

library("tvGarchKF")
n = 2000
seed = 321
type = c( "polynomial","trigonometric","polynomial")
model<-tvGarch_Sim(n=n,c=c(0.18,0.10),alpha=c(0.45,0.4),beta=c(0,1.05,-1.05,0.1),
                      type=type,trig="cos",arg="2*pi*u",seed=seed)
tv_curve<-tvParameter(model[,1],S=120,N=600,plot=TRUE,cex.axis=3,cex.main=3,cex.points=3,
                        mar=c(5,6,4,2),mfrow=c(1,3))
fit_par<-auto.tvGarchKF(object=tv_curve,data=model[,1],k_grid=3:8,bs_gam="ps",gam_method="REML",
                          spar_grid=seq(0.3,1.3,by=0.01),span_grid=seq(0.50,2.50,by=0.05),
                          loess_degree=c(1,2,2), bs_degree_grid=c(2,2,2),tol=1e-3)

Fitted method.

Description

Fitted method.

Usage

## S4 method for signature 'tvGarchKF'
fitted(object)

## S4 method for signature 'tvGarchKFNonParam'
fitted(object)

## S4 method for signature 'tvGarchKFNonParam_Fit'
fitted(object)

Arguments

Selective Stock Price Index

Description

The data covers the period from March 2000 to October 2014, totaling 3186 observations.

Usage

indipsa

Format

A time series object with 3186 elements

It's a stock market index that tracks the performance of a select group of Chilean companies.

Source

Yahoo Finance

Number of Observations

Description

Generic function to extract the number of observations from an object.

Usage

## S4 method for signature 'tvGarchKF'
nobs(object, ...)

Arguments

Value

Number of observations.

Plot method for tvGarchKF, tvGarchKFNonParam and tvGarchKFNonParam_Fit objects.

Description

This method provides graphical summaries for fitted tv-GARCH models. By default, it displays the estimated conditional volatility. Alternatively, setting which = "tvparameters" produces a three-panel plot of the estimated time-varying GARCH parameters c(u), \alpha(u) and \beta(u). The function also allows the user to superimpose fitted parametric curves and stationary GARCH benchmarks, and to customize the main graphical elements.

Usage

## S4 method for signature 'tvGarchKF,ANY'
plot(
  x,
  y,
  which = c("volatility", "tvparameters"),
  x.index = NULL,
  tvParameter = NULL,
  type = NULL,
  trig = NULL,
  arg = NULL,
  col.points = "black",
  col.fit = NULL,
  col.stationary = "black",
  lwd.y = 3,
  lwd.fit = 3,
  lwd.stationary = 2,
  lty.fit = 1,
  lty.stationary = 2,
  pch = 20,
  cex.points = 3,
  cex.axis = 2,
  cex.lab = 2,
  cex.main = 2,
  cex.legend = 1.2,
  mfrow = c(1, 3),
  mar = c(5, 5, 4, 2),
  xlab = "Time",
  ylab = "Log-Returns",
  main = NULL,
  legend = TRUE,
  legend.pos = "bottomleft",
  legend.label = NULL,
  bty = "n",
  las = 1,
  ann = FALSE,
  ...
)

## S4 method for signature 'tvGarchKFNonParam_Fit,ANY'
plot(
  x,
  y,
  which = c("volatility", "tvparameters"),
  x.index = NULL,
  tvParameter = NULL,
  type = NULL,
  trig = NULL,
  arg = NULL,
  col.points = "black",
  col.fit = NULL,
  col.stationary = "black",
  lwd.y = 3,
  lwd.fit = 3,
  lwd.stationary = 2,
  lty.fit = 1,
  lty.stationary = 2,
  pch = 20,
  cex.points = 3,
  cex.axis = 2,
  cex.lab = 2,
  cex.main = 2,
  cex.legend = 1.2,
  mfrow = c(1, 3),
  mar = c(5, 5, 4, 2),
  xlab = "Time",
  ylab = "Log-Returns",
  main = NULL,
  legend = TRUE,
  legend.pos = "bottomleft",
  legend.label = NULL,
  bty = "n",
  las = 1,
  ann = FALSE,
  ...
)

