Package 'GARCH.X' reference manual

Title:	Estimation and Exogenous Covariate Selection for ARCH-m(X), Additive ARCH-m(x), and GARCH-X Models
Description:	Estimates the parameters and nonparametric functions of an ARCH-m(X) model with exogenous covariates, estimates the parameters and nonparametric functions of an Additive ARCH-m(X) model with exogenous covariates, estimates the parameters of a GARCH-X model with exogenous covariates, performs hypothesis tests for the covariates returning the p-values, and performs stepwise variable selection on the exogenous covariates, and uses False Discovery Rate p-value corrections to select the exogenous variables.
Authors:	Adriano Zambom [aut, cre], Vincent Alegrete [aut], Elijah Sagaran [aut], Avni Israni [aut]
Maintainer:	Adriano Zambom <[email protected]>
License:	GPL (>= 2)
Version:	2.0
Built:	2026-05-24 06:08:16 UTC
Source:	https://github.com/cran/GARCH.X

Fitting an ARCH-m(X) model

Description

Fits an ARCH-m(X) model with given data and estimates the alphas and the nonparametric function m(X).

Usage

ARCHmX(
  eps, 
  X, 
  p, 
  delta=2,
  covariates="all",
  hypothesisTest=TRUE,
  bandwidth = "plugin", 
  k.choice = "plugin"
)
ARCHmX(
  eps, 
  X, 
  p, 
  delta=2,
  covariates="all",
  hypothesisTest=TRUE,
  bandwidth = "plugin", 
  k.choice = "plugin"
)

Arguments

eps

Time series

X

Matrix with exogenous covariates where the number of rows is equal to the length of eps and columns represent the covariates.

p

Order of the ARCH-m(X) model. Value of p cannot be 0.

delta

Value of the power of the main time series for a POWER-GARCH, default is 2.

covariates

Instead of slicing X when building partial models, you can specify a vector of column indices. Defaults to "all".

hypothesisTest

Whether to perform covariate hypothesis testing, default is TRUE.

bandwidth

Choice of bandwidth for the Nadaraya-Watson Kernel regression estimation. Options are "plugin" or "CV". Default is "plugin".

k.choice

Choice of window size k for the ANOVA hypotghesis test. Options are "plugin" or "CV". Default is "plugin".

Details

Performs semiparametric estimation of $\alpha_1,\cdots,\alpha_p$ and $m(X_{t})$ for the ARCH-m(X) model

$\epsilon_t=z_t\sigma_t,$

$\sigma_t^2=\sum_{j=1}^p\alpha_j\epsilon_{t-j}^2+m(\mathbf{X}_{t-1}).$

Under the model, it can be shown that

$\mathbb{E}\left[\epsilon_t^2-\sum_{j=1}^p\alpha_j\epsilon_{t-j}^2\right]=m(\mathbf{X}_{t-1}).$

A Nadaraya-Watson kernel is used to estimate $m(\mathbf{x}_{t-1})$ as

$\hat{m}(\mathbf{x}_{t-1})=\hat{m}(\mathbf{x}_{t-1},\mathbf{\alpha})=\sum_{\ell=p+1}^{T}W_{\ell}(\mathbf{x}_{t-1})\left(\epsilon_\ell^2-\sum_{j=1}^p\alpha_j\epsilon_{\ell-j}^2\right).$

This gives rise to a residual sum of squares

$\text{RSS}=\sum_{t=p+1}^T\left(\tilde{\epsilon}_{t,0}^2-\sum_{k=1}^p\alpha_k\tilde{\epsilon}_{t,k}^2\right)^2=\|\tilde{\mathbf{Y}}-\tilde{\mathbf{X}}\alpha\|^2$

where

$\tilde{\epsilon}_{t,k}^2=\epsilon_{t-k}^2-\sum_{\ell=p+1}^TW_\ell(\mathbf{x}_{t-1})\epsilon_{\ell-k}^2.$

Non-negative least squares regression is used to obtain the final estimates of $\alpha_1,\cdots,\alpha_p$ .

No variable selection is done in this function, but an ANOVA-type hypothesis test may be performed for the significance of each covariate. This is enabled by default.

