Nothing
R/FCVAR_spec.R
In FCVAR: Estimation and Inference for the Fractionally Cointegrated VAR
Defines functions
FCVARbootRank
summary.FCVAR_ranks
FCVARrankTests
summary.FCVAR_lags
FCVARlagSelect
Documented in
FCVARbootRank
FCVARlagSelect
FCVARrankTests
summary.FCVAR_lags
summary.FCVAR_ranks
#' Select Lag Order
#'
#' \code{FCVARlagSelect} takes a matrix of variables and performs lag
#' selection on it by using the likelihood ratio test. Output and test
#' results are printed to the screen.
#'
#' @param x A matrix of variables to be included in the system.
#' @param kmax The maximum number of lags in the system.
#' @param r The cointegrating rank.This is often set equal to \code{p},
#' the number of variables in the system, since it is better to overspecify
#' than underspecify the model.
#' @param order The order of serial correlation for white noise tests.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return An S3 object of type \code{FCVAR_lags} containing the results
#' from repeated estimation of the FCVAR model with different orders
#' of the autoregressive lag length.
#' Note that row \code{j} of each of the vectors in the \code{FCVAR_lags} object
#' contains the associated results for lag length \code{j+1}.
#' The \code{FCVAR_lags} object includes the following parameters:
#' \describe{
#' \item{\code{D}}{A (\code{kmax} + 1) x 2 vector of estimates of d and b.}
#' \item{\code{loglik}}{A (\code{kmax} + 1) x 1 vector of log-likelihood values.}
#' \item{\code{LRtest}}{A (\code{kmax} + 1) x 1 vector of likelihood ratio test statistics for tests of significance of \eqn{\Gamma_{j+1}}.}
#' \item{\code{pvLRtest}}{A (\code{kmax} + 1) x 1 vector of P-values for the likelihood ratio tests of significance of \eqn{\Gamma_{j+1}}.}
#' \item{\code{i_aic}}{The lag corresponding to the minimum value of the Akaike information criteria.}
#' \item{\code{aic}}{A (\code{kmax} + 1) x 1 vector of values of the Akaike information criterion.}
#' \item{\code{i_bic}}{The lag corresponding to the minimum value of the Bayesian information criteria.}
#' \item{\code{bic}}{A (\code{kmax} + 1) x 1 vector of values of the Bayesian information criterion.}
#' \item{\code{pvMVq}}{A scalar P-value for the Q-test for multivariate residual white noise.}
#' \item{\code{pvWNQ}}{A (\code{kmax} + 1) x 1 vector of P-values for the Q-tests for univariate residual white noise.}
#' \item{\code{pvWNLM}}{A (\code{kmax} + 1) x 1 vector of P-values for the LM-tests for univariate residual white noise.}
#' \item{\code{kmax}}{The maximum number of lags in the system.}
#' \item{\code{r}}{The cointegrating rank. This is often set equal to \code{p},
#' the number of variables in the system, since it is better to overspecify
#' than underspecify the model.}
#' \item{\code{p}}{The number of variables in the system.}
#' \item{\code{cap_T}}{The sample size.}
#' \item{\code{order}}{The order of serial correlation for white noise tests.}
#' \item{\code{opt}}{An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.}
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' FCVARlagSelectStats <- FCVARlagSelect(x, kmax = 3, r = 3, order = 12, opt)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate lag order.
#' \code{summary.FCVAR_lags} prints a summary of the output of \code{FCVARlagSelect} to screen.
#' @export
#'
FCVARlagSelect <- function(x, kmax, r, order, opt ) {
# Determine (initial) dimensions of system.
cap_T <- nrow(x) # Length of sample (before truncation for initial values).
p <- ncol(x) # Number of variables.
# Do not print output for each WN test.
printWN <- 0
# Do not print FCVAR estimation for each lag in the loop.
print2screen <- opt$print2screen
opt$print2screen <- 0
# Do not plot roots of characteristic polynomial for each lag in the loop.
opt$plotRoots <- 0
# Do not calculate standard errors.
opt$CalcSE <- 0
#--------------------------------------------------------------------------------
# Create storage bins for output
#--------------------------------------------------------------------------------
# Row j of each of the following contains the associated results for
# lag length j+1
D <- matrix(0, nrow = kmax+1, ncol = 2) # estimates of d and b
loglik <- matrix(0, nrow = kmax+1, ncol = 1) # log-likelihood
LRtest <- matrix(0, nrow = kmax+1, ncol = 1) # likelihood ratio test statistic for
# significance of Gamma_{j+1}
pvLRtest <- matrix(0, nrow = kmax+1, ncol = 1) # likelihood ratio test P-value
aic <- matrix(0, nrow = kmax+1, ncol = 1) # Akaike information criterion
bic <- matrix(0, nrow = kmax+1, ncol = 1) # Bayesian information criterion
pvMVq <- matrix(0, nrow = kmax+1, ncol = 1) # multivariate residual white noise Q-test P-value
pvWNQ <- matrix(0, nrow = kmax+1, ncol = p) # univariate residual white noise Q-test P-values
# for each residual
pvWNLM <- matrix(0, nrow = kmax+1, ncol = p) # univariate residual white noise LM-test P-values
# for each residual
#--------------------------------------------------------------------------------