## S4 method for signature 'tvGarchKFNonParam,ANY'
plot(
  x,
  y,
  which = c("volatility", "tvparameters"),
  x.index = NULL,
  tvParameter = NULL,
  type = NULL,
  trig = NULL,
  arg = NULL,
  col.points = "black",
  col.fit = NULL,
  col.stationary = "black",
  lwd.y = 3,
  lwd.fit = 3,
  lwd.stationary = 2,
  lty.fit = 1,
  lty.stationary = 2,
  pch = 20,
  cex.points = 3,
  cex.axis = 2,
  cex.lab = 2,
  cex.main = 2,
  cex.legend = 1.2,
  mfrow = c(1, 3),
  mar = c(5, 5, 4, 2),
  xlab = "Time",
  ylab = "Log-Returns",
  main = NULL,
  legend = TRUE,
  legend.pos = "bottomleft",
  legend.label = NULL,
  bty = "n",
  las = 1,
  ann = FALSE,
  ...
)

Arguments

Examples

library("tvGarchKF")
n = 2000
seed = 321
type = c("polynomial","trigonometric","polynomial")
model<-tvGarch_Sim(n=n,c=c(0.18,0.10),alpha=c(0.45,0.4),beta=c(0,1.05,-1.05,0.1),
                      type=type,trig="cos",arg="2*pi*u",seed=seed)
tv_curve<-tvParameter(model[,1],S=120,N=600,plot=TRUE,cex.axis=3,cex.main=3,cex.points=3,
                        mar=c(5,6,4,2),mfrow=c(1,3))
# Parametric
c0     <- c(0.20,0.08)
alpha0 <- c(0.40,0.45)
beta0  <- c(0.00,0.95,-0.95,0.1)
fit<-tvGarchKalmanFit(data=model[,1],c=c0,alpha=alpha0,beta=beta0,type=type,trig="cos",arg="2*pi*u")
plot(fit, which = "volatility", mar = c(5, 4, 4, 2))
plot(fit,which = "tvparameters",tvParameter = tv_curve,
        type = c("linear", "trigonometric", "cubic"),trig = "cos",arg = "2*pi*u")

# Smoothing Spline
FitNonParam_ss<-tvGarchNonParamFit(data=model[,1],tv_curve,smoothing=list(spar=c(0.5,0.68,0.8),
                                    nknots=c(3,3,3)),plot=TRUE,method="Smoothing Spline")
plot(FitNonParam_ss,which = "tvparameters",tvParameter = tv_curve)

TVparameter_ss<-predict(FitNonParam_ss)
FitNonParamKF_ss<-tvGarchNonParamKF(model[, 1], TVparameter_ss)
plot(FitNonParamKF_ss)

# Local Polynomial
FitNonParam_lp<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                    localPoly=list(span=c(0.70,2.5,1.4),degree=c(1,2,2)),
                                    plot=TRUE,method="Local Polynomial")
plot(FitNonParam_lp,which = "tvparameters",tvParameter = tv_curve)

TVparameter_lp<-predict(FitNonParam_lp)
FitNonParamKF_lp<-tvGarchNonParamKF(model[, 1], TVparameter_lp)
plot(FitNonParamKF_lp)

# B-Spline
FitNonParam_bs<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                     bspline=list(degree=c(2,2,2),df=c(3,4,2)),
                                     plot=TRUE,method="B-Spline")
plot(FitNonParam_bs,which = "tvparameters",tvParameter = tv_curve)

TVparameter_bs<-predict(FitNonParam_bs)
FitNonParamKF_bs<-tvGarchNonParamKF(model[,1],TVparameter_bs)
plot(FitNonParamKF_bs)

Forecast conditional volatility from a tv-GARCH model.

Description

This method computes out-of-sample forecasts of the conditional volatility from a fitted tv-GARCH model using the Kalman Filter prediction recursions. Approximate confidence intervals for the forecasted volatility are constructed from the state prediction error variance.

Usage

## S4 method for signature 'tvGarchKF'
predict(
  object,
  n.ahead = 10,
  conf = 0.95,
  crit_val = NULL,
  interval = TRUE,
  Plot = TRUE,
  ...
)

## S4 method for signature 'tvGarchKFNonParam'
predict(object, n.ahead = 10, ...)

## S4 method for signature 'tvGarchKFNonParam_Fit'
predict(
  object,
  n.ahead = 10,
  conf = 0.95,
  crit_val = NULL,
  interval = TRUE,
  Plot = TRUE,
  ...
)

Arguments

Value

An object of class "tvGarchKF_predict" containing the forecast of the conditional volatility and the quantities required to asses their uncertainty. The returned object includes:

• n.ahead: Forecast horizon.

