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10_bvar.R
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10_bvar.R
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#' Hierarchical Bayesian vector autoregression
#'
#' Used to estimate hierarchical Bayesian Vector Autoregression (VAR) models in
#' the fashion of Giannone, Lenza and Primiceri (2015).
#' Priors are adjusted and added via \code{\link{bv_priors}}.
#' The Metropolis-Hastings step can be modified with \code{\link{bv_mh}}.
#'
#' The model can be expressed as:
#' \deqn{y_t = a_0 + A_1 y_{t-1} + ... + A_p y_{t-p} + \epsilon_t}{y_t = a_0 +
#' A_1 y_{t-1} + ... + A_p y_{t-p} + e_t}
#' See Kuschnig and Vashold (2021) and Giannone, Lenza and Primiceri (2015)
#' for further information.
#' Methods for a \code{bvar} object and its derivatives can be used to:
#' \itemize{
#' \item predict and analyse scenarios;
#' \item evaluate shocks and the variance of forecast errors;
#' \item visualise forecasts and impulse responses, parameters and residuals;
#' \item retrieve coefficents and the variance-covariance matrix;
#' \item calculate fitted and residual values;
#' }
#' Note that these methods generally work by calculating quantiles from the
#' posterior draws. The full posterior may be retrieved directly from the
#' objects. The function \code{\link[utils]{str}} can be very helpful for this.
#'
#' @author Nikolas Kuschnig, Lukas Vashold
#'
#' @param data Numeric matrix or dataframe. Note that observations are expected
#' to be ordered from earliest to latest, and variables in the columns.
#' @param lags Integer scalar. Lag order of the model.
#' @param n_draw,n_burn Integer scalar. The number of iterations to (a) cycle
#' through and (b) burn at the start.
#' @param n_thin Integer scalar. Every \emph{n_thin}'th iteration is stored.
#' For a given memory requirement thinning reduces autocorrelation, while
#' increasing effective sample size.
#' @param priors Object from \code{\link{bv_priors}} with prior settings.
#' Used to adjust the Minnesota prior, add custom dummy priors, and choose
#' hyperparameters for hierarchical estimation.
#' @param mh Object from \code{\link{bv_mh}} with settings for the
#' Metropolis-Hastings step. Used to tune automatic adjustment of the
#' acceptance rate within the burn-in period, or manually adjust the proposal
#' variance.
#' @param fcast Object from \code{\link{bv_fcast}} with forecast settings.
#' Options include the horizon and settings for conditional forecasts i.e.
#' scenario analysis.
#' May also be calculated ex-post using \code{\link{predict.bvar}}.
#' @param irf Object from \code{\link{bv_irf}} with settings for the calculation
#' of impulse responses and forecast error variance decompositions. Options
#' include the horizon and different identification schemes.
#' May also be calculated ex-post using \code{\link{irf.bvar}}.
#' @param verbose Logical scalar. Whether to print intermediate results and
#' progress.
#' @param ... Not used.
#'
#' @return Returns a list of class \code{bvar} with the following elements:
#' \itemize{
#' \item \code{beta} - Numeric array with draws from the posterior of the VAR
#' coefficients. Also see \code{\link{coef.bvar}}.
#' \item \code{sigma} - Numeric array with draws from the posterior of the
#' variance-covariance matrix. Also see \code{\link{vcov.bvar}}.
#' \item \code{hyper} - Numeric matrix with draws from the posterior of the
#' hierarchically treated hyperparameters.
#' \item \code{ml} - Numeric vector with the marginal likelihood (with respect
#' to the hyperparameters), that determines acceptance probability.
#' \item \code{optim} - List with outputs of \code{\link[stats]{optim}},
#' which is used to find starting values for the hyperparameters.
#' \item \code{prior} - Prior settings from \code{\link{bv_priors}}.
#' \item \code{call} - Call to the function. See \code{\link{match.call}}.
#' \item \code{meta} - List with meta information. Includes the number of
#' variables, accepted draws, number of iterations, and data.
#' \item \code{variables} - Character vector with the column names of
#' \emph{data}. If missing, variables are named iteratively.
#' \item \code{explanatories} - Character vector with names of explanatory
#' variables. Formatting is akin to: \code{"FEDFUNDS-lag1"}.
#' \item \code{fcast} - Forecasts from \code{\link{predict.bvar}}.
#' \item \code{irf} - Impulse responses from \code{\link{irf.bvar}}.
