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"
)

Arguments

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)

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"
)

Arguments

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:

1. Updating the ARCH parameters \alpha_1,\ldots,\alpha_p (subject to nonnegativity constraints), and

2. 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)

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"
)

Arguments

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)

Simulating data from an ARCH-m(X) process

Description

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

Usage

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

Arguments

Details

The user's specified \alphas, \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 \epsilons 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 \epsilons 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")

Simulate data from an ARCHmXAdditive model

Description

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

Usage

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

Arguments

Value

A vector of simulated \epsilons 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")

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
)

Arguments

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)

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
)

Arguments

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")