• nsample: Number of observations in the fitted sample.

• sigma_forecast: Forecasted conditional volatility (\widehat{\sigma}_{t+h}, for h = 1,\ldots, \texttt{n.ahead}).

• se_sigma: Standard errors of the forecasted conditional volatility.

• X_forecast: Predicted state values.

• P_forecast: State prediction error variances.

• method: Estimation method used in the fitted tv-GARCH model.

• conf: Confidence level used to construct intervals.

• crit_val: Critical values used to construct the confidence intervals.

• interval: A data frame containing the forecast horizon, forecasted volatility, standard error, and lower and upper confidence limits. This components is included only when interval = TRUE.

If Plot = TRUE, the forecasted volatility and its confidence intervals are also displayed.

Examples

library("tvGarchKF")
n = 2000
seed = 321
type = c("polynomial","trigonometric","polynomial")
model<-tvGarch_Sim(n=n,c=c(0.18,0.10),alpha=c(0.45,0.4),beta=c(0,1.05,-1.05,0.1),
                      type=type,trig="cos",arg="2*pi*u",seed=seed)
tv_curve<-tvParameter(model[,1],S=120,N=600,plot=TRUE,cex.axis=3,cex.main=3,cex.points=3,
                        mar=c(5,6,4,2),mfrow=c(1,3))
# Parametric
c0     <- c(0.20,0.08)
alpha0 <- c(0.40,0.45)
beta0  <- c(0.00,0.95,-0.95,0.1)
fit<-tvGarchKalmanFit(data=model[,1],c=c0,alpha=alpha0,beta=beta0,type=type,trig="cos",arg="2*pi*u")
predict(fit,n.ahead=30,Plot=TRUE,legend.cex=1.5,legend.pos="topleft")
# Smoothing Spline
FitNonParam_ss<-tvGarchNonParamFit(data=model[,1],tv_curve,smoothing=list(spar=c(0.5,0.68,0.8),
                                    nknots=c(3,3,3)),plot=TRUE,method="Smoothing Spline")
TVparameter_ss<-predict(FitNonParam_ss)
FitNonParamKF_ss<-tvGarchNonParamKF(model[, 1], TVparameter_ss)
predict(FitNonParamKF_ss, n.ahead = 20, Plot=TRUE)
# Local Polynomial
FitNonParam_lp<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                    localPoly=list(span=c(0.70,2.5,1.4),degree=c(1,2,2)),
                                    plot=TRUE,method="Local Polynomial")
TVparameter_lp<-predict(FitNonParam_lp)
FitNonParamKF_lp<-tvGarchNonParamKF(model[, 1], TVparameter_lp)
predict(FitNonParamKF_lp, n.ahead = 20, Plot=TRUE)
# B-Spline
FitNonParam_bs<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                     bspline=list(degree=c(2,2,2),df=c(3,4,2)),
                                     plot=TRUE,method="B-Spline")
TVparameter_bs<-predict(FitNonParam_bs)
FitNonParamKF_bs<-tvGarchNonParamKF(model[,1],TVparameter_bs)
predict(FitNonParamKF_bs,n.ahead=20,Plot=TRUE)

Residuals method.

Description

Residuals method.

Usage

## S4 method for signature 'tvGarchKF'
residuals(object, as.zoo = TRUE, ...)

Arguments

Summary method.

Description

Summary method.

Usage

## S4 method for signature 'tvGarchKF'
summary(object, ...)

## S4 method for signature 'tvGarchKFNonParam'
summary(object, ...)

Arguments

Clase tvGarchKF

Description

Define principal S4 class of parametrics methods.

Slots

call

User call.

coef

Estimation of coefficients c, \alpha and beta.

var.coef

Matrix of covariance.

residuals

Residuals

sigma2

Conditional variance.

data

A univariate time series, where could be an object of class "ts" or "zoo".

loglik

Log-likelihood.

aic

AIC value.

title

Title.

description

Description.

method

A character string specifying the parametric estimation approach used to the tv-parameter curves.

type

A charactertype=c("polynomial","NoLineal","trigonometric"), which indicates what type of function will be assigned to each tv-parameters.

pars

A vector with initial coefficients (c, \alpha and beta).

exponentes

A vector for the exponents for the "NoLinear" case.

trig

A character parameter indicating the trigonometric function to be used, "cos" or "sin".

arg

This is the value of the argument to be used in the trigonometric function.

stationary_coef

A vector with stationary coefficients (c, \alpha and beta).