#' }
#'
#' @references
#' Giannone, D. and Lenza, M. and Primiceri, G. E. (2015) Prior Selection for
#' Vector Autoregressions. \emph{The Review of Economics and Statistics},
#' \bold{97:2}, 436-451, \doi{10.1162/REST_a_00483}.
#'
#' Kuschnig, N. and Vashold, L. (2021) BVAR: Bayesian Vector Autoregressions
#' with Hierarchical Prior Selection in R.
#' \emph{Journal of Statistical Software}, \bold{14}, 1-27,
#' \doi{10.18637/jss.v100.i14}.
#'
#' @seealso \code{\link{bv_priors}}; \code{\link{bv_mh}};
#' \code{\link{bv_fcast}}; \code{\link{bv_irf}};
#' \code{\link{predict.bvar}}; \code{\link{irf.bvar}}; \code{\link{plot.bvar}};
#'
#' @keywords BVAR Metropolis-Hastings MCMC priors hierarchical
#'
#' @export
#'
#' @importFrom utils setTxtProgressBar txtProgressBar
#' @importFrom stats optim runif quantile
#' @importFrom mvtnorm rmvnorm
#'
#' @examples
#' # Access a subset of the fred_qd dataset
#' data <- fred_qd[, c("CPIAUCSL", "UNRATE", "FEDFUNDS")]
#' # Transform it to be stationary
#' data <- fred_transform(data, codes = c(5, 5, 1), lag = 4)
#'
#' # Estimate a BVAR using one lag, default settings and very few draws
#' x <- bvar(data, lags = 1, n_draw = 1000L, n_burn = 200L, verbose = FALSE)
#'
#' # Calculate and store forecasts and impulse responses
#' predict(x) <- predict(x, horizon = 8)
#' irf(x) <- irf(x, horizon = 8, fevd = FALSE)
#'
#' \dontrun{
#' # Check convergence of the hyperparameters with a trace and density plot
#' plot(x)
#' # Plot forecasts and impulse responses
#' plot(predict(x))
#' plot(irf(x))
#' # Check coefficient values and variance-covariance matrix
#' summary(x)
#' }
bvar <- function(
data, lags,
n_draw = 10000L, n_burn = 5000L, n_thin = 1L,
priors = bv_priors(),
mh = bv_mh(),
fcast = NULL,
irf = NULL,
verbose = TRUE, ...) {
cl <- match.call()
start_time <- Sys.time()
# Setup and checks -----
# Data
if(!all(vapply(data, is.numeric, logical(1L))) ||
any(is.na(data)) || ncol(data) < 2) {
stop("Problem with the data. Make sure it is numeric, without any NAs.")
}
Y <- as.matrix(data)
# Integers
lags <- int_check(lags, min = 1L, max = nrow(Y) - 1, msg = "Issue with lags.")
n_draw <- int_check(n_draw, min = 10L, msg = "Issue with n_draw.")
n_burn <- int_check(n_burn, min = 0L, max = n_draw - 1L,
msg = "Issue with n_burn. Is n_burn < n_draw?")
n_thin <- int_check(n_thin, min = 1L, max = ((n_draw - n_burn) / 10),
msg = "Issue with n_thin. Maximum allowed is (n_draw - n_burn) / 10.")
n_save <- int_check(((n_draw - n_burn) / n_thin), min = 1L)
verbose <- isTRUE(verbose)
# Constructors, required
if(!inherits(priors, "bv_priors")) {
stop("Please use `bv_priors()` to configure the priors.")
}
if(!inherits(mh, "bv_metropolis")) {
stop("Please use `bv_mh()` to configure the Metropolis-Hastings step.")
}
# Not required
if(!is.null(fcast) && !inherits(fcast, "bv_fcast")) {
stop("Please use `bv_fcast()` to configure forecasts.")
}
if(!is.null(irf) && !inherits(irf, "bv_irf")) {
stop("Please use `bv_irf()` to configure impulse responses.")