# Estimate FCVAR for each lag order.
#--------------------------------------------------------------------------------
for (k in 0:kmax) {
message(sprintf('Estimating for k = %d and r = %d.', k, r))
# ----- Estimation ---------#
results <- FCVARestn(x, k, r, opt)
message(sprintf('Finished Estimation for k = %d and r = %d.', k, r))
# ----- Record relevant output ---------%
loglik[k+1] <- results$like
D[k+1, ] <- results$coeffs$db
aic[k+1] <- -2*loglik[k+1] + 2*results$fp
bic[k+1] <- -2*loglik[k+1] + results$fp*log(cap_T - opt$N)
# ----- White noise tests ---------%
MVWNtest_stats <- MVWNtest(results$Residuals, order, printWN)
pvWNQ[k+1, ] <- MVWNtest_stats$pvQ
pvWNLM[k+1, ] <- MVWNtest_stats$pvLM
pvMVq[k+1, ] <- MVWNtest_stats$pvMVQ
# ----- LR test of lag <- k vs lag <- k-1 -----%
if (k > 0) {
LRtest[k+1] <- 2*(loglik[k+1] - loglik[k])
pvLRtest[k+1] <- 1 - stats::pchisq(LRtest[k+1], p^2)
}
}
# Find lag corresponding to min of information criteria
i_aic <- which.min(aic)
i_bic <- which.min(bic)
# Return list of lag selection statistics.
FCVARlagSelectStats <- list(
D = D,
loglik = loglik,
LRtest = LRtest,
pvLRtest = pvLRtest,
i_aic = i_aic,
aic = aic,
i_bic = i_bic,
bic = bic,
pvMVq = pvMVq,
pvWNQ = pvWNQ,
pvWNLM = pvWNLM,
kmax = kmax,
r = r,
p = p,
cap_T = cap_T,
order = order,
opt = opt
)
class(FCVARlagSelectStats) <- 'FCVAR_lags'
# Print output if required, restoring original settings.
opt$print2screen <- print2screen
if (opt$print2screen) {
summary(object = FCVARlagSelectStats)
}
return(FCVARlagSelectStats)
}
#' Summarize Statistics from Lag Order Selection
#'
#' \code{summary.FCVAR_lags} prints a summary of the table of statistics from
#' the output of \code{FCVARlagSelect}.
#' \code{FCVARlagSelect} takes a matrix of variables and performs lag
#' selection on it by using the likelihood ratio test.
#'
#' @param object An S3 object of type \code{FCVAR_lags} containing the results
#' from repeated estimation of the FCVAR model with different orders
#' of the autoregressive lag length. It is the output of \code{FCVARlagSelect}.
#' @param ... additional arguments affecting the summary produced.
#' @return NULL
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' FCVAR_lag_1 <- FCVARlagSelect(x, kmax = 3, r = 3, order = 12, opt)
#' summary(object = FCVAR_lag_1)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate lag order.
#' \code{summary.FCVAR_lags} prints a summary of the output of \code{FCVARlagSelect} to screen.
#' @export
#'
summary.FCVAR_lags <- function(object, ...) {
#--------------------------------------------------------------------------------
# Print output
#--------------------------------------------------------------------------------
# create a variable for output strings
yesNo <- c('No','Yes') # Ironic order, No?
cat(sprintf('\n--------------------------------------------------------------------------------\n'))
cat(sprintf(' Lag Selection Results \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Dimension of system: %6.0f Number of observations in sample: %6.0f \n',
object$p, object$cap_T))
cat(sprintf('Order for WN tests: %6.0f Number of observations for estimation: %6.0f \n',
object$order, object$cap_T - object$opt$N))
cat(sprintf('Restricted constant: %6s Initial values: %6.0f\n',
yesNo[object$opt$rConstant+1], object$opt$N ) )
cat(sprintf('Unrestricted constant:%6s Level parameter: %6s\n',
yesNo[object$opt$unrConstant+1], yesNo[object$opt$levelParam+1] ))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Parameter Estimates and Information Criteria:\n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf(' k r d b LogL LR pv AIC BIC'))
cat(sprintf('\n'))
for (k in seq(object$kmax, 0, by = -1) ) {
cat(sprintf('%2.0f %2.0f %4.3f %4.3f %7.2f %6.2f %5.3f %8.2f',
k, object$r, object$D[k+1, 1], object$D[k+1, 2], object$loglik[k+1],
object$LRtest[k+1], object$pvLRtest[k+1], object$aic[k+1]))
# For AIC add asterisk if min value
if(k+1 == object$i_aic) {cat(sprintf('*'))} else {cat(sprintf(' '))}
# Print BIC information criteria and add asterisk if min value
cat(sprintf(' %8.2f', object$bic[k+1]))
if(k+1 == object$i_bic) {cat(sprintf('*'))} else {cat(sprintf(' '))}
cat(sprintf('\n'))
}
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Tests for Serial Correlation of Residuals: \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf(' k pmvQ'))
for (i in 1:object$p) {
cat(sprintf(' pQ%1.0f pLM%1.0f', i, i))
}
cat(sprintf('\n'))
for (k in seq(object$kmax, 0, by = -1) ) {
cat(sprintf('%2.0f ', k))
# Print multivariate white noise test P-values
cat(sprintf(' %4.2f', object$pvMVq[k+1, ]))
# Print the individual series white noise test P-values
for (i in 1:object$p) {
cat(sprintf(' %4.2f %4.2f', object$pvWNQ[k+1,i], object$pvWNLM[k+1,i]))
}
cat(sprintf('\n'))
}
cat(sprintf('--------------------------------------------------------------------------------\n'))
}
#' Test for Cointegrating Rank
#'
#' \code{FCVARrankTests} performs a sequence of likelihood ratio tests
#' for cointegrating rank.
#'
#' @param x A matrix of variables to be included in the system.
#' @param k The number of lags in the system.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return An S3 object of type \code{FCVAR_ranks} containing the results
#' from cointegrating rank tests, containing the following \code{(p+1)} vectors
#' with \code{i}th element corresponding to \code{rank = i-1},
#' including the following parameters:
#' \describe{
#' \item{\code{dHat}}{Estimates of \code{d}.}
#' \item{\code{bHat}}{Estimates of \code{b}.}
#' \item{\code{LogL}}{Maximized log-likelihood.}
#' \item{\code{LRstat}}{LR trace statistic for testing rank \code{r} against rank \code{p}.}
#' \item{\code{pv}}{The p-value of LR trace test, or "999" if p-value is not available.}
#' \item{\code{k}}{The number of lags in the system.}
#' \item{\code{p}}{The number of variables in the system.}
#' \item{\code{cap_T}}{The sample size.}
#' \item{\code{opt}}{An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.}
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' rankTestStats <- FCVARrankTests(x, k = 2, opt)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate cointegrating rank.