Clase tvGarchKFNonParam

Description

Define principal S4 class of non-parametrics methods.

Slots

call

User call.

Check_condition

Stationarity condition

fitted.values

A data frame containing the estimated non-parametric curves c(u), \alpha(u) and \beta(u).

tvparameter

The original time-varying parameter structure used as input.

data

A univariate time series, where could be an object of class "ts" or \"zoo".

ParamStationary

Estimation of stationary coefficients.

Omega

Estimation of c parameter.

Alpha

Estimation of \alpha parameter.

Beta

Estimation of \beta parameter.

title

Title.

HattvParameters

Predicted values of the non-parametric parameter curves over the complete time domain.

Param_model

This variable it's the stationary model.

best_degree

The degree of the spline basis functions.

best_df

The degrees of freedom used to estimate the tv-parameter curves c(u), \alpha(u) and \beta(u).

local_used

A list containing local polynomial tuning parameters.

smoothing_used

A list containing the smoothing spline tuning parameters.

method

The selected non-parametric estimation method.

description

Description.

Clase tvGarchKFNonParam

Description

Define S4 class of non-parametrics methods applied Filter Kalman.

Slots

title

Title.

call

User call.

fitted.values

This value it's the conditional variance.

tvparameter

The original time-varying parameter structure used as input.

data

A univariate time series, where could be an object of class "ts" or "zoo".

Omega

Estimation of c parameter.

Alpha

Estimation of \alpha parameter.

Beta

Estimation of \beta parameter.

Param_model

This variable it's the stationary model.

method

The selected non-parametric estimation method.

description

Description.

Estimate a tv-GARCH(1,1) by using the Kalman Filter.

Description

This function estimates a time-varying GARCH(1,1) (tv-GARCH(1,1)) model using the Kalman Filter (KF). The model parameters are allowed to evolve smoothly over rescaled time (u=t/T) according to user-specified functional forms. The time-varying parameters c(t), \alpha(t) and \beta(t) determine the dynamics of the condition variance through c(u_t) +\alpha(u_t) r^2_{t-1} +\beta(u_t)\sigma_{t-1}, u_t = \frac{t}{T}. The functional forms of the time-varying parameters are specified by the user and the corresponding coefficients are estimated using the KF-based procedure.

Usage

tvGarchKalmanFit(
  data,
  c,
  alpha,
  beta,
  type = c("polynomial", "NoLineal", "trigonometric"),
  exponentes = NULL,
  trig = NULL,
  arg = NULL,
  predict = 0,
  trace.log = FALSE,
  title = NULL,
  description = NULL
)

Arguments

Details

The time-varying parameters can be represented using polynomial, nonlinear or trigonometric functional forms. For the polynomial specification,

c(t) = c_0 + c_1u + c_2 u^2 + \ldots + c_p u^p,

\alpha(t) = a_0 + a_1u + a_2 u^2 + \ldots + a_p u^p,

\beta(t) = b_0 + b_1u + b_2 u^2 + \ldots + b_p u^p,

where u=t/T, with t=1, 2, \ldots, T. For the nonlinear specification,

c(t) = c_0 + \sum_{j=1}^k c_j u^{q_{c,j}},

\alpha(t) = a_0 + \sum_{j=1}^k a_j u^{q_{\alpha,j}},

\beta(t) = b_0 + \sum_{j=1}^k b_j u^{q_{\beta,j}},

where the exponents q_{c,j}, q_{\alpha,j} and q_{\beta,j} are specified through the exponentes argument For the trigonometric specification,

c(t) = c_0 + c_1 g(u),

\alpha(t) = a_0 + a_1 g(u),

\beta(t) = b_0 + b_1 g(u),

where (g(u)) denotes the trigonometric function specified through trig, with its argument defined by arg.

Value

A list of class "tvGarchKF".

