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# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# A copy of the GNU General Public License is available at
# http://www.r-project.org/Licenses/
bwWilhelm <- function(x, order.by = NULL, kernel = c("Quadratic Spectral",
"Bartlett", "Parzen", "Tukey-Hanning"), approx = c("AR(1)", "ARMA(1,1)"),
weights = NULL, prewhite = 1, ar.method = "ols", data=list())
{
## ensure that NAs are omitted
if(is.list(x) && !is.null(x$na.action)) class(x$na.action) <- "omit"
# ensure that x is a gmm object
stopifnot(attr(x,"class") == "gmm")
kernel <- match.arg(kernel)
approx <- match.arg(approx)
prewhite <- as.integer(prewhite)
# the following line applies only to the case class(res)=="gmm"
umat <- x$gt
n <- nrow(umat)
k <- ncol(umat)
# the following line applies only to the case class(res)=="gmm"
G <- x$G
p <- ncol(G)
# return Andrews' bandwidth in the just-identified case
if (p==k)
{
class(umat) <- "gmmFct"
return(bwAndrews(umat, order.by=order.by, kernel=kernel,
approx=approx, weights=weights, prewhite=prewhite,
ar.method=ar.method, data=data))
}
if(!is.null(order.by))
{
if(inherits(order.by, "formula")) {
z <- model.matrix(order.by, data = data)
z <- as.vector(z[,ncol(z)])
} else {
z <- order.by
}
index <- order(z)
} else {
index <- 1:n
}
umat <- umat[index, , drop = FALSE]
## compute weights (w_a, see p. 834)
## (try to set the intercept weight to 0)
#### could be ignored by using: weights = 1
if(is.null(weights)) {
weights <- rep(1, p)
unames <- colnames(umat)
if(!is.null(unames) && "(Intercept)" %in% unames)
weights[which(unames == "(Intercept)")] <- 0
else {
res <- try(as.vector(rowMeans(estfun(x)/model.matrix(x), na.rm = TRUE)), silent = TRUE)
if(inherits(res, "try-error")) res <- try(residuals(x), silent = TRUE)
if(!inherits(res, "try-error")) weights[which(colSums((umat - res)^2) < 1e-16)] <- 0
}
if(isTRUE(all.equal(weights, rep(0, p)))) weights <- rep(1, p)
} else {
weights <- rep(weights, length.out = p)
}
if(length(weights) < 2) weights <- 1
## pre-whitening
if(prewhite > 0) {
var.fit <- ar(umat, order.max = prewhite, demean = FALSE, aic = FALSE, method = ar.method)
if(inherits(var.fit, "try-error")) stop(sprintf("VAR(%i) prewhitening of estimating functions failed", prewhite))
umat <- as.matrix(na.omit(var.fit$resid))
n <- n - prewhite
}
# define kernel constants
kernConst <- switch(kernel,
"Bartlett" = list(q = 1, g_q = 1, mu1 = 1, mu2 = 2/3),
"Parzen" = list(q = 2, g_q = 6, mu1 = 0.75, mu2 = 0.539286),
"Tukey-Hanning" = list(q = 2, g_q = pi^2/4, mu1 = 1, mu2 = 0.75),
"Quadratic Spectral" = list(q = 2, g_q = 1.421223, mu1 = 1.25003, mu2 = 0.999985))
# fit approximating model to moments
if(approx == "AR(1)") {
fitAR1 <- function(x) {
rval <- ar(x, order.max = 1, aic = FALSE, method = "ols")
rval <- c(rval$ar, sqrt(rval$var.pred))
names(rval) <- c("rho", "sigma")
return(rval)
}
ar.coef <- apply(umat, 2, fitAR1)
Omega0 <- ar.coef["sigma", ]^2 / (1-ar.coef["rho", ])^2
Omega_q <- if(kernConst$q == 1) {
diag(2 * ar.coef["sigma", ]^2 * ar.coef["rho", ] / ((1 - ar.coef["rho", ])^3 * (1 + ar.coef["rho", ])))
} else {
diag(2 * ar.coef["sigma", ]^2 * ar.coef["rho", ] / (1 - ar.coef["rho", ])^4)
}
} else {
fitARMA11 <- function(x) {
rval <- arima(x, order = c(1, 0, 1), include.mean = FALSE)
rval <- c(rval$coef, sqrt(rval$sigma2))
names(rval) <- c("rho", "psi", "sigma")
return(rval)
}
arma.coef <- apply(umat, 2, fitARMA11)
Omega0 <- (1 + ar.coef["psi", ])^2 * ar.coef["sigma", ]^2 / (1 - ar.coef["rho", ])^2
Omega_q <- if(kernConst$q == 1) {
diag(2 * (1 + ar.coef["psi", ] * ar.coef["rho", ]) * (ar.coef["psi", ] + ar.coef["rho", ]) * ar.coef["sigma", ]^2 / ((1 - ar.coef["rho", ])^3 * (1 + ar.coef["rho", ])))
} else {
diag(2 * (1 + ar.coef["psi", ] * ar.coef["rho", ]) * (ar.coef["psi", ] + ar.coef["rho", ]) * ar.coef["sigma", ]^2 / (1 - ar.coef["rho", ])^4)
}
}
# compute remaining bandwidth components
W <- diag(weights)
Omega0inv <- diag(1/Omega0)
Sigma0 <- solve( crossprod(Omega0inv%*%G, G) )
H0 <- Sigma0 %*% t(G) %*% Omega0inv
P0 <- Omega0inv - Omega0inv %*% G %*% H0
# compute optimal bandwidth
nu2 <- (2 * kernConst$mu1 + kernConst$mu2) * (k - p) * sum(diag(Sigma0 %*% W))
nu3 <- kernConst$g_q^2 * sum(diag(t(Omega_q) %*% t(H0) %*% W %*% H0 %*% Omega_q %*% P0))
c0 <- if(nu2 * nu3 > 0) 2 * kernConst$q else -1
rval <- (c0 * nu3/nu2 * n)^(1/(1 + 2 * kernConst$q))
return(rval)
}
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