#' \code{summary.FCVAR_ranks} prints a summary of the output of \code{FCVARrankTests} to screen.
#' @export
#'
FCVARrankTests <- function(x, k, opt) {
cap_T <- nrow(x) - opt$N
p <- ncol(x)
# Store user specified options for printing to screen because it will be
# turned off while looping over ranks.
tempPrint2Screen <- opt$print2screen
# Create output storage to be filled.
bHat <- matrix(0, nrow = p+1, ncol = 1)
dHat <- matrix(0, nrow = p+1, ncol = 1)
LogL <- matrix(0, nrow = p+1, ncol = 1)
LRstat <- matrix(0, nrow = p+1, ncol = 1)
pv <- matrix(NA, nrow = p+1, ncol = 1)
# Do not print FCVAR estimation for each rank in the loop.
opt$print2screen <- 0
# Do not plot roots of characteristic polynomial for each lag in the loop.
opt$plotRoots <- 0
# Do not calculate standard errors.
opt$CalcSE <- 0
# For calculation of P-values
if(opt$rConstant | opt$levelParam) {
consT <- 1
} else {
consT <- 0
}
# Estimate models for all ranks
for (r in 0 : p) {
message(sprintf('Estimating for k = %d and r = %d.', k, r))
results <- FCVARestn(x, k, r, opt)
message(sprintf('Finished Estimation for k = %d and r = %d.', k, r))
dHat[r+1] <- results$coeffs$db[1]
bHat[r+1] <- results$coeffs$db[2]
LogL[r+1] <- results$like
}
# Calculate the LR statistics and P-values
for (r in 0 : (p-1)) {
LRstat[r+1] <- - 2*( LogL[r+1] - LogL[p+1] )
p_val <- NULL
# Get P-values, if
# (1) no deterministic terms, or
# (2) there is only restricted constant and d=b, or
# (3) there is only a level parameter and d=b.
# In all cases, p-values are only calculated
# rank of system (iq) is an integer from 1 through 12.
if (bHat[r+1] > 0 & bHat[r+1] < 2 & (
(!opt$rConstant & !opt$unrConstant & !opt$levelParam) |
(opt$rConstant & !opt$unrConstant & opt$restrictDB) |
(opt$levelParam & !opt$unrConstant & opt$restrictDB) ) ) {
# Call function from fracdist package for p-values.
if (p - r <= 12) {
p_val <- fracdist::fracdist_values(iq = p - r,
iscon = consT,
bb = bHat[r+1],
stat = LRstat[r+1])
} else {
warning(sprintf('P-values not calculated for the rank test with rank %d.\n', r),
'Tables of simulated quantiles are only avaiable for rank 12 or lower.',
'P-values can be calculated by simulation with the FCVARbootRank() function.')
}
} else {
warning(sprintf('P-values not calculated for the rank test with rank %d.\n', r),
'P-values are only calculated if:\n',
'1. there are no deterministic terms, or\n',
'2. there is only restricted constant and d = b, or\n',
'3. there is only a level parameter and d = b.\n')
}
# Store P-values, if calculated.
if(!is.null(p_val)) {
pv[r+1] <- p_val
}
}
# Return list of rank test results.
rankTestStats <- list(
dHat = dHat,
bHat = bHat,
LogL = LogL,
LRstat = LRstat,
pv = pv,
k = k,
p = p,
cap_T = cap_T,
opt = opt
)
class(rankTestStats) <- 'FCVAR_ranks'
# Restore settings.
opt$print2screen <- tempPrint2Screen
# Print the results to screen.
if (opt$print2screen) {
summary(object = rankTestStats)
}
return(rankTestStats)
}
#' Summarize Results of Tests for Cointegrating Rank
#'
#' \code{summary.FCVAR_ranks} prints the table of statistics from
#' the output of \code{FCVARrankTests}.
#' \code{FCVARrankTests} performs a sequence of likelihood ratio tests
#' for cointegrating rank.
#'
#' @param object An S3 object of type \code{FCVAR_ranks} containing the results
#' from repeated estimation of the FCVAR model with different
#' cointegrating ranks. It is the output of \code{FCVARrankTests}.
#' @param ... additional arguments affecting the summary produced.
#' @return NULL
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' rankTestStats <- FCVARrankTests(x, k = 2, opt)
#' summary(object = rankTestStats)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate cointegrating rank.
#' \code{summary.FCVAR_ranks} prints a summary of the output of \code{FCVARrankTests} to screen.