}
if(mh[["adjust_acc"]]) {n_adj <- as.integer(n_burn * mh[["adjust_burn"]])}
# Preparation ---
X <- lag_var(Y, lags = lags)
Y <- Y[(lags + 1):nrow(Y), ]
X <- X[(lags + 1):nrow(X), ]
X <- cbind(1, X)
XX <- crossprod(X)
K <- ncol(X)
M <- ncol(Y)
N <- nrow(Y)
variables <- name_deps(variables = colnames(data), M = M)
explanatories <- name_expl(variables = variables, M = M, lags = lags)
# Priors -----
# Minnesota prior ---
b <- priors[["b"]]
if(length(b) == 1 || length(b) == M) {
priors[["b"]] <- matrix(0, nrow = K, ncol = M)
priors[["b"]][2:(M + 1), ] <- diag(b, M)
} else if(!is.matrix(b) || !all(dim(b) == c(K, M))) {
stop("Issue with the prior mean b. Please reconstruct.")
}
if(any(priors[["psi"]][["mode"]] == "auto")) {
psi_temp <- auto_psi(Y, lags)
priors[["psi"]][["mode"]] <- psi_temp[["mode"]]
priors[["psi"]][["min"]] <- psi_temp[["min"]]
priors[["psi"]][["max"]] <- psi_temp[["max"]]
}
if(!all(vapply(priors[["psi"]][1:3],
function(x) length(x) == M, logical(1L)))) {
stop("Dimensions of psi do not fit the data.")
}
# Parameters ---
pars_names <- names(priors)[ # Exclude reserved names
!grepl("^hyper$|^var$|^b$|^psi[0-9]+$|^dummy$", names(priors))]
pars_full <- do.call(c, lapply(pars_names, function(x) priors[[x]][["mode"]]))
names(pars_full) <- name_pars(pars_names, M)
# Hierarchical priors ---
hyper_n <- length(priors[["hyper"]]) +
sum(priors[["hyper"]] == "psi") * (M - 1)
if(hyper_n == 0) {stop("Please provide at least one hyperparameter.")}
get_priors <- function(name, par) {priors[[name]][[par]]}
hyper <- do.call(c, lapply(priors[["hyper"]], get_priors, par = "mode"))
hyper_min <- do.call(c, lapply(priors[["hyper"]], get_priors, par = "min"))
hyper_max <- do.call(c, lapply(priors[["hyper"]], get_priors, par = "max"))
names(hyper) <- name_pars(priors[["hyper"]], M)
# Split up psi ---
for(i in seq_along(priors[["psi"]][["mode"]])) {
priors[[paste0("psi", i)]] <- vapply(c("mode", "min", "max"), function(x) {
priors[["psi"]][[x]][i]}, numeric(1L))
}
# Optimise and draw -----
opt <- optim(par = hyper, bv_ml, gr = NULL,
hyper_min = hyper_min, hyper_max = hyper_max, pars = pars_full,
priors = priors, Y = Y, X = X, XX = XX, K = K, M = M, N = N, lags = lags,
opt = TRUE, method = "L-BFGS-B", lower = hyper_min, upper = hyper_max,
control = list("fnscale" = -1))
names(opt[["par"]]) <- names(hyper)
if(verbose) {
cat("Optimisation concluded.",
"\nPosterior marginal likelihood: ", round(opt[["value"]], 3),
"\nHyperparameters: ", paste(names(hyper), round(opt[["par"]], 5),
sep = " = ", collapse = "; "), "\n", sep = "")
}
# Hessian ---
if(length(mh[["scale_hess"]]) != 1 &&
length(mh[["scale_hess"]]) != length(hyper)) {
stop("Length of scale_hess does not match the ", length(hyper),
" hyperparameters. Please provide a scalar or an element for every ",
"hyperparameter (see `?bv_mn()`).")
}
H <- diag(length(opt[["par"]])) * mh[["scale_hess"]]
J <- unlist(lapply(names(hyper), function(name) {
exp(opt[["par"]][[name]]) / (1 + exp(opt[["par"]][[name]])) ^ 2 *
(priors[[name]][["max"]] - priors[[name]][["min"]])
}))
if(any(is.nan(J))) {
stop("Issue with parameter(s) ",
paste0(names(hyper)[which(is.nan(J))], collapse = ", "), ". ",
"Their mode(s) may be too large to exponentiate.")