Value

An object of class ARCHmX

Examples

# We simulate a time series as in the simulate_ARCHmX example.
set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)

p <- length(alpha)

# Then we fit the model
model <- ARCHmX(eps = data$eps, X = data$X, p=p)

# Model summary shows Covariate 1 significant at the <0.001 level. 
summary(model)
# We simulate a time series as in the simulate_ARCHmX example.
set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)

p <- length(alpha)

# Then we fit the model
model <- ARCHmX(eps = data$eps, X = data$X, p=p)

# Model summary shows Covariate 1 significant at the <0.001 level. 
summary(model)

Creating and fitting an Additive ARCH-m(X) model for variable selection

Description

Fits an Additive ARCH-m(X) model with given data and estimates the alphas and nonparametric functions m(X).

Usage

ARCHmXAdditive(
  eps, 
  X, 
  p, 
  delta=2,
  covariates="all",
  method="smooth_means_Positive_Constraint_Exp",
  train_end = NULL, 
  n_burn = 10, 
  max_outer_iter = 30, 
  tol = 1e-4,
  hypothesisTest=TRUE,
  k.choice = "plugin"
)
ARCHmXAdditive(
  eps, 
  X, 
  p, 
  delta=2,
  covariates="all",
  method="smooth_means_Positive_Constraint_Exp",
  train_end = NULL, 
  n_burn = 10, 
  max_outer_iter = 30, 
  tol = 1e-4,
  hypothesisTest=TRUE,
  k.choice = "plugin"
)

Arguments

eps

Time series.

X

Matrix with exogenous covariates where the number of rows is equal to the length of eps and columns represent the covariates

p

Order of the Additive ARCH-m(X) model. Value of p cannot be 0.

delta

Value of the power of the main time series for a POWER-GARCH. Default is 2.

covariates

Instead of slicing X when building partial models, you can specify a vector of column indices. Defaults to "all".

method

Backfitting strategy used for the additive nonparametric component. One of "smooth_means_Positive_Constraint_Exp", "smooth_means_lowes", or "smooth_variance".

`"smooth_means_Positive_Constraint_Exp"`	Smooth log-partial-residuals and exponentiate (log/exp positivity constraint).
`"smooth_means_lowes"`	Smooth partial residuals and shift upward by subtracting the minimum (smooth + shift).
`"smooth_variance"`	Smooth a local weighted variance curve of partial residuals, take a square root, then shift upward (variance-smooth + shift).

train_end

Optional integer. If provided, fit on 1:train_end and compute test metrics on the remainder. If not provided, fit on the full sample.

n_burn

Integer burn-in. Iterations and fitted quantities are evaluated only after burn-in has passed. Default is 10.

max_outer_iter

Maximum number of outer backfitting iterations. Default is 30.

tol

Convergence tolerance for the outer iterations. Default is 1e-4.

hypothesisTest

Whether to perform covariate hypothesis testing, default is TRUE.

k.choice

Choice of window size k for the ANOVA hypotghesis test. Options are "plugin" or "CV". Default is "plugin".

Details

This function fits an additive semiparametric volatility model of the form

$\epsilon_t = \sigma_t w_t,$

$\sigma_t^2 = \sum_{i=1}^{p}\alpha_i |\epsilon_{t-i}|^2 + \sum_{\ell=1}^{d} m_{\ell}(X_{t-1,\ell}).$

Estimation proceeds by iterating between:

Updating the ARCH parameters $\alpha_1,\ldots,\alpha_p$ (subject to nonnegativity constraints), and
Updating the additive components $m_{\ell}$ using backfitting with a positivity-enforcing strategy controlled by method.

The outer loop stops when the change in the objective (or updates) is below tol or when max_outer_iter is reached.

No variable selection is done in this function, but an ANOVA-type hypothesis test may be performed for the significance of each covariate. This is enabled by default.