#' @export
#'
summary.FCVAR_ranks <- function(object, ...) {
# create a variable for output strings
yesNo <- c('No','Yes')
cat(sprintf('\n--------------------------------------------------------------------------------\n'))
cat(sprintf(' Likelihood Ratio Tests for Cointegrating Rank \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Dimension of system: %6.0f Number of observations in sample: %6.0f \n',
object$p, object$cap_T + object$opt$N))
cat(sprintf('Number of lags: %6.0f Number of observations for estimation: %6.0f \n',
object$k, object$cap_T))
cat(sprintf('Restricted constant: %6s Initial values: %6.0f\n',
yesNo[object$opt$rConstant+1], object$opt$N ))
cat(sprintf('Unestricted constant: %6s Level parameter: %6s\n',
yesNo[object$opt$unrConstant+1], yesNo[object$opt$levelParam+1] ))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Rank d b Log-likelihood LR statistic P-value\n'))
for (i in 1:object$p) {
if (!is.na(object$pv[i])) {
cat(sprintf('%2.0f %5.3f %5.3f %15.3f %13.3f %8.3f\n',
i-1, object$dHat[i], object$bHat[i], object$LogL[i], object$LRstat[i], object$pv[i]))
}
else {
cat(sprintf('%2.0f %5.3f %5.3f %15.3f %13.3f ----\n',
i-1, object$dHat[i], object$bHat[i], object$LogL[i], object$LRstat[i]))
}
}
cat(sprintf('%2.0f %5.3f %5.3f %15.3f ---- ----\n',
i, object$dHat[i+1], object$bHat[i+1], object$LogL[i+1]))
cat(sprintf('--------------------------------------------------------------------------------\n'))
}
#' Distribution of LR Test Statistic for the Rank Test
#'
#' \code{FCVARbootRank} generates a distribution of a likelihood ratio
#' test statistic for the rank test using a wild bootstrap,
#' following the method of Cavaliere, Rahbek, and Taylor (2010). It
#' takes the two ranks as inputs to estimate the model under the
#' null and the model under the alternative.
#' @param x A matrix of variables to be included in the system.
#' If \code{k>0}, actual data is used for initial values.
#' @param k The number of lags in the system.
#' @param r1 The cointegrating rank under the null hypothesis.
#' @param r2 The cointegrating rank under the alternative hypothesis.
#' @param B The number of bootstrap samples.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return A list \code{FCVARbootRank_stats} containing the test results,
#' including the following parameters:
#' \describe{
#' \item{\code{LRbs}}{A B x 1 vector of simulated likelihood ratio statistics.}
#' \item{\code{pv}}{An approximate p-value for the LR statistic based on the bootstrap distribution. }
#' \item{\code{H}}{A list containing LR test results. It is
#' identical to the output from \code{HypoTest}, with one addition,
#' namely \code{H$pvBS} which is the bootstrap p-value)}
#' \item{\code{mBS}}{Model estimates under the null hypothesis. }
#' \item{\code{mUNR}}{Model estimates under the alternative hypothesis. }
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' opt$plotRoots <- 0
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' set.seed(42)
#' FCVARbootRank_stats <- FCVARbootRank(x, k = 2, opt, r1 = 0, r2 = 1, B = 2)
#' # In practice, set the number of bootstraps so that (B+1)*alpha is an integer,
#' # where alpha is the chosen level of significance.
#' # For example, set B = 999 (but it takes a long time to compute).
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{HypoTest} for the format of a hypothesis test results.
#' \code{FCVARestn} for the estimates from a rectricted and unrestricted model within a hypothesis test.
#' @export
#' @references Cavaliere, G., A. Rahbek, and A. M. R. Taylor (2010).
#' "Testing for co-integration in vector autoregressions
#' with non-stationary volatility," Journal of Econometrics 158, 7-24.
#'
FCVARbootRank <- function(x, k, opt, r1, r2, B) {
# Calculate length of sample to generate, adjusting for initial values
cap_T <- nrow(x) - opt$N
# Use first k+1 observations for initial values
data <- x[1:(k+1), ]
LR <- matrix(0, nrow = B, ncol = 1)
# Turn off output and calculation of standard errors for faster computation
print2screen <- opt$print2screen
opt$print2screen <- 0
opt$CalcSE <- 0
opt$plotRoots <- 0
mBS <- FCVARestn(x, k, r1, opt)
mUNR <- FCVARestn(x, k, r2, opt)
# Initialize H (a list containing LR test results, it is
# identical to the output from HypoTest, with one addition,
# namely H$pvBS which is the Bootstrap P-value)
H <- list(LRstat = NA,
pvBS = NA)
H$LRstat <- -2*(mBS$like - mUNR$like)
# How often should the number of iterations be displayed
show_iters <- 10
for (j in 1:B) {
# Display replication count every show_iters Bootstraps
if(round((j+1)/show_iters) == (j+1)/show_iters) {
# cat(sprintf('iteration: %1.0f\n', j))
message(sprintf('Completed bootstrap replication %d of %d.', j, B))
}
# (1) generate bootstrap DGP under the null
xBS <- FCVARsimBS(data, mBS, cap_T)
# Append initial values to bootstrap sample
BSs <- rbind(data, xBS)
# (2) estimate unrestricted model
mUNRbs <- FCVARestn(BSs, k, r2, opt)
# (3) estimate restricted model (under the null)
mRES <- FCVARestn(BSs, k, r1, opt)
# (4) calculate test statistic
LR[j] <- -2*(mRES$like - mUNRbs$like)
}
# Return sorted LR stats
LRbs <- LR[order(LR)]
# Calculate Bootstrap P-value (see ETM p.157 eq 4.62)
H$pvBS <- sum(LRbs > H$LRstat)/B
# Print output, if required, after restoring settings.
opt$print2screen <- print2screen
if (opt$print2screen) {
cat(sprintf('Bootstrap rank test results:'))
cat(sprintf('\nUnrestricted log-likelihood: %3.3f\nRestricted log-likelihood: %3.3f\n',
mUNR$like, mBS$like))
cat(sprintf('Test results:\nLR statistic: \t %3.3f\nP-value (BS): \t %1.3f\n',
H$LRstat, H$pvBS))
}
# Return a list of bootstrap test results.
FCVARbootRank_stats <- list(
LRbs = LRbs,
H = H,
mBS = mBS,
mUNR = mUNR
)
return(FCVARbootRank_stats)
}
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Defines functions FCVARbootRank summary.FCVAR_ranks FCVARrankTests summary.FCVAR_lags FCVARlagSelect
Documented in FCVARbootRank FCVARlagSelect FCVARrankTests summary.FCVAR_lags summary.FCVAR_ranks
#' Select Lag Order
#'
#' \code{FCVARlagSelect} takes a matrix of variables and performs lag
#' selection on it by using the likelihood ratio test. Output and test
#' results are printed to the screen.