Examples

ipsa<-diff(log(indipsa))
c <- c(0.05,0.05)
alpha <- c(0.05,0.05)
beta <- c(0.05,0.05)
type_fit <- c("trigonometric","trigonometric","trigonometric")
fit<-tvGarchKalmanFit(ipsa,c=c,alpha=alpha,beta=beta,type=type_fit,trig="cos",arg="3*(1-log(u))")

Models tv-Garch Filter Kalman LogLikehood.

Description

It is the function to use in the process to fit coefficients.

Usage

tvGarchKalmanLoglike(
  x,
  data,
  c,
  alpha,
  beta,
  nsample = length(data),
  type = c("polynomial", "NoLineal", "trigonometric"),
  exponentes = NULL,
  trig = NULL,
  arg = NULL,
  predict,
  trace.log = FALSE
)

Arguments

Value

Value of loglike in model.

Models tv-Garch Filter Kalman print outputs.

Description

This function is designed to print the outputs of the tv-Garch model, which include the returns, conditional variance, log-likelihood value, and mean squared error (MSE).

Usage

tvGarchKalmanPrint(
  x,
  data,
  c,
  alpha,
  beta,
  nsample = length(data),
  type = c("polynomial", "NoLineal", "trigonometric"),
  exponentes,
  trig,
  arg,
  trace.log = FALSE,
  predict
)

Arguments

Value

A data frame containing the following columns:

• X: State vector of Kalman equations.

• Fm: Value of MSE

• sigma: Conditional variance.

• loglike: Value of the loglike.

Examples

data(ipsa)
ipsa<-diff(log(indipsa))
c<-c(0.05,0.05)
alpha<-c(0.05,0.05)
beta<-c(0.05,0.05)
type_fit<-c("trigonometric","trigonometric","trigonometric")
fit<-tvGarchKalmanFit(ipsa,c=c,alpha=alpha,beta=beta,type=type_fit,trig="cos",arg="3*(1-log(u))")
arg<-"3*(1-log(u))"
model<-tvGarchKalmanPrint(fit@coef,ipsa,c=c,alpha=alpha,beta=beta,type=type_fit,trig="cos",arg=arg)
plot(ipsa,ylab="",xlim=c(2000,2015))
lines(ts(model$sigma, star=2000, freq=225), col="red", lwd=2)
lines(ts(model$sigma*(-1), star=2000, freq=225), col="red", lwd=2)

EStimate tv-GARCH(1,1) parameter curves using non-parametric smoothing methods.

Description

Given a set of initial local estimates of the time-varying parameters, this function applies smoothing spline, local polynomial, or B-spline to obtain smooth estimates tv-GARCH(1,1) parameter curves (c(u), \alpha(u), \beta(u)). The tuning parameters can be specified directly by the user or selected automatically using auto.tvGarchKF().

Usage

tvGarchNonParamFit(
  data,
  tvparameter,
  smoothing = list(spar = c(0.5, 0.5, 0.5), nknots = c(12L, 12L, 12L), methods = NULL,
    control.spar = list(lower = 0.5, upper = 1.2), max_knots = NULL),
  bspline = list(degree = 1:3, df = c(3, 3, 4)),
  localPoly = list(span = seq(0.1, 0.45, by = 0.05), degree = 1:2),
  plot = TRUE,
  method = c("Smoothing Spline", "Local Polynomial", "B-Spline"),
  title = NULL,
  description = NULL
)

Arguments

Value

An object of class "tvGarchKFNonParam" containing the non-parametric estimates of the tv-GARCH parameter curves. The returned object includes:

• tvparameter: The initial estimates of the time-varying GARCH parameters used as input.

• fitted.values: A data frame containing the fitted non-parametric curves c(u), \alpha(u) and \beta(u) at the observed time points.

• HattvParameters: The estimated non-parametric curves evaluated over the complete time domain.

• method: The selected non-parametric smoothing method.

• C, Alpha, Beta: Fitted objects associated with the non-parametric estimation of (c(u), \alpha(u), \beta(u)), respectively.

• Method-specific tuning parameters: The tuning parameters associated with the selected non-parametric method, such as spar, nknots, span, degree or df.

• Check_condition: Diagnostic information for the local stationarity condition (\alpha(u) + \beta(u) < 1), including the maximum value of (\alpha(u)+\beta(u)) over the observed time domain and an indicator of whether the condition is satisfied.

The resulting object can be used with predict() to obtain the estimated time-varying parameter curves required as input to tvGarchNonParamKF().