Value

An object of class ARCHmXAdditive

Examples


# We simulate a time series as in the simulate.ARCHmXAdditive example.
set.seed(1)
n <- 2000
alpha <- c(0.2, 0.1)
valinit <- 200
X <- matrix(rnorm((n+valinit) * 3), ncol = 3)

m <- list(
  function(x) 4 * abs(sin(x)),
  function(x) 2 * x^2,
  function(x) 1.5
)

data <- sim.ARCHmXAdditive(n = n, alpha = alpha, X = X, m = m)
p <- length(alpha)

# Then we fit the model
model <- ARCHmXAdditive(X = data$X, p = p, eps = data$eps, 
                        method = "smooth_means_lowes", max_outer_iter = 15)
model$alphas
model$converged

# We inspect the model summary
summary(model)

# We simulate a time series as in the simulate.ARCHmXAdditive example.
set.seed(1)
n <- 2000
alpha <- c(0.2, 0.1)
valinit <- 200
X <- matrix(rnorm((n+valinit) * 3), ncol = 3)

m <- list(
  function(x) 4 * abs(sin(x)),
  function(x) 2 * x^2,
  function(x) 1.5
)

data <- sim.ARCHmXAdditive(n = n, alpha = alpha, X = X, m = m)
p <- length(alpha)

# Then we fit the model
model <- ARCHmXAdditive(X = data$X, p = p, eps = data$eps, 
                        method = "smooth_means_lowes", max_outer_iter = 15)
model$alphas
model$converged

# We inspect the model summary
summary(model)

Fitting a GARCH-X model

Description

Fits a GARCHX model with given data and estimates the coefficients for omega, alpha, beta, and gamma

Usage

GARCHX(
  eps,
  X,
  order = c(1, 1),
  delta = 2,
  covariates = "all",
  optim.method = "NR"
)
GARCHX(
  eps,
  X,
  order = c(1, 1),
  delta = 2,
  covariates = "all",
  optim.method = "NR"
)

Arguments

eps

Time series

X

Matrix with exogenous covariates where the number of rows is equal to the length of eps and columns represent the covariates.

order

Order of the GARCH model. Value of p cannot be 0.

delta

Value of the power of the main time series to allow for Power GARCHX, default is 2 for GARCHX.

covariates

Instead of slicing X when building partial models, you can specify a vector of column indices. Defaults to "all".

optim.method

Optimization method for maximizing quasi-likelihood function. Options: "NR", "L-BFGS-B", "GA", "PS", "SA". Default value is "NR".

Details

Uses the GARCHX model

$\mathcal{E}_t = \sigma_tw_t$

$\sigma^2_t = \omega_0 + \sum^{p}_{i=1}\alpha_i\mathcal{E}_{t-i}^2 + \sum^q_{j=1}\beta_j\sigma^2_{t-j}+\mathbf{\gamma}^T\mathbf{x}_{t-1}$

To estimate the coefficients for

$\omega, \alpha, \beta, \gamma$

Value

An object of class GARCHX

Examples

# We simulate a time-series as in the simulate.GARCHX example. 
set.seed(1)
n <- 200
d <- 4
valinit <- 100
n2 <- n + d + 1
omega <- 0.05
alpha <- 0.05
beta <- 0.05
delta <- 2
gamma <- rep(0.05, d)
e<-rnorm(n2+valinit)
Y<-e
for (t in 2:n2)
 Y[t]<- 0.2*Y[t-1]+e[t]
x<-exp(Y)
X <- matrix(0, nrow = (n+valinit), ncol = length(gamma))
for(j in 1:d)
 X[, j] <- x[(d+2-j):(n+d+1-j+valinit)]
data <- sim.GARCHX(n = n, omega = omega, alpha = alpha, beta = beta, 
                   delta = delta, X = X, gamma = gamma, valinit = valinit)

# Then we fit the model
model <- GARCHX(data$eps, data$X, order = c(1, 1), delta = 2, 
                covariates = "all", optim.method = "NR")

# We inspect the model summary
summary(model)
# We simulate a time-series as in the simulate.GARCHX example. 
set.seed(1)
n <- 200
d <- 4
valinit <- 100
n2 <- n + d + 1
omega <- 0.05
alpha <- 0.05
beta <- 0.05
delta <- 2
gamma <- rep(0.05, d)
e<-rnorm(n2+valinit)
Y<-e
for (t in 2:n2)
 Y[t]<- 0.2*Y[t-1]+e[t]
x<-exp(Y)
X <- matrix(0, nrow = (n+valinit), ncol = length(gamma))
for(j in 1:d)
 X[, j] <- x[(d+2-j):(n+d+1-j+valinit)]
data <- sim.GARCHX(n = n, omega = omega, alpha = alpha, beta = beta, 
                   delta = delta, X = X, gamma = gamma, valinit = valinit)