#'
#' @param x A matrix of variables to be included in the system.
#' @param kmax The maximum number of lags in the system.
#' @param r The cointegrating rank.This is often set equal to \code{p},
#' the number of variables in the system, since it is better to overspecify
#' than underspecify the model.
#' @param order The order of serial correlation for white noise tests.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return An S3 object of type \code{FCVAR_lags} containing the results
#' from repeated estimation of the FCVAR model with different orders
#' of the autoregressive lag length.
#' Note that row \code{j} of each of the vectors in the \code{FCVAR_lags} object
#' contains the associated results for lag length \code{j+1}.
#' The \code{FCVAR_lags} object includes the following parameters:
#' \describe{
#' \item{\code{D}}{A (\code{kmax} + 1) x 2 vector of estimates of d and b.}
#' \item{\code{loglik}}{A (\code{kmax} + 1) x 1 vector of log-likelihood values.}
#' \item{\code{LRtest}}{A (\code{kmax} + 1) x 1 vector of likelihood ratio test statistics for tests of significance of \eqn{\Gamma_{j+1}}.}
#' \item{\code{pvLRtest}}{A (\code{kmax} + 1) x 1 vector of P-values for the likelihood ratio tests of significance of \eqn{\Gamma_{j+1}}.}
#' \item{\code{i_aic}}{The lag corresponding to the minimum value of the Akaike information criteria.}
#' \item{\code{aic}}{A (\code{kmax} + 1) x 1 vector of values of the Akaike information criterion.}
#' \item{\code{i_bic}}{The lag corresponding to the minimum value of the Bayesian information criteria.}
#' \item{\code{bic}}{A (\code{kmax} + 1) x 1 vector of values of the Bayesian information criterion.}
#' \item{\code{pvMVq}}{A scalar P-value for the Q-test for multivariate residual white noise.}
#' \item{\code{pvWNQ}}{A (\code{kmax} + 1) x 1 vector of P-values for the Q-tests for univariate residual white noise.}
#' \item{\code{pvWNLM}}{A (\code{kmax} + 1) x 1 vector of P-values for the LM-tests for univariate residual white noise.}
#' \item{\code{kmax}}{The maximum number of lags in the system.}
#' \item{\code{r}}{The cointegrating rank. This is often set equal to \code{p},
#' the number of variables in the system, since it is better to overspecify
#' than underspecify the model.}
#' \item{\code{p}}{The number of variables in the system.}
#' \item{\code{cap_T}}{The sample size.}
#' \item{\code{order}}{The order of serial correlation for white noise tests.}
#' \item{\code{opt}}{An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.}
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' FCVARlagSelectStats <- FCVARlagSelect(x, kmax = 3, r = 3, order = 12, opt)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate lag order.
#' \code{summary.FCVAR_lags} prints a summary of the output of \code{FCVARlagSelect} to screen.
#' @export
#'
FCVARlagSelect <- function(x, kmax, r, order, opt ) {
# Determine (initial) dimensions of system.
cap_T <- nrow(x) # Length of sample (before truncation for initial values).
p <- ncol(x) # Number of variables.
# Do not print output for each WN test.
printWN <- 0
# Do not print FCVAR estimation for each lag in the loop.
print2screen <- opt$print2screen
opt$print2screen <- 0
# Do not plot roots of characteristic polynomial for each lag in the loop.
opt$plotRoots <- 0
# Do not calculate standard errors.
opt$CalcSE <- 0
#--------------------------------------------------------------------------------
# Create storage bins for output
#--------------------------------------------------------------------------------
# Row j of each of the following contains the associated results for
# lag length j+1
D <- matrix(0, nrow = kmax+1, ncol = 2) # estimates of d and b
loglik <- matrix(0, nrow = kmax+1, ncol = 1) # log-likelihood
LRtest <- matrix(0, nrow = kmax+1, ncol = 1) # likelihood ratio test statistic for
# significance of Gamma_{j+1}
pvLRtest <- matrix(0, nrow = kmax+1, ncol = 1) # likelihood ratio test P-value
aic <- matrix(0, nrow = kmax+1, ncol = 1) # Akaike information criterion
bic <- matrix(0, nrow = kmax+1, ncol = 1) # Bayesian information criterion
pvMVq <- matrix(0, nrow = kmax+1, ncol = 1) # multivariate residual white noise Q-test P-value
pvWNQ <- matrix(0, nrow = kmax+1, ncol = p) # univariate residual white noise Q-test P-values
# for each residual
pvWNLM <- matrix(0, nrow = kmax+1, ncol = p) # univariate residual white noise LM-test P-values
# for each residual
#--------------------------------------------------------------------------------
# Estimate FCVAR for each lag order.
#--------------------------------------------------------------------------------
for (k in 0:kmax) {
message(sprintf('Estimating for k = %d and r = %d.', k, r))
# ----- Estimation ---------#
results <- FCVARestn(x, k, r, opt)
message(sprintf('Finished Estimation for k = %d and r = %d.', k, r))
# ----- Record relevant output ---------%
loglik[k+1] <- results$like
D[k+1, ] <- results$coeffs$db
aic[k+1] <- -2*loglik[k+1] + 2*results$fp
bic[k+1] <- -2*loglik[k+1] + results$fp*log(cap_T - opt$N)
# ----- White noise tests ---------%
MVWNtest_stats <- MVWNtest(results$Residuals, order, printWN)
pvWNQ[k+1, ] <- MVWNtest_stats$pvQ
pvWNLM[k+1, ] <- MVWNtest_stats$pvLM
pvMVq[k+1, ] <- MVWNtest_stats$pvMVQ
# ----- LR test of lag <- k vs lag <- k-1 -----%
if (k > 0) {
LRtest[k+1] <- 2*(loglik[k+1] - loglik[k])
pvLRtest[k+1] <- 1 - stats::pchisq(LRtest[k+1], p^2)
}
}
# Find lag corresponding to min of information criteria
i_aic <- which.min(aic)
i_bic <- which.min(bic)