Examples

library("tvGarchKF")
n = 2000
seed = 321
type = c("polynomial","trigonometric","polynomial")
model<-tvGarch_Sim(n=n,c=c(0.18,0.10),alpha=c(0.45,0.4),beta=c(0,1.05,-1.05,0.1),
                      type=type,trig="cos",arg="2*pi*u",seed=seed)
tv_curve<-tvParameter(model[,1],S=120,N=600,plot=TRUE,cex.axis=3,cex.main=3,cex.points=3,
                        mar=c(5,6,4,2),mfrow=c(1,3))
fit_par<-auto.tvGarchKF(object=tv_curve,data=model[,1],k_grid=3:8,bs_gam="ps",gam_method="REML",
                          spar_grid=seq(0.3,1.3,by=0.01),span_grid=seq(0.50,2.50,by=0.05),
                          loess_degree=c(1,2,2), bs_degree_grid=c(2,2,2),tol=1e-3)
# Smoothing Spline
FitNonParam_ss<-tvGarchNonParamFit(data=model[,1],tv_curve,smoothing=list(spar=c(0.5,0.68,0.8),
                                    nknots=c(3,3,3)),plot=TRUE,method="Smoothing Spline")
# Local Polynomial
FitNonParam_lp<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                    localPoly=list(span=c(0.70,2.5,1.4),degree=c(1,2,2)),
                                    plot=TRUE,method="Local Polynomial")
# B-Spline
FitNonParam_bs<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                     bspline=list(degree=c(2,2,2),df=c(3,4,2)),
                                     plot=TRUE,method="B-Spline")

Kalman Filter estimation with non-parametric tv-GARCH parameters.

Description

This function applies the Kalman Filter (KF) to a tv-GARCH(1,1) model using non-parametric estimates ofthe time-varying parameter curves c(u), \alpha(u) and \beta(u). These curves are typically obtained by applying predict() to an object returned by tvGarchNonParamFit(). The resulting parameter estimates are then incorporated into the KF recursion to update the state vector (X_{t+1} and obtain estimates of the conditional variance (\widehat{\sigma}_{t+1}^2).

Usage

tvGarchNonParamKF(data, TVparameter, title = NULL, description = NULL)

Arguments

Value

An object of class "tvGarchKFNonParam_Fit" containing the Kalman-filter-based estimates of the conditional variance obtained using non-parametric estimates of the tv-GARCH parameters. The returned object includes the fitted conditional variance, the original time series, the estimated time-varying parameter curve, the selected non-parametric smoothing method and the corresponding fitted model objects.

The returned objectinherits from the general tv-GARCH Kalman Filter class and can therefore be used with the S3/S4 methods provided by the package. In particular, fitted() extracts the fitted conditional variance, plot() displays either the estimated conditional volatility forecasts and their corresponding confidence intervals and summary() provides a summary of the fitted model and the selected non-parametric specification.

Examples

library("tvGarchKF")
n = 2000
seed = 321
type = c("polynomial","trigonometric","polynomial")
model<-tvGarch_Sim(n=n,c=c(0.18,0.10),alpha=c(0.45,0.4),beta=c(0,1.05,-1.05,0.1),
                      type=type,trig="cos",arg="2*pi*u",seed=seed)
tv_curve<-tvParameter(model[,1],S=120,N=600,plot=TRUE,cex.axis=3,cex.main=3,cex.points=3,
                        mar=c(5,6,4,2),mfrow=c(1,3))
fit_par<-auto.tvGarchKF(object=tv_curve,data=model[,1],k_grid=3:8,bs_gam="ps",gam_method="REML",
                          spar_grid=seq(0.3,1.3,by=0.01),span_grid=seq(0.50,2.50,by=0.05),
                          loess_degree=c(1,2,2), bs_degree_grid=c(2,2,2),tol=1e-3)
# Smoothing Spline
FitNonParam_ss<-tvGarchNonParamFit(data=model[,1],tv_curve,smoothing=list(spar=c(0.5,0.68,0.8),
                                    nknots=c(3,3,3)),plot=TRUE,method="Smoothing Spline")
TVparameter_ss<-predict(FitNonParam_ss)
FitNonParamKF_ss<-tvGarchNonParamKF(model[, 1], TVparameter_ss)
# Local Polynomial
FitNonParam_lp<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                    localPoly=list(span=c(0.70,2.5,1.4),degree=c(1,2,2)),
                                    plot=TRUE,method="Local Polynomial")
TVparameter_lp<-predict(FitNonParam_lp)
FitNonParamKF_lp<-tvGarchNonParamKF(model[, 1], TVparameter_lp)
# B-Spline
FitNonParam_bs<-tvGarchNonParamFit(data=model[,1],tvparameter=tv_curve,
                                     bspline=list(degree=c(2,2,2),df=c(3,4,2)),
                                     plot=TRUE,method="B-Spline")
TVparameter_bs<-predict(FitNonParam_bs)
FitNonParamKF_bs<-tvGarchNonParamKF(model[, 1],TVparameter_bs)