# Then we fit the model
model <- GARCHX(data$eps, data$X, order = c(1, 1), delta = 2, 
                covariates = "all", optim.method = "NR")

# We inspect the model summary
summary(model)

Simulating data from an ARCH-m(X) process

Description

Returns a sample ARCH-m(X) time series given $\alpha$ s, data $X$ , and a function $m(X)$ provided by the user. A vector of simulated $\epsilon$ s will be returned for every row in $X$ .

Usage

sim.ARCHmX(n, 
  alpha, 
  m, 
  X, 
  delta=2, 
  shock.distr = "Normal", 
  valinit=200
)
sim.ARCHmX(n, 
  alpha, 
  m, 
  X, 
  delta=2, 
  shock.distr = "Normal", 
  valinit=200
)

Arguments

n

Desired length of simulated time series data

alpha

Set of parameters which determines the model's $p$ -step time dependence.

m

The function to apply to $X$ . m must accept the matrix $X$ and produce a vector of outputs for every row of $X$ .

X

Matrix with exogenous covariates where each row is an observation. Must be compatible with $m$

delta

Value of the power of the main time series for a POWER-GARCH, default is 2

shock.distr

The distribution from which the epsilons will be simulated. Default is 'Normal', and must be one of 'Normal', 't5', 't7', or 'Laplace'.

valinit

How many cycles the simulation should spend warming up before returning samples. Default is 200

Details

The user's specified $\alpha$ s, $\delta$ , function $m(X)$ , data $X$ , and shock distribution $Z$ partially define an ARCH-m(X) time series

$\begin{cases} \epsilon_t=z_t\sigma_t\\ \sigma_t^\delta=\sum_{j=1}^p\alpha_j\epsilon_{t-j}^\delta+m(\mathbf{X}_{t-1}) \end{cases}$

To get the $\epsilon_t$ simulations off the ground, a series of burn-in $\epsilon$ s are generated and used to build each $\sigma_t^\delta$ . From here, the shock distribution is sampled to produce each $\epsilon_t$ .

Value

A vector of simulated $\epsilon$ s corresponding to each observation (row) in $X$ .

Examples

set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)

summary(data$eps)
plot(data$eps, type = "l", main = "Simulated ARCH-m(X) series")
set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)

summary(data$eps)
plot(data$eps, type = "l", main = "Simulated ARCH-m(X) series")

Simulate data from an ARCHmXAdditive model

Description

Returns a sample Additive ARCH-m(X) time series given $\alpha$ s, data $X$ , and list of functions m provided by the user. A vector of simulated $\epsilon$ s will be returned for every row in $X$ .

Usage

sim.ARCHmXAdditive(n, 
  alpha, 
  m, 
  X, 
  delta = 2, 
  valinit = 200, 
  shock.distr = "Normal"
)
sim.ARCHmXAdditive(n, 
  alpha, 
  m, 
  X, 
  delta = 2, 
  valinit = 200, 
  shock.distr = "Normal"
)

Arguments

n

Desired length of simulated time series data.

alpha

Set of parameters which determines the model's $p$ -step time dependence.

m

List of length ncol(X) containing univariate functions $m_1,\ldots,m_d$ . Each m[[ell]] must accept a single numeric input and return a single finite nonnegative numeric value. The additive term is $\sum_{\ell=1}^d m_\ell(X_{t-1,\ell})$ .

X

Matrix with exogenous covariates where each row is an observation. Must be compatible with m.

delta

Value of the power of the main time series for a POWER-GARCH, default is 2.

valinit

Integer burn-in length used to stabilize the recursion before returning the simulated sample. Default is 200.

shock.distr

Innovation (shock) distribution for $z_t$ . One of "Normal", "t5", "t7", or "Laplace".

Value

A vector of simulated $\epsilon$ s corresponding to each observation (row) in $X$ .