# Return list of lag selection statistics.
FCVARlagSelectStats <- list(
D = D,
loglik = loglik,
LRtest = LRtest,
pvLRtest = pvLRtest,
i_aic = i_aic,
aic = aic,
i_bic = i_bic,
bic = bic,
pvMVq = pvMVq,
pvWNQ = pvWNQ,
pvWNLM = pvWNLM,
kmax = kmax,
r = r,
p = p,
cap_T = cap_T,
order = order,
opt = opt
)
class(FCVARlagSelectStats) <- 'FCVAR_lags'
# Print output if required, restoring original settings.
opt$print2screen <- print2screen
if (opt$print2screen) {
summary(object = FCVARlagSelectStats)
}
return(FCVARlagSelectStats)
}
#' Summarize Statistics from Lag Order Selection
#'
#' \code{summary.FCVAR_lags} prints a summary of the table of statistics from
#' the output of \code{FCVARlagSelect}.
#' \code{FCVARlagSelect} takes a matrix of variables and performs lag
#' selection on it by using the likelihood ratio test.
#'
#' @param object An S3 object of type \code{FCVAR_lags} containing the results
#' from repeated estimation of the FCVAR model with different orders
#' of the autoregressive lag length. It is the output of \code{FCVARlagSelect}.
#' @param ... additional arguments affecting the summary produced.
#' @return NULL
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' FCVAR_lag_1 <- FCVARlagSelect(x, kmax = 3, r = 3, order = 12, opt)
#' summary(object = FCVAR_lag_1)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate lag order.
#' \code{summary.FCVAR_lags} prints a summary of the output of \code{FCVARlagSelect} to screen.
#' @export
#'
summary.FCVAR_lags <- function(object, ...) {
#--------------------------------------------------------------------------------
# Print output
#--------------------------------------------------------------------------------
# create a variable for output strings
yesNo <- c('No','Yes') # Ironic order, No?
cat(sprintf('\n--------------------------------------------------------------------------------\n'))
cat(sprintf(' Lag Selection Results \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Dimension of system: %6.0f Number of observations in sample: %6.0f \n',
object$p, object$cap_T))
cat(sprintf('Order for WN tests: %6.0f Number of observations for estimation: %6.0f \n',
object$order, object$cap_T - object$opt$N))
cat(sprintf('Restricted constant: %6s Initial values: %6.0f\n',
yesNo[object$opt$rConstant+1], object$opt$N ) )
cat(sprintf('Unrestricted constant:%6s Level parameter: %6s\n',
yesNo[object$opt$unrConstant+1], yesNo[object$opt$levelParam+1] ))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Parameter Estimates and Information Criteria:\n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf(' k r d b LogL LR pv AIC BIC'))
cat(sprintf('\n'))
for (k in seq(object$kmax, 0, by = -1) ) {
cat(sprintf('%2.0f %2.0f %4.3f %4.3f %7.2f %6.2f %5.3f %8.2f',
k, object$r, object$D[k+1, 1], object$D[k+1, 2], object$loglik[k+1],
object$LRtest[k+1], object$pvLRtest[k+1], object$aic[k+1]))
# For AIC add asterisk if min value
if(k+1 == object$i_aic) {cat(sprintf('*'))} else {cat(sprintf(' '))}
# Print BIC information criteria and add asterisk if min value
cat(sprintf(' %8.2f', object$bic[k+1]))
if(k+1 == object$i_bic) {cat(sprintf('*'))} else {cat(sprintf(' '))}
cat(sprintf('\n'))
}
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Tests for Serial Correlation of Residuals: \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf(' k pmvQ'))
for (i in 1:object$p) {
cat(sprintf(' pQ%1.0f pLM%1.0f', i, i))
}
cat(sprintf('\n'))
for (k in seq(object$kmax, 0, by = -1) ) {
cat(sprintf('%2.0f ', k))
# Print multivariate white noise test P-values
cat(sprintf(' %4.2f', object$pvMVq[k+1, ]))
# Print the individual series white noise test P-values
for (i in 1:object$p) {
cat(sprintf(' %4.2f %4.2f', object$pvWNQ[k+1,i], object$pvWNLM[k+1,i]))
}
cat(sprintf('\n'))
}
cat(sprintf('--------------------------------------------------------------------------------\n'))
}
#' Test for Cointegrating Rank
#'
#' \code{FCVARrankTests} performs a sequence of likelihood ratio tests
#' for cointegrating rank.
#'
#' @param x A matrix of variables to be included in the system.
#' @param k The number of lags in the system.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return An S3 object of type \code{FCVAR_ranks} containing the results
#' from cointegrating rank tests, containing the following \code{(p+1)} vectors
#' with \code{i}th element corresponding to \code{rank = i-1},
#' including the following parameters:
#' \describe{
#' \item{\code{dHat}}{Estimates of \code{d}.}
#' \item{\code{bHat}}{Estimates of \code{b}.}
#' \item{\code{LogL}}{Maximized log-likelihood.}
#' \item{\code{LRstat}}{LR trace statistic for testing rank \code{r} against rank \code{p}.}
#' \item{\code{pv}}{The p-value of LR trace test, or "999" if p-value is not available.}
#' \item{\code{k}}{The number of lags in the system.}
#' \item{\code{p}}{The number of variables in the system.}
#' \item{\code{cap_T}}{The sample size.}
#' \item{\code{opt}}{An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.}
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' rankTestStats <- FCVARrankTests(x, k = 2, opt)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate cointegrating rank.
#' \code{summary.FCVAR_ranks} prints a summary of the output of \code{FCVARrankTests} to screen.