}
if(hyper_n != 1) {J <- diag(J)}
HH <- J %*% H %*% t(J)
# Make sure HH is positive definite
if(hyper_n != 1) {
HH_eig <- eigen(HH)
HH_eig[["values"]] <- abs(HH_eig[["values"]])
HH <- HH_eig
} else {HH <- list("values" = abs(HH))}
# Initial draw ---
while(TRUE) {
hyper_draw <- rmvn_proposal(n = 1, mean = opt[["par"]], sigma = HH)[1, ]
ml_draw <- bv_ml(hyper = hyper_draw,
hyper_min = hyper_min, hyper_max = hyper_max, pars = pars_full,
priors = priors, Y = Y, X = X, XX = XX, K = K, M = M, N = N, lags = lags)
if(ml_draw[["log_ml"]] > -1e16) {break}
}
# Sampling -----
# Storage ---
accepted <- 0 -> accepted_adj # Beauty
ml_store <- vector("numeric", n_save)
hyper_store <- matrix(NA, nrow = n_save, ncol = length(hyper_draw),
dimnames = list(NULL, names(hyper)))
beta_store <- array(NA, c(n_save, K, M))
sigma_store <- array(NA, c(n_save, M, M))
if(verbose) {pb <- txtProgressBar(min = 0, max = n_draw, style = 3)}
# Start loop ---
for(i in seq.int(1 - n_burn, n_draw - n_burn)) {
# Metropolis-Hastings
hyper_temp <- rmvn_proposal(n = 1, mean = hyper_draw, sigma = HH)[1, ]
ml_temp <- bv_ml(hyper = hyper_temp,
hyper_min = hyper_min, hyper_max = hyper_max, pars = pars_full,
priors = priors, Y = Y, X = X, XX = XX, K = K, M = M, N = N, lags = lags)
if(runif(1) < exp(ml_temp[["log_ml"]] - ml_draw[["log_ml"]])) { # Accept
ml_draw <- ml_temp
hyper_draw <- hyper_temp
accepted_adj <- accepted_adj + 1
if(i > 0) {accepted <- accepted + 1}
}
# Tune acceptance during burn-in phase
if(mh[["adjust_acc"]] && i <= -n_adj && (i + n_burn) %% 10 == 0) {
acc_rate <- accepted_adj / (i + n_burn)
if(acc_rate < mh[["acc_lower"]]) {
HH[["values"]] <- HH[["values"]] * mh[["acc_tighten"]]
} else if(acc_rate > mh[["acc_upper"]]) {
HH[["values"]] <- HH[["values"]] * mh[["acc_loosen"]]
}
}
if(i > 0 && i %% n_thin == 0) { # Store draws
ml_store[(i / n_thin)] <- ml_draw[["log_ml"]]
hyper_store[(i / n_thin), ] <- hyper_draw
# Draw parameters, i.e. beta_draw and sigma_draw
# These need X and N with the dummy observations from `ml_draw`
draws <- draw_post(XX = ml_draw[["XX"]], N = ml_draw[["N"]],
M = M, lags = lags, b = priors[["b"]], psi = ml_draw[["psi"]],
sse = ml_draw[["sse"]], beta_hat = ml_draw[["beta_hat"]],
omega_inv = ml_draw[["omega_inv"]])
beta_store[(i / n_thin), , ] <- draws[["beta_draw"]]
sigma_store[(i / n_thin), , ] <- draws[["sigma_draw"]]
} # End store
if(verbose) {setTxtProgressBar(pb, (i + n_burn))}
} # End loop
timer <- Sys.time() - start_time
if(verbose) {
close(pb)
cat("Finished MCMC after ", format(round(timer, 2)), ".\n", sep = "")
}
# Outputs -----
out <- structure(list(
"beta" = beta_store, "sigma" = sigma_store,
"hyper" = hyper_store, "ml" = ml_store,
"optim" = opt, "priors" = priors, "call" = cl,
"variables" = variables, "explanatories" = explanatories,
"meta" = list("accepted" = accepted, "timer" = timer,
"Y" = Y, "X" = X, "N" = N, "K" = K, "M" = M, "lags" = lags,
"n_draw" = n_draw, "n_burn" = n_burn, "n_save" = n_save,
"n_thin" = n_thin)
), class = "bvar")
if(!is.null(irf)) {
if(verbose) {cat("Calculating impulse responses.")}
out[["irf"]] <- tryCatch(irf.bvar(out, irf), error = function(e) {
warning("\nImpulse response calculation failed with:\n", e)
return(NULL)})
if(verbose) {cat("..Done!\n")}
}
if(!is.null(fcast)) {
if(verbose) {cat("Calculating forecasts.")}
out[["fcast"]] <- tryCatch(predict.bvar(out, fcast), error = function(e) {
warning("\nForecast calculation failed with:\n", e)
return(NULL)})
if(verbose) {cat("..Done!\n")}
}
return(out)
}