Examples

set.seed(1)
n <- 2000
alpha <- c(0.2, 0.1)
valinit <- 200
X <- matrix(rnorm((n+valinit) * 3), ncol = 3)

m <- list(
  function(x) 4 * abs(sin(x)),
  function(x) 2 * x^2,
  function(x) 1.5
)

data <- sim.ARCHmXAdditive(n = n, alpha = alpha, X = X, m = m)
summary(data$eps)
plot(data$eps, type = "l", main = "Simulated additive ARCH-m(X) series")
set.seed(1)
n <- 2000
alpha <- c(0.2, 0.1)
valinit <- 200
X <- matrix(rnorm((n+valinit) * 3), ncol = 3)

m <- list(
  function(x) 4 * abs(sin(x)),
  function(x) 2 * x^2,
  function(x) 1.5
)

data <- sim.ARCHmXAdditive(n = n, alpha = alpha, X = X, m = m)
summary(data$eps)
plot(data$eps, type = "l", main = "Simulated additive ARCH-m(X) series")

Simulate GARCHX model

Description

Simulates Time series data from GARCH model with exogenous covariates

Usage

sim.GARCHX(n, 
  omega, 
  alpha, 
  beta, 
  gamma, 
  X, 
  delta = 2, 
  shock.distr = "Normal", 
  valinit = 200
)
sim.GARCHX(n, 
  omega, 
  alpha, 
  beta, 
  gamma, 
  X, 
  delta = 2, 
  shock.distr = "Normal", 
  valinit = 200
)

Arguments

n

Desired length of simulated time series data.

omega

Coefficient value for omega, required to be

$\omega_0 > 0$

alpha

ARCH Coefficient value, required to be

$\alpha_0 \geq 0$

beta

GARCH Coefficient value, required to be

$\beta_0 \geq 0$

gamma

Vector containing coefficients for exogenous covariates.

X

Matrix with exogenous covariates where the number of rows is equal to the length of n + valinit

delta

Value of the power of the time series to allow for Power GARCHX, default is 2 for GARCHX

shock.distr

Distribution of the shock eta_t that multiply w_t in the GARCH-X model eps_ = w_t*eta_t.

valinit

Initialization value, default value is 200

Value

A named list containing vector of Time Series data and X covariates used

Examples

set.seed(1)
n <- 200
d <- 4
valinit <- 100
n2 <- n + d + 1
omega <- 0.05
alpha <- 0.05
beta <- 0.05
delta <- 2
gamma <- rep(0.05, d)
e<-rnorm(n2+valinit)
Y<-e
for (t in 2:n2)
 Y[t]<- 0.2*Y[t-1]+e[t]
x<-exp(Y)
X <- matrix(0, nrow = (n+valinit), ncol = length(gamma))
for(j in 1:d)
 X[, j] <- x[(d+2-j):(n+d+1-j+valinit)]
data <- sim.GARCHX(n = n, omega = omega, alpha = alpha, beta = beta, 
                   delta = delta, X = X, gamma = gamma, valinit = valinit)
set.seed(1)
n <- 200
d <- 4
valinit <- 100
n2 <- n + d + 1
omega <- 0.05
alpha <- 0.05
beta <- 0.05
delta <- 2
gamma <- rep(0.05, d)
e<-rnorm(n2+valinit)
Y<-e
for (t in 2:n2)
 Y[t]<- 0.2*Y[t-1]+e[t]
x<-exp(Y)
X <- matrix(0, nrow = (n+valinit), ncol = length(gamma))
for(j in 1:d)
 X[, j] <- x[(d+2-j):(n+d+1-j+valinit)]
data <- sim.GARCHX(n = n, omega = omega, alpha = alpha, beta = beta, 
                   delta = delta, X = X, gamma = gamma, valinit = valinit)

Performing Stepwise Variable Selection for an ARCH-m(X) or GARCHX Process

Description

Returns the fitted ARCH-m(X), additive ARCH-m(X), or GARCHX model resulting from the specified stepwise selection algorithm.