#' @export
#'
FCVARrankTests <- function(x, k, opt) {
cap_T <- nrow(x) - opt$N
p <- ncol(x)
# Store user specified options for printing to screen because it will be
# turned off while looping over ranks.
tempPrint2Screen <- opt$print2screen
# Create output storage to be filled.
bHat <- matrix(0, nrow = p+1, ncol = 1)
dHat <- matrix(0, nrow = p+1, ncol = 1)
LogL <- matrix(0, nrow = p+1, ncol = 1)
LRstat <- matrix(0, nrow = p+1, ncol = 1)
pv <- matrix(NA, nrow = p+1, ncol = 1)
# Do not print FCVAR estimation for each rank in the loop.
opt$print2screen <- 0
# Do not plot roots of characteristic polynomial for each lag in the loop.
opt$plotRoots <- 0
# Do not calculate standard errors.
opt$CalcSE <- 0
# For calculation of P-values
if(opt$rConstant | opt$levelParam) {
consT <- 1
} else {
consT <- 0
}
# Estimate models for all ranks
for (r in 0 : p) {
message(sprintf('Estimating for k = %d and r = %d.', k, r))
results <- FCVARestn(x, k, r, opt)
message(sprintf('Finished Estimation for k = %d and r = %d.', k, r))
dHat[r+1] <- results$coeffs$db[1]
bHat[r+1] <- results$coeffs$db[2]
LogL[r+1] <- results$like
}
# Calculate the LR statistics and P-values
for (r in 0 : (p-1)) {
LRstat[r+1] <- - 2*( LogL[r+1] - LogL[p+1] )
p_val <- NULL
# Get P-values, if
# (1) no deterministic terms, or
# (2) there is only restricted constant and d=b, or
# (3) there is only a level parameter and d=b.
# In all cases, p-values are only calculated
# rank of system (iq) is an integer from 1 through 12.
if (bHat[r+1] > 0 & bHat[r+1] < 2 & (
(!opt$rConstant & !opt$unrConstant & !opt$levelParam) |
(opt$rConstant & !opt$unrConstant & opt$restrictDB) |
(opt$levelParam & !opt$unrConstant & opt$restrictDB) ) ) {
# Call function from fracdist package for p-values.
if (p - r <= 12) {
p_val <- fracdist::fracdist_values(iq = p - r,
iscon = consT,
bb = bHat[r+1],
stat = LRstat[r+1])
} else {
warning(sprintf('P-values not calculated for the rank test with rank %d.\n', r),
'Tables of simulated quantiles are only avaiable for rank 12 or lower.',
'P-values can be calculated by simulation with the FCVARbootRank() function.')
}
} else {
warning(sprintf('P-values not calculated for the rank test with rank %d.\n', r),
'P-values are only calculated if:\n',
'1. there are no deterministic terms, or\n',
'2. there is only restricted constant and d = b, or\n',
'3. there is only a level parameter and d = b.\n')
}
# Store P-values, if calculated.
if(!is.null(p_val)) {
pv[r+1] <- p_val
}
}
# Return list of rank test results.
rankTestStats <- list(
dHat = dHat,
bHat = bHat,
LogL = LogL,
LRstat = LRstat,
pv = pv,
k = k,
p = p,
cap_T = cap_T,
opt = opt
)
class(rankTestStats) <- 'FCVAR_ranks'
# Restore settings.
opt$print2screen <- tempPrint2Screen
# Print the results to screen.
if (opt$print2screen) {
summary(object = rankTestStats)
}
return(rankTestStats)
}
#' Summarize Results of Tests for Cointegrating Rank
#'
#' \code{summary.FCVAR_ranks} prints the table of statistics from
#' the output of \code{FCVARrankTests}.
#' \code{FCVARrankTests} performs a sequence of likelihood ratio tests
#' for cointegrating rank.
#'
#' @param object An S3 object of type \code{FCVAR_ranks} containing the results
#' from repeated estimation of the FCVAR model with different
#' cointegrating ranks. It is the output of \code{FCVARrankTests}.
#' @param ... additional arguments affecting the summary produced.
#' @return NULL
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' rankTestStats <- FCVARrankTests(x, k = 2, opt)
#' summary(object = rankTestStats)
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{FCVARestn} is called repeatedly within this function
#' for each candidate cointegrating rank.
#' \code{summary.FCVAR_ranks} prints a summary of the output of \code{FCVARrankTests} to screen.
#' @export
#'
summary.FCVAR_ranks <- function(object, ...) {
# create a variable for output strings
yesNo <- c('No','Yes')
cat(sprintf('\n--------------------------------------------------------------------------------\n'))
cat(sprintf(' Likelihood Ratio Tests for Cointegrating Rank \n'))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Dimension of system: %6.0f Number of observations in sample: %6.0f \n',
object$p, object$cap_T + object$opt$N))
cat(sprintf('Number of lags: %6.0f Number of observations for estimation: %6.0f \n',
object$k, object$cap_T))
cat(sprintf('Restricted constant: %6s Initial values: %6.0f\n',
yesNo[object$opt$rConstant+1], object$opt$N ))
cat(sprintf('Unestricted constant: %6s Level parameter: %6s\n',
yesNo[object$opt$unrConstant+1], yesNo[object$opt$levelParam+1] ))
cat(sprintf('--------------------------------------------------------------------------------\n'))
cat(sprintf('Rank d b Log-likelihood LR statistic P-value\n'))
for (i in 1:object$p) {
if (!is.na(object$pv[i])) {
cat(sprintf('%2.0f %5.3f %5.3f %15.3f %13.3f %8.3f\n',
i-1, object$dHat[i], object$bHat[i], object$LogL[i], object$LRstat[i], object$pv[i]))
}
else {
cat(sprintf('%2.0f %5.3f %5.3f %15.3f %13.3f ----\n',
i-1, object$dHat[i], object$bHat[i], object$LogL[i], object$LRstat[i]))
}
}
cat(sprintf('%2.0f %5.3f %5.3f %15.3f ---- ----\n',
i, object$dHat[i+1], object$bHat[i+1], object$LogL[i+1]))
cat(sprintf('--------------------------------------------------------------------------------\n'))
}
#' Distribution of LR Test Statistic for the Rank Test
#'
#' \code{FCVARbootRank} generates a distribution of a likelihood ratio
#' test statistic for the rank test using a wild bootstrap,
#' following the method of Cavaliere, Rahbek, and Taylor (2010). It
#' takes the two ranks as inputs to estimate the model under the
#' null and the model under the alternative.