Generating Simulations using a tv-Garch Model

Description

Simulate from a tv-Garch(1,1) model, using specific functions for the tv-parameters.

Usage

tvGarch_Sim(
  n,
  c,
  alpha,
  beta,
  type = c("polynomial", "NoLineal", "trigonometric"),
  exponentes = NULL,
  trig = NULL,
  arg = NULL,
  seed = NULL
)

Arguments

Value

An object of class "zoo" with two components: the first component represents returns "x", while the second component denotes conditional variance "sigma".

Examples

## Simulate from a tv-GARCH(1,1) model lineal:
alpha_sim <- c(0.2, 0.2)
beta_sim <- c(0.45, 0.5, -0.85)
type_sim <- c("polynomial","polynomial","polynomial")
Sim1 <- tvGarch_Sim(n = 6000, c = 0.1, alpha = alpha_sim, beta = beta_sim, type = type_sim)
plot(Sim1[,1], type="l", main="Simulated tvGARCH(1, 1) process",
    ylim=c(-max(Sim1[,2]), max(Sim1[,2])))
lines(Sim1[,2], type="l", col="red")
legend("topright",legend=c("tvGARCH(1,1)",expression(sigma(u))),
      col=c("black","red"),lty=1,bty="n",lwd=1)
## Simulate from a tv-GARCH(1,1) model non linear:
alpha_sim2 <- c(0.75, 0.08)
beta_sim2 <- c(0.05, 0.03, 0.06)
type_sim2 <- c("polynomial","polynomial","NoLineal")
expo <- c(0, 1, 1/2)
Sim2<-tvGarch_Sim(n=6000,c=0.05,alpha=alpha_sim2,beta=beta_sim2,type=type_sim2,exponentes=expo)
plot(Sim2[,1], type="l", main="Simulated tvGARCH(1, 1) process",
     ylim=c(-max(Sim2[,2]),max(Sim2[,2])))
lines(Sim2[,2], type="l", col="red")
legend("topright",legend=c("tvGARCH(1,1)",expression(sigma(u))),
       col=c("black","red"),lty=1,bty="n",lwd=1)

Estimate the local structure of tv-GARCH(1,1) parameters.

Description

This function performs an exploratory analysis of the dynamic structure of the time-varying GARCH(1,1) parameters. Specifically, the observation domain \{1, \ldots,T\} is partitioned into M overlapping blocks, each of length N, with a constant shift of size S between consecutive blocks. These quantities satisfy T = S(M-1)+N. The midpoint of the j-th block, for j=1,\ldots,M, is defined as t_j=S(j-1)+N/2. For each block, a stationary GARCH(1,1) model is fitted using the obervations contained in that block. The resulting sequence of local estimates provides an empirical trajectory of the time-varying parameters (c(u)), \alpha(u) and \beta(u). These trajectories can can be used as an exploratory tool for identifying suitable functional forms or as initial estimates for subsequent parametric or non-parametric estimation of the tv-GARCH parameter curves.

Usage

tvParameter(
  data,
  S,
  N,
  plot = TRUE,
  title = NULL,
  description = NULL,
  cex.axis = 2,
  cex.main = 2.5,
  cex.points = 2,
  mar = c(5, 6, 4, 2),
  mfrow = c(1, 3)
)

Arguments

Value

Data frame who contains c, alpha, beta of GARCH(1,1) model and midpoint each block.

Examples

ipsa<-diff(log(indipsa))*100
S = 100
N = 800
tv <- tvParameter(ipsa,S,N)

Vcov method.

Description

Vcov method.

Usage

## S4 method for signature 'tvGarchKF'
vcov(object, ...)

Arguments