Usage

stepwise(
  target_model, 
  direction="forward",
  criterion="pvalue",
  adjust_method="fdr",
  alpha.level=0.05,
  trace = TRUE
)
stepwise(
  target_model, 
  direction="forward",
  criterion="pvalue",
  adjust_method="fdr",
  alpha.level=0.05,
  trace = TRUE
)

Arguments

target_model

A model object of class ARCHmX, ARCHmXAdditive, or GARCHX. This model must be fit on all available covariates and, in the case of ARCHmX and ARCHmXAdditive, the covariate hypothesis testing must have been set to TRUE. This model defines all the relevant attributes of the candidate models the stepwise selection process will consider.

direction

The mode of stepwise search, can be one of "forward", "backward", "both_null" (both directions starting with the empty model), "both_full" (both directions starting with the full model), "bestsubset", and "onepass", with a default of "forward".

criterion

The criterion for taking additional steps, default is "pvalue".

adjust_method

Correction for multiple testing accumulation of error. See p.adjust.

alpha.level

Defines the adjusted p-value significance level, default is 0.05.

trace

If 'TRUE', information is printed for each step of variable selection. Default is 'FALSE'. Offers summaries of prospective models as each predictor in the scope is added to or removed from the model. 'max_pvalue' indicates the maximum p-value from the multiple tests for each predictor in the model.

Details

The forward selection routine starts with a (null or empty) ARCH model. At each iteration, the current model is compared to an array of all possible models with one additional covariate. A forward step is taken to the candidate model with all significant covariates providing the most improvement in the specified criterion. If no such model exists, or if all covariates are exhausted, the current model is returned.

The backward selection routine starts with the full ARCH-m(X), additive ARCH-m(X), or GARCHX model with all possible covariates. At each iteration, the current model is compared to an array of all possible models with one covariate removed. A backward step is taken to the candidate model providing the most improvement in the specified criterion until a model is found with all significant covariates, or until the logic reaches the empty model.

The both_null and both_full selection routines start with the null and full models, respectively. At each iteration, both forward and backward steps are attempted as outlined above. This continues until no step is taken, and the current model will be returned. The pvalue criterion can not be used for these routines since the p-value contstraints for forward and backward selection are incompatible with one another.

The best_subset selection routine will build models for every possible combination of covariates and return the one with the best value for the specified criterion.

The onepass selection routine builds the full model and returns a new model containing every covariate identified as significant within the full model. The only criterion available for onepass is pvalue.

The user may specify p-value corrections to apply to the p-values used within the desired stepwise selection routine. All options provided by p.adjust are exposed to the user; the default is set to fdr.

Value

The final model chosen by the stepwise-selection process.

Examples


# We simulate a time series as in the simulate_ARCHmX example.
set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)
p <- length(alpha)

# Output will show the trace from the empty model to the model with Covariate 1, 
# which is returned as stepwise_model.
target_ARCHmX_model <- ARCHmX(X = data$X, p = p, eps = data$eps)
stepwise_model <- stepwise(target_ARCHmX_model, direction="forward", criterion="pvalue") 

# We simulate a time series as in the simulate_ARCHmX example.
set.seed(1)
n <- 2000
alpha <- c(0.01, 0.02, 0.03)
valinit <- 200
X <- matrix(rnorm((n + valinit) * 5), ncol = 5)

m <- function(X) {
  abs(X[, 1])
}

data <- sim.ARCHmX(n = n, alpha = alpha, m = m, X = X, delta = 2, valinit = valinit)
p <- length(alpha)

# Output will show the trace from the empty model to the model with Covariate 1, 
# which is returned as stepwise_model.
target_ARCHmX_model <- ARCHmX(X = data$X, p = p, eps = data$eps)
stepwise_model <- stepwise(target_ARCHmX_model, direction="forward", criterion="pvalue")

Package 'GARCH.X'

Help Index

Fitting an ARCH-m(X) model

Description

Usage

Arguments

Details

Value

Examples

Creating and fitting an Additive ARCH-m(X) model for variable selection

Description

Usage

Arguments

Details

Value

Examples

Fitting a GARCH-X model

Description

Usage

Arguments

Details

Value

Examples

Simulating data from an ARCH-m(X) process

Description

Usage

Arguments

Details

Value

Examples

Simulate data from an ARCHmXAdditive model

Description

Usage

Arguments

Value

Examples

Simulate GARCHX model

Description

Usage

Arguments

Value

Examples

Performing Stepwise Variable Selection for an ARCH-m(X) or GARCHX Process

Description

Usage

Arguments

Details

Value

Examples