#' @param x A matrix of variables to be included in the system.
#' If \code{k>0}, actual data is used for initial values.
#' @param k The number of lags in the system.
#' @param r1 The cointegrating rank under the null hypothesis.
#' @param r2 The cointegrating rank under the alternative hypothesis.
#' @param B The number of bootstrap samples.
#' @param opt An S3 object of class \code{FCVAR_opt} that stores the chosen estimation options,
#' generated from \code{FCVARoptions()}.
#' @return A list \code{FCVARbootRank_stats} containing the test results,
#' including the following parameters:
#' \describe{
#' \item{\code{LRbs}}{A B x 1 vector of simulated likelihood ratio statistics.}
#' \item{\code{pv}}{An approximate p-value for the LR statistic based on the bootstrap distribution. }
#' \item{\code{H}}{A list containing LR test results. It is
#' identical to the output from \code{HypoTest}, with one addition,
#' namely \code{H$pvBS} which is the bootstrap p-value)}
#' \item{\code{mBS}}{Model estimates under the null hypothesis. }
#' \item{\code{mUNR}}{Model estimates under the alternative hypothesis. }
#' }
#' @examples
#' \donttest{
#' opt <- FCVARoptions()
#' opt$gridSearch <- 0 # Disable grid search in optimization.
#' opt$dbMin <- c(0.01, 0.01) # Set lower bound for d,b.
#' opt$dbMax <- c(2.00, 2.00) # Set upper bound for d,b.
#' opt$constrained <- 0 # Impose restriction dbMax >= d >= b >= dbMin ? 1 <- yes, 0 <- no.
#' opt$plotRoots <- 0
#' x <- votingJNP2014[, c("lib", "ir_can", "un_can")]
#' set.seed(42)
#' FCVARbootRank_stats <- FCVARbootRank(x, k = 2, opt, r1 = 0, r2 = 1, B = 2)
#' # In practice, set the number of bootstraps so that (B+1)*alpha is an integer,
#' # where alpha is the chosen level of significance.
#' # For example, set B = 999 (but it takes a long time to compute).
#' }
#' @family FCVAR specification functions
#' @seealso \code{FCVARoptions} to set default estimation options.
#' \code{HypoTest} for the format of a hypothesis test results.
#' \code{FCVARestn} for the estimates from a rectricted and unrestricted model within a hypothesis test.
#' @export
#' @references Cavaliere, G., A. Rahbek, and A. M. R. Taylor (2010).
#' "Testing for co-integration in vector autoregressions
#' with non-stationary volatility," Journal of Econometrics 158, 7-24.
#'
FCVARbootRank <- function(x, k, opt, r1, r2, B) {
# Calculate length of sample to generate, adjusting for initial values
cap_T <- nrow(x) - opt$N
# Use first k+1 observations for initial values
data <- x[1:(k+1), ]
LR <- matrix(0, nrow = B, ncol = 1)
# Turn off output and calculation of standard errors for faster computation
print2screen <- opt$print2screen
opt$print2screen <- 0
opt$CalcSE <- 0
opt$plotRoots <- 0
mBS <- FCVARestn(x, k, r1, opt)
mUNR <- FCVARestn(x, k, r2, opt)
# Initialize H (a list containing LR test results, it is
# identical to the output from HypoTest, with one addition,
# namely H$pvBS which is the Bootstrap P-value)
H <- list(LRstat = NA,
pvBS = NA)
H$LRstat <- -2*(mBS$like - mUNR$like)
# How often should the number of iterations be displayed
show_iters <- 10
for (j in 1:B) {
# Display replication count every show_iters Bootstraps
if(round((j+1)/show_iters) == (j+1)/show_iters) {
# cat(sprintf('iteration: %1.0f\n', j))
message(sprintf('Completed bootstrap replication %d of %d.', j, B))
}
# (1) generate bootstrap DGP under the null
xBS <- FCVARsimBS(data, mBS, cap_T)
# Append initial values to bootstrap sample
BSs <- rbind(data, xBS)
# (2) estimate unrestricted model
mUNRbs <- FCVARestn(BSs, k, r2, opt)
# (3) estimate restricted model (under the null)
mRES <- FCVARestn(BSs, k, r1, opt)
# (4) calculate test statistic
LR[j] <- -2*(mRES$like - mUNRbs$like)
}
# Return sorted LR stats
LRbs <- LR[order(LR)]
# Calculate Bootstrap P-value (see ETM p.157 eq 4.62)
H$pvBS <- sum(LRbs > H$LRstat)/B
# Print output, if required, after restoring settings.
opt$print2screen <- print2screen
if (opt$print2screen) {
cat(sprintf('Bootstrap rank test results:'))
cat(sprintf('\nUnrestricted log-likelihood: %3.3f\nRestricted log-likelihood: %3.3f\n',
mUNR$like, mBS$like))
cat(sprintf('Test results:\nLR statistic: \t %3.3f\nP-value (BS): \t %1.3f\n',
H$LRstat, H$pvBS))
}
# Return a list of bootstrap test results.
FCVARbootRank_stats <- list(
LRbs = LRbs,
H = H,
mBS = mBS,
mUNR = mUNR
)
return(FCVARbootRank_stats